Generalization of Pólya's zero distribution theory for exponential polynomials, plus sharp results for asymptotic growth
aa r X i v : . [ m a t h . C V ] J un Generalization of P´olya’s zero distribution theory forexponential polynomials, plus sharp results forasymptotic growth
Janne M. Heittokangas and Zhi-Tao Wen ∗ Abstract
An exponential polynomial of order q is an entire function of the form f ( z ) = P ( z ) e Q ( z ) + · · · + P k ( z ) e Q k ( z ) , where the coefficients P j ( z ) , Q j ( z ) are polynomials in z such thatmax { deg( Q j ) } = q. In 1977 Steinmetz proved that the zeros of f lying outside of finitely many loga-rithmic strips around so called critical rays have exponent of convergence ≤ q − f in each individuallogarithmic strip. Here, it is shown that the asymptotic growth of the non-integratedcounting function of zeros of f is asymptotically comparable to r q in each logarithmicstrip. The result generalizes the first order results by P´olya and Schwengeler fromthe 1920’s, and it shows, among other things, that the critical rays of f are preciselythe Borel directions of order q of f . The error terms in the asymptotic equations for T ( r, f ) and N ( r, /f ) originally due to Steinmetz are also improved. Key Words:
Asymptotic growth, critical ray, error term, exponential polynomial,logarithmic strip, number of zeros, value distribution.
An exponential sum is an entire function of the form f ( z ) = P ( z ) e w z + · · · + P n ( z ) e w m z , (1.1)where the coefficients P j ( z ) are polynomials such that P j ( z ) j = 1 , . . . , m . If P ( z )
0, we set w = 0. The constants w j ∈ C are pairwise distinct, and they are calledthe leading coefficients of f . Thus w j = 0 for j = 1 , . . . , m . ∗ Corresponding author.Heittokangas was partially supported by the Academy of Finland ero distribution theory for exponential polynomials z = i ( e iz − e − iz ) and cos z = ( e iz + e − iz ) are typical examples ofexponential sums, their quotient being tan z . In 1929 Ritt [19] showed that if the ratio g of two exponential sums with constant coefficients is entire, then g is an exponential sumalso. The proof of this result was simplified by Lax in 1949 [14]. In 1958 Shapiro [21]conjectured that if two exponential sums with polynomial coefficients have infinitely manyzeros in common, then they are both multiples of some third exponential sum. This claimis known as Shapiro’s conjecture, and it remains unsolved.We note that distribution of a -points and zero distribution of exponential sums areessentially the same thing because f is an exponential sum if and only if f − a is. Zerodistribution of functions f of the form (1.1) has been investigated for roughly a centuryby several authors, starting from the case where the coefficients P j are constants and theleading coefficients w j are real and commensurable. The latter means that w j = αp j ,where α ∈ R and p j ∈ Z . The early developments, with references, are surveyed in [13].Zero distribution in the general case is based on the work of P´olya [17, 18] and Schwen-geler [20]. We summarize these findings in Theorem A below, for which Dickson [3] hasgiven a generalization in the sense that the coefficients are asymptotically polynomials.Before stating the actual result, we need to agree on the notation.If P ( z )
0, set W f = { w = 0 , w , . . . , w m } , while if P ( z ) ≡
0, set W f = { w , . . . , w m } .Moreover, set W f = W f ∪ { } . Thus, if P ( z )
0, then W f = W f . For any finite set G ⊂ C , let C (co( G )) denote the circumference of the convex hull co( G ). Around a givenray arg( z ) = ϕ , we define a logarithmic stripΛ( ϕ, c ) = (cid:26) re iθ : r > , | θ − ϕ | < c log rr (cid:27) , (1.2)where c >
0. Finally, we let θ ⊥ denote the argument of a ray parallel to one of the outernormals of co ( W f ). Thus the ray arg( z ) = θ ⊥ is orthogonal to a line passing through oneof the sides of the polygon co ( W f ). Theorem A ([3, 17, 18, 20])
Let f be an exponential sum of the form (1.1) . Then f hasat most finitely many zeros outside of the finitely many domains Λ( θ ⊥ , c ) . Moreover, let w j , w k be two consecutive vertex points of co ( W f ) , and let θ ⊥ denote the argument of theray parallel to the outer normal of co ( W f ) corresponding to its side [ w j , w k ] . Then thenumber n ( r, Λ) of zeros of f in Λ( θ ⊥ , c ) ∩ {| z | < r } satisfies n ( r, Λ) = | w j − w k | r π + O (1) . (1.3)By summing over all sides of co( W f ), we get P | w j − w k | = C (co( W f )). Hence Theo-rem A yields the asymptotic equations n ( r, /f ) = C (co( W f )) r π + O (1) , (1.4) N ( r, /f ) = C (co( W f )) r π + O (log r ) , (1.5) ero distribution theory for exponential polynomials r → ∞ without an exceptional set. Finally, Wittich [25] has proved the asymptoticequation T ( r, f ) = C (co( W f )) r π + o ( r ) . (1.6)Exponential sums can be generalized to exponential polynomials, which are higherorder entire functions of the form f ( z ) = P ( z ) e Q ( z ) + · · · + P n ( z ) e Q n ( z ) , (1.7)where P j ( z ), Q j ( z ) are polynomials for 0 ≤ j ≤ n . A polynomial can be consideredas a special case of an exponential polynomial. We assume that the polynomials Q j ( z )are pairwise different and normalized such that Q j (0) = 0. This convention forces thefunctions g j ( z ) = P j ( z ) e Q j ( z ) to be linearly independent. Let q = max { deg( Q j ) } denotethe order of f . As observed in [22], we may then write f in the normalized form f ( z ) = H ( z ) e w z q + H ( z ) e w z q + · · · + H m ( z ) e w m z q , (1.8)where the functions H ( z ) , H ( z ) , . . . , H m ( z ) are nonvanishing exponential polynomials oforder ≤ q − p for some 1 ≤ p ≤ q , and m ≤ n . The leading coefficients w j (0 ≤ j ≤ m )are pairwise distinct and nonzero, except possibly w . Thus H ( z ) ≡ q = deg( Q j ) holds for every j .Steinmetz [22] has proved analogues of (1.5) and (1.6) for exponential polynomials.We will improve the error terms in these asymptotic equations. So far there has been noattempt on finding a higher order analogue of (1.3). The main focus of this paper is tofind one. Then a higher order analogue of (1.4) will be found in two different ways assimple consequences.The main results are stated and further motivated in Sections 2–3. The proofs of themain results are given in Sections 4–8. So far there has been no attempts to prove a higher order analogue of Theorem A. Themajor part of this paper focuses in proving such a result. Before stating the actual result,we need to recall a few concepts.If f is an exponential polynomial of the form (1.8), then its Phragm´en-Lindel¨of indi-cator function h f ( θ ) = lim sup r →∞ r − q log | f ( re iθ ) | takes the form h f ( θ ) = max j (cid:8) ℜ (cid:0) w j e iqθ (cid:1)(cid:9) , see [10, p. 993]. The indicator h f ( θ ) is 2 π -periodic and differentiable everywhere exceptfor cusps θ ∗ at which h f ( θ ∗ ) = ℜ (cid:0) w j e iqθ ∗ (cid:1) = ℜ (cid:0) w k e iqθ ∗ (cid:1) (2.1)for a pair of indices j, k such that j = k . These cusps are called the critical angles for f ,and the corresponding rays are called critical rays for f [9, 22].An alternative way to find the critical angles is by means of the convex hull co( W f )of the conjugated leading coefficients of f . Namely, let arg( z ) = θ ⊥ j,k be the ray that is ero distribution theory for exponential polynomials W f ) determined by two successive vertex points w j , w k of co( W f ). Then the rays arg( z ) = ( θ ⊥ j,k + 2 πn ) /q, n ∈ Z , (2.2)are critical rays for f , see [9, Lemma 3.2]. Thus, if co( W f ) has s sides/vertices, then f has sq critical rays all together. In the particular case when q = 1 the critical rays andthe orthogonal rays are one and the same.Let f be given in the normalized form (1.8), where we suppose that ρ ( H j ) ≤ q − p for1 ≤ p ≤ q . The case p = 1 corresponds to the standard case, while the identity p = q reduces the coefficients H j ( z ) to be polynomials. In [9] it is proved that most of the zerosof f are inside of modified logarithmic stripsΛ p ( θ ∗ , c ) = (cid:26) z = re iθ : r > , | arg( z ) − θ ∗ | < c log rr p (cid:27) , where c > z ) = θ ∗ represents one of the critical rays for f . For p ≥ p = 1, see [9, Section 2]. More precisely, let N Λ p ( r ) denote theintegrated counting function of the zeros of f in | z | ≤ r lying outside of the union of thefinitely many domains Λ( θ ∗ , c ). Then N Λ p ( r ) = O ( r q − p + log r ) . (2.3)The case p = 1 was proved earlier in [22], while partial results for the general case wereproved in [5, 16]. For (2.3) to hold, the constant c > f ). In what follows, we will assume that this is always the case without further noticewhenever discussing the domains Λ p ( θ ∗ , c ).The discussion above says nothing about the zero distribution in each individual modi-fied logarithmic strip Λ p . This motivates us to state and prove the following generalizationof Theorem A. Theorem 2.1
Let f be an exponential polynomial of the form (1.8) , where the coefficients H j ( z ) are exponential polynomials of growth ρ ( H j ) ≤ q − p for some integer ≤ p ≤ q .Then the number n λ p ( r ) of zeros of f outside of the finitely many domains Λ p ( θ ∗ , c ) satisfies n λ p ( r ) = O (cid:0) r q − p + log r (cid:1) . Moreover, let w j , w k be two consecutive vertex points of co ( W f ) , and let θ ⊥ denote the ar-gument of the ray parallel to the outer normal of co ( W f ) corresponding to its side [ w j , w k ] .Let ε > , and let θ ∗ be any critical angle for f associated with the angle θ ⊥ by means of (2.2) . Then the number of zeros n ( r, Λ p ) of f in Λ p ( θ ∗ , c ) ∩ {| z | < r } satisfies n ( r, Λ p ) = | w j − w k | r q π + O (cid:0) r q − p log ε r (cid:1) . (2.4)We make some comments regarding Theorem 2.1, and discuss some immediate con-sequences of it. First, according to (2.1), at least two leading coefficients are needed toinduce a critical ray. The example f ( z ) = n X j =0 e jz , n ≥ , (2.5) ero distribution theory for exponential polynomials xy HITWSchwengelerP´olyaFigure 1: Assuming that the positive real axis is a critical ray for f , thezeros of f have been considered in different regions as follows: P´olya in sec-tors | arg( z ) | < ε , Schwengeler in logarithmic strips | arg( z ) | ≤ c log | z || z | , andHeittokangas-Ishizaki-Tohge-Wen in modified logarithmic strips | arg( z ) | ≤ c log | z || z | p . Here ε > c > p ≥ z ) = ± π/ W f ). Indeed, if (2.1) holds, then ℜ (cid:0) w j e iqθ ∗ (cid:1) = ℜ (cid:0) w k e iqθ ∗ (cid:1) ≥ max l = k,j ℜ (cid:0) w l e iqθ ∗ (cid:1) . From (2.2) we get that arg( z ) = qθ ∗ is an orthogonal ray for a line through the points w j and w k . Hence this line must be a support line for co( W f ), and so w j and w k are boundarypoints of co( W f ). However, being a boundary point of co ( W f ) is not enough. In provingTheorem 2.1 it is necessary that in between two consecutive Λ p -domains precisely oneexponential term is dominant. This property is explained in more detail in Lemma 7.2below. Only those exponential terms that are associated with vertex points of co ( W f )have this property. For the function f in (2.5), the dominant term in the right half-planeis e nz , while the dominant term in the left half-plane is 1.Second, keeping in mind that w j , w k are two consecutive vertex points of co ( W f ), weget P | w j − w k | = C (co( W f )), where the summation is taken over all sides of co( W f ). From(2.2) we see that the orthogonal ray associated with the pair w j , w k induces q critical rays.This gives raise to the asymptotic equation n ( r, /f ) = qC (co( W f )) r q π + O (cid:0) r q − p log ε r (cid:1) . (2.6)Third, since a domain Λ p ( θ ∗ , c ) for any p is essentially contained in an arbitrary ε -angle { z : | arg( z ) − θ ∗ | < ε } , it follows that the critical rays of f are precisely the Boreldirections of order q of f . Note that f has no Borel directions of order ρ ∈ ( q − p, q ). Inthe case q = 1 the critical rays of f (which are the same as the orthogonal rays of f ) areprecisely the Julia directions for f in a strong sense. Indeed, for any a ∈ C , f has at mostfinitely many a -points outside of the domains Λ ( θ ∗ , c ) by Theorem A. ero distribution theory for exponential polynomials O ( r q − p ), but ourmethod does not give this. T ( r, f ) and N ( r, /f ) For an exponential polynomial f of the normalized form (1.8), Steinmetz [22] has provedthe asymptotic equations N (cid:18) r, f (cid:19) = C (co( W f )) r q π + o ( r q ) , (3.1) T ( r, f ) = C (co( W f )) r q π + o ( r q ) . (3.2)These are higher order analogues of (1.5) and (1.6). The next result improves the errorterm in (3.2) even in the standard case p = 1. It is obvious that the logarithmic term in(3.3) below is only needed when q = p , that is, when the coefficients H j ( z ) are ordinarypolynomials. Theorem 3.1
Let f be an exponential polynomial in the normalized form (1.8) , where wesuppose that ρ ( H j ) ≤ q − p for ≤ p ≤ q . Then T ( r, f ) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) . (3.3)Let f be the exponential polynomial in (1.7), and suppose that f can be written in thenormalized form (1.8), where ρ ( H j ) ≤ q − p for 1 ≤ p ≤ q . If one of the polynomials Q j ( z )vanishes identically, we may suppose that it is Q ( z ), and that no other polynomial Q j ( z )vanishes identically. In this case H ( z ) reduces to an ordinary polynomial in z . Underthese conditions, we have the following result that improves the error term in (3.1). Theorem 3.2 If H ( z ) , we have m (cid:18) r, f (cid:19) = O (cid:0) r q − p + log r (cid:1) . (3.4) If Q ( z ) ≡ , then we have the stronger estimate m (cid:18) r, f (cid:19) = O (log r ) . (3.5) Moreover, we have N (cid:18) r, f (cid:19) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) . (3.6)We conclude this section by the following quick consequence of (3.1). This is a weakerform than (2.6), but it is needed in proving Theorem 2.1, whereas (2.6) is a consequenceof Theorem 2.1. ero distribution theory for exponential polynomials Corollary 3.3
Let f be an exponential polynomial of order q such that C = C (co( W f )) > . Then n ( r, /f ) = qC (co( W f )) r q π + o ( r q ) . Proof.
Write (3.1) in the form C (co( W f )) r q π − ε ( r ) r q ≤ N (cid:18) r, f (cid:19) ≤ C (co( W f )) r q π + ε ( r ) r q , where 0 < ε ( r ) < ε ( r ) → r → ∞ . Let 0 < δ ( r ) < δ ( r ) → r → ∞ . Using log x ≥ x − x for x ≥ n ( r ) = n ( r )log (1 + δ ( r )) Z r (1+ δ ( r )) r dtt ≤ δ ( r ) δ ( r ) Z r (1+ δ ( r )) r n ( t ) t dt = 1 + δ ( r ) δ ( r ) (cid:0) N ( r (1 + δ ( r )) , /f ) − N ( r, /f ) (cid:1) = 1 + δ ( r ) δ ( r ) Cr q π q X j =1 (cid:18) qj (cid:19) δ ( r ) j + O ( ε ( r ) r q ) ! = qC r q π + O ( δ ( r ) r q ) + O ( ε ( r ) r q ) + O (cid:18) ε ( r ) r q δ ( r ) (cid:19) . Similarly, using log x ≤ x − x ≥
1, we get n ( r ) = n ( r ) − log (1 − δ ( r )) Z rr (1 − δ ( r )) dtt ≥ − δ ( r ) δ ( r ) Z rr (1 − δ ( r )) n ( t ) t dt = qC r q π + O ( δ ( r ) r q ) + O ( ε ( r ) r q ) + O (cid:18) ε ( r ) r q δ ( r ) (cid:19) . Choose δ ( r ) = p ε ( r ), and all error terms above are of the form o ( r q ). ✷ Let f be an exponential polynomial with no zeros. Then f = e P , where P is a polynomialof degree p , and there exist constants C >
R > | f ( z ) | ≥ log e −| P ( z ) | ≥ − Cr p (4.1)for | z | > R . If f has finitely many zeros, then f = Qe P , where Q is also a polynomial.The conclusion in (4.1) holds in this case also, but possibly for different constants C and R . For an exponential polynomial f with infinitely many zeros, a lower bound for log | f ( z ) | is given in [9, Lemma 5.1], which originates from [23, Theorem V. 19]. This estimate isvalid for all z outside of certain discs centered at the zeros of f . In our applications,however, we need to be more flexible with the size of these discs, as is described next. ero distribution theory for exponential polynomials Lemma 4.1
Let f be an entire function of order ρ = ρ ( f ) , let λ = λ ( f ) denote theexponent of convergence of zeros of f , and let k : (0 , ∞ ) → [1 , ∞ ) be any non-decreasingfunction. Suppose that λ ≥ and that the number of zeros of f in | z | < r satisfies n ( r, /f ) = C ( r ) r λ , where C ( r ) is bounded as r → ∞ . Then there exists a constant A > such that log | f ( z ) | ≥ − A max (cid:8) r ρ , r λ log ( rk (2 r )) (cid:9) (4.2) whenever z = re iθ lies outside of the discs | z − z n | ≤ /k ( | z n | ) and | z | ≤ e .Proof. We may factorize the entire f as f = Qe P , where P is a polynomial and Q isa canonical product formed with the zeros z n of f . Set p = deg( P ), in which case ρ =max { p, λ } , and q = ⌊ λ ⌋ . Since the integral R ∞ N ( r, /f ) r q +1 dr diverges, the sum P ∞ n =1 | z n | − q also diverges, see [8, Lemma 1.4]. Hence q is the genus of the sequence { z n } , see [23,p. 216] for the definition. This allows us to write Q in the form Q ( z ) = ∞ Y n =1 (cid:18) − zz n (cid:19) exp (cid:18) zz n + · · · + 1 q (cid:18) zz n (cid:19) q (cid:19) . We also write Ψ( z, z n ) = log z n z n − z − (cid:18) zz n + · · · + 1 q (cid:18) zz n (cid:19) q (cid:19) , so that log 1 Q ( z ) = ∞ X n =1 Ψ( z, z n ) , z = z n . Taking positive real parts on both sides of the above equation results inlog + | Q ( z ) | ≤ X r n ≤ r ℜ + (Ψ( z, z n )) + X r n > r | Ψ( z, z n ) | =: Σ + Σ , where r n = | z n | . Similarly as in the proof of [9, Lemma 5.1], we obtain Σ = O ( r q ). Thisestimate has nothing to do with the size of the discs around the points z n . In Σ , we have ℜ + (Ψ( z, z n )) ≤ log + (cid:12)(cid:12)(cid:12)(cid:12) z n z − z n (cid:12)(cid:12)(cid:12)(cid:12) + q X j =1 (cid:18) rr n (cid:19) j ≤ log + (cid:12)(cid:12)(cid:12)(cid:12) z n z − z n (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) rr n (cid:19) q q − X j =0 (cid:16) r n r (cid:17) j ≤ log + (cid:12)(cid:12)(cid:12)(cid:12) z n z − z n (cid:12)(cid:12)(cid:12)(cid:12) + 2 q (cid:18) rr n (cid:19) q . If z lies outside of the discs | z − z n | ≤ /k ( | z n | ) and | z | ≤ e , it follows thatlog + (cid:12)(cid:12)(cid:12)(cid:12) z n z − z n (cid:12)(cid:12)(cid:12)(cid:12) ≤ log ( r n k ( r n )) ≤ log (2 rk (2 r )) , ero distribution theory for exponential polynomials r n ≤ r and k is non-decreasing. ThusΣ ≤ n (2 r ) log (2 rk (2 r )) + 2 q r q Z re dn ( t ) t q + O (1) ≤ O (cid:0) r λ log ( rk (2 r )) (cid:1) + 2 q qr q Z re n ( t ) t q +1 dt = O (cid:0) r λ log ( rk (2 r )) (cid:1) . The reasoning above shows that there exists a constant
B > | Q ( z ) | ≥− Br λ log ( rk (2 r )) whenever z = re iθ lies outside of the aforementioned closed discs. Since P is a polynomial of degree p , there is a constant C > (cid:12)(cid:12) e P ( z ) (cid:12)(cid:12) ≥ − Cr p for | z | > e . The assertion now follows by choosing A = B + C . ✷ Corollary 4.2
Let f be an exponential polynomial of order ρ = ρ ( f ) , and let λ = λ ( f ) ≥ be the exponent of convergence of zeros of f . Suppose that k : (0 , ∞ ) → [1 , ∞ ) is any non-decreasing function. Then there exists a constant A > such that log | f ( z ) | ≥ − A max (cid:8) r ρ , r λ log ( rk (2 r )) (cid:9) whenever z = re iθ lies outside of the discs | z − z n | ≤ /k ( | z n | ) and | z | ≤ e . Remark.
We may choose k ( x ) = x λ + ε , ε >
0, in Lemma 4.1 and k ( x ) = x q log x inCorollary 4.2. For these choices of k , it follows by Riemann-Stieltjes integration that X | z n |≥ e k ( | z n | ) = Z ∞ e dn ( t ) k ( t ) < ∞ . Thus the projection I of the discs | z − z n | ≤ /k ( | z n | ) on the positive real axis has finitelinear measure.The following lemma follows implicitly from the discussions in [11, Section 3]. For theconvenience of the reader, we give a direct proof. Lemma 4.3
Suppose that T : R + → R + is eventually a non-decreasing function. Supposefurther that there exist constants C > , q > , ≤ p < q and s ≥ such that, as r → ∞ , T can be written in the form T ( r ) = Cr q + O ( r p log s r ) , r E ∪ [0 , , (4.3) where E ⊂ [1 , ∞ ) has finite logarithmic measure (or finite linear measure). Then theasymptotic equality in (4.3) holds for all r large enough.Proof. Let ε ∈ (0 , min { , C } ). Then there exist constants R = R ( ε ) > C > T ( r ) is increasing for r > R and( C − ε ) r q − C r p log s r ≤ T ( r ) ≤ ( C + ε ) r q + C r p log s r, where r ∈ ( R , ∞ ) \ E . Moreover, due to p < q we may suppose that R is chosen largeenough so that the lower bound and the upper bound for T ( r ) are increasing functions ero distribution theory for exponential polynomials r . Then, using [7, Lemma 5], we can find an R = R ( ε ) > max { , R } such that, for r > R , T ( r ) ≤ ( C + ε )(1 + ε ) q r q + C (2 r ) p log s (2 r ) ≤ ( C + ε )(1 + ε ) q r q + 2 p + s C r p log s r, and similarly T ( r ) ≥ C − ε (1 + ε ) q r q − C r p log s r. In other words,( C − ε ′ ) r q − C r p log s r ≤ T ( r ) ≤ ( C + ε ′ ) r q + C r p log s r, r > R , where R = R ( ε ′ ) > C = 2 p + s C . This yields the assertion. ✷ Lemma 4.4 ([9])
Let f be an exponential polynomial of the form (1.8) , ψ k ( θ ) = ℜ (cid:0) w k e iqθ (cid:1) , p ≤ q and a = min (cid:8) | ψ ′ k ( θ ) − ψ ′ j ( θ ) | : ψ k ( θ ) = ψ j ( θ ) , k = j (cid:9) . (4.4) For a given c > there exists an R > , with the following property: If z = re iθ Λ p ( θ ∗ , c ) for any θ ∗ and any r ≥ R , and if h f ( θ ) = ψ j ( θ ) , then h f ( θ ) > max k = j ψ k ( θ ) + ac log rr p . We divide the proof into four steps for clarity.
Let arg( z ) = θ j , j = 1 , . . . , k , be the critical rays of f organizedsuch that 0 ≤ θ < θ < · · · < θ k < π . Set θ k +1 = θ + 2 π . For j = 1 , . . . , k , define F j = ( θ j , θ j +1 ) andΛ p ( θ j , c ) = (cid:26) z = re iθ : r > , | arg( z ) − θ j | < c log rr p (cid:27) ,F j ( r ) = (cid:18) θ j + c log rr p , θ j +1 − c log rr p (cid:19) , where c > p ( θ j , c ) curves asymptotically towardsthe critical ray arg( z ) = θ j , provided that p ≥
2. The interval F j ( r ) corresponds tothe arguments in between two consequtive domains Λ p ( θ j , c ) and Λ p ( θ j +1 , c ). Clearly F j ( r ) → ( θ j , θ j +1 ) = F j as r → ∞ for any p ≥ f has no poles, we have T ( r, f ) = m ( r, f ) = 12 π Z π log + | f ( re iθ ) | dθ. For j = 0 , . . . , m , we define E j = (cid:26) θ ∈ [0 , π ) : ℜ (cid:0) w j e iqθ (cid:1) > max k = j ℜ (cid:0) w k e iqθ (cid:1)(cid:27) ,E j ( r ) = (cid:26) θ ∈ [0 , π ) : ℜ (cid:0) w j e iqθ (cid:1) > max k = j ℜ (cid:0) w k e iqθ (cid:1) + c log rr p (cid:27) . ero distribution theory for exponential polynomials E j ( r ) is a finite union of intervals F i ( r ). Thus set E j is a finiteunion of intervals F i . If E = ∅ , then P ( z ) w = 0, so the definition of the set E j implies that the terms | H j ( z ) e w j z q | , j = 1 , . . . , m , are bounded for arg( z ) = θ ∈ E . Onthe other hand, if θ E j for some j , then | H j ( z ) e w j z q | ≤ M ( r, H j ). Keeping in mind that ρ ( H j ) ≤ q − p , we conclude in both cases E = ∅ and E = ∅ that T ( r, f ) = m X j =1 π Z E j log + | f ( re iθ ) | dθ + O (cid:0) r q − p + log r (cid:1) . (5.1) T ( r, f ) . To estimate the integrals in (5.1) over the sets E j , wewrite f ( z ) = e w j z q H j ( z ) + X k = j H k ( z ) e ( w k − w j ) z q ! . (5.2)If z = re iθ and θ ∈ E j , it follows thatlog + | f ( z ) | ≤ ℜ (cid:0) w j e iqθ (cid:1) + m X j =0 log + M ( r, H j ) + log( m + 1)= k f ( qθ ) r q + O (cid:0) r q − p + log r (cid:1) , (5.3)where k f ( θ ) is the support function for the convex set co( W f ) [15, p. 74]. Therefore T ( r, f ) ≤ m X j =1 r q π Z E j k f ( qθ ) dθ + O (cid:0) r q − p + log r (cid:1) ≤ r q π Z π k f ( qθ ) dθ + O (cid:0) r q − p + log r (cid:1) = r q π Z π k f ( ϕ ) dϕ + O (cid:0) r q − p + log r (cid:1) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) , where the last identity follows by Cauchy’s formula for convex curves. This formula canbe found in many books on integral geometry. T ( r, f ) . If q = 1 or if p = q , then the coefficients H j ( z ) arepolynomials, and no exceptional set other than | z | is large enough occurs when estimating T ( r, f ) downwards. Hence we suppose that 1 ≤ p ≤ q −
1. Observe thatlog + ( xy ) ≥ max (cid:26) log + x − log + y , (cid:27) for all x ≥ y >
0. So, if z = re iθ and θ ∈ E j ( r ), by applying this observation to(5.2), we havelog + | f ( z ) | ≥ max ℜ (cid:0) w j e iqθ (cid:1) − log + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | H j ( z ) | − X k = j M ( r, H k ) (cid:12)(cid:12) e ( w k − w j ) z q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − , ≥ max h f ( θ ) r q − log + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | H j ( z ) | − X k = j M ( r, H k ) e − cr q − p log r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − , . ero distribution theory for exponential polynomials H j ( z ) are of order ≤ q − p ≤ q − A > M ( r, H j ) ≤ Ar q − p log r and log | H j ( z ) | ≥ − Ar q − p log r. The latter inequality is valid for all | z | = r outside of a set I ⊂ (0 , ∞ ) of finite linearmeasure. This follows by applying Lemma 4.1 with k ( x ) = x q to each of the coefficients H j ( z ), and then taking the union of all the exceptional sets involved. See also the remarkfollowing Lemma 4.1. We note that k ( x ) could be chosen to have less growth at this point,but the particular growth rate fixed here is needed later on.Choosing c > a in Lemma 4.4, we havelog + | f ( z ) | ≥ max { h f ( θ ) r q , } − O (cid:0) r q − p + log r (cid:1) , (5.4)where | z | = r I . Recall that each set E j is a finite union of intervals F i . In fact, E ( r ) := ∪ j E j ( r ) = ∪ i F i ( r ), so that Z [0 , π ] \ E ( r ) log + | f ( re iθ ) | dθ = n X j =1 Z θ j +1 − c log rrp θ j + c log rrp log + | f ( re iθ ) | dθ = O (cid:0) r q − p + log r (cid:1) . From (5.1), (5.3) and (5.4), it then follows that T ( r, f ) ≥ m X j =1 r q π Z E j ( r ) max { h f ( θ ) , } dθ + O (cid:0) r q − p + log r (cid:1) = m X j =1 r q π Z E j max { h f ( θ ) , } dθ + O (cid:0) r q − p + log r (cid:1) = r q π Z π max { h f ( θ ) , } dθ + O (cid:0) r q − p + log r (cid:1) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) , r I, where the last identity follows again by Cauchy’s formula and by the fact that h f ( θ ) = k f ( qθ ). The upper and lower estimates just obtained for T ( r, f )have the same magnitude of growth, but the lower estimate is valid outside of an excep-tional set I of finite linear measure. The set I can be avoided by means of Lemma 4.3.This completes the proof of (3.3). Let us begin with the particular case Q ( z ) ≡
0. As the functions g j ( z ) = P j ( z ) e Q j ( z ) , j = 1 , . . . , n , are linearly independent, they form a fundamental solution base for a lineardifferential equation L n ( g ) := g ( n ) + a n − ( z ) g ( n − + · · · + a ( z ) g = 0 , (6.1) ero distribution theory for exponential polynomials a j ( z ) can be expressed as quotientsof Wronskian and modified Wronskian determinants of the functions g j ( z ). Note that g ′ j ( z ) = R j ( z ) e Q j ( z ) , where R j ( z ) = P ′ j ( z ) + Q ′ j ( z ) P j ( z ) . This means that the exponential term e Q j ( z ) remains in differentiation, and further differ-entiations will not change that. Therefore, from each column of each determinant we canpull out the single exponential term as a common factor. Thus exp (cid:0) Q ( z ) + · · · + Q n ( z ) (cid:1) isa common factor for each determinant, and they cancel out in the quotients. The entriesof the remaining determinants are polynomials. Hence the coefficients a j ( z ) in (6.1) arerational functions.If P ( z ) would be a solution of (6.1), it would have to be linearly dependent with thesolutions g , . . . , g n . Thus P ( z ) would be transcendental, which is a contradiction. Itfollows that a n ( z ) := L n ( P ) f is a solutionof the non-homogeneous equation f ( n ) + a n − ( z ) f ( n − + · · · + a ( z ) f = a n ( z ) . Dividing both sides by f a n ( z ) and using the lemma on the logarithmic derivative togetherwith the fact that the coefficients are rational gives us (3.5).Suppose next that Q ( z )
0. Define g ( z ) = e − Q ( z ) f ( z ) = P ( z ) + · · · + P n ( z ) e Q n ( z ) − Q ( z ) . (6.2)Now the discussion above can be applied to g giving us m ( r, /g ) = O (log r ) and, a fortiori, N (cid:18) r, f (cid:19) = N (cid:18) r, g (cid:19) = T ( r, g ) + O (log r ) . (6.3)Suppose in addition that H ( z )
0. Due to the assumption ρ ( H j ) ≤ q − p , we mayassume, without loss of generality, that deg( Q ) ≤ q − p . Thus, using Theorem 3.1 to f and using (6.2), we get T ( r, f ) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) = T ( r, g ) + O (cid:0) r q − p + log r (cid:1) . (6.4)We deduce the estimate (3.4) by combining (6.3) and (6.4). Finally, we suppose that H ( z ) ≡
0, in which case deg( Q j ) = q for every j and m = n . Let U = { w − w , . . . , w m − w } be the set of conjugate leading coefficients of g , and set U = U ∪ { } . By a simplevector calculus, we conclude the following: If w is a boundary point of co( W f ), then 0 isa boundary point of co( U ), while if w is an interior point of co( W f ), then 0 is an interiorpoint of co( U ). Thus, applying Theorem 3.1 to g , we get T ( r, g ) = C (co( U )) r q π + O (cid:0) r q − p + log r (cid:1) = C (co( W f )) r q π + O (cid:0) r q − p + log r (cid:1) . The final assertion (3.6) follows from this and (6.3). ero distribution theory for exponential polynomials The following lemma is considered for more general curves in [4] but in Cartesian coordi-nates. However, our situation is very delicate, and we need the representation particularlyin polar coordinates.
Lemma 7.1
Let p ≥ be any real number, and let U be any collection of Euclidean discs D n = D ( z n , r n ) , where the center points z n ∈ C are ordered according to increasing moduli, | z n | → ∞ , r n > , r n → and X | z n | >e | z n | p − r n log | z n | < ∞ . (7.1) Then the set C ⊂ (0 , ∞ ) of values c for which the curve ∂ Λ p (0 , c ) = (cid:26) z = re iθ : r ≥ , | arg( z ) | = c log rr p (cid:27) meets infinitely many discs D n has linear measure zero.Proof. If z ∈ ∂ Λ(0 , c ), then z = r cos (cid:18) c log rr p (cid:19) ± ir sin (cid:18) log rr p (cid:19) ∼ r ± ic log rr p . Asymptotically this corresponds to the Cartesian curves y = ± c log xx p − . From this represen-tation we see that the cases p = 1 and p > z n lie in the righthalf-plane in between the lines y = ± x . Since | z n | → ∞ and r n →
0, there exists apositive integer N such that | z n | − r n ≥ | z n | / ≥ e, n ≥ N. Note that the function x log xx p is strictly decreasing for x ≥ e . Moreover, it suffices toconsider the portion of ∂ Λ p (0 , c ) located in the upper half-plane, call it Λ + ( c ) for short.Suppose that a disc D n lies asymptotically in between the curves Λ + ( c ) and Λ + ( c ),where c < c , and let ζ and ζ denote the respective intersection points. The disc D n can be seen from the origin at an angle 2 θ n , where sin θ n = r n / | z n | .We have θ n ≤ r n / | z n | → n → ∞ , and c − c = | ζ | p arg( ζ )log | ζ | − | ζ | p arg( ζ )log | ζ |≤ ( | z n | + r n ) p arg( ζ )log( | z n | + r n ) − ( | z n | − r n ) p arg( ζ )log( | z n | − r n ) ≤ | z n | p (cid:16) r n | z n | (cid:17) p (arg( z n ) + θ n )log( | z n | + r n ) − | z n | p (cid:16) − r n | z n | (cid:17) p (arg( z n ) − θ n )log( | z n | − r n ) ≤ | z n | p arg( z n ) + 2 | z n | p θ n log( | z n | − r n ) (cid:18)(cid:18) r n | z n | (cid:19) p − (cid:18) − r n | z n | (cid:19) p (cid:19) . ero distribution theory for exponential polynomials xy arg z = c | z || z | arg z = c | z || z | z n ζ ζ r n r n arg z = arg z n + θ n arg z = arg z n − θ n Figure 2: The disc D n and the asymptotic curves in the case p = 1.Using the estimate (cid:18) r n | z n | (cid:19) p − (cid:18) − r n | z n | (cid:19) p ≤ p (cid:18) − (cid:18) | z n | − r n | z n | + r n (cid:19) p (cid:19) = 2 p p Z | zn |− rn | zn | + rn x p − dx ≤ p +1 pr n | z n | , we conclude that c − c ≤ p +1 p | z n | p − r n (arg( z n ) + 2 θ n )log( | z n | − r n ) ≤ p pπ | z n | p − r n log( | z n | / . Let ε >
0. Then there exists an N ( ε ) ∈ N such that P ∞ n = N ( ε ) 2 p pπ | z n | p − r n log( | z n | / < ε . Let C ε ⊂ (0 , ∞ ) denote the set of values c such that the curve Λ + ( c ) meets at least one of thediscs D n , where n ≥ N ( ε ). Then C ε has linear measure < ε . Since C is contained in allthe sets C ε , ε >
0, it follows that C has linear measure zero. ✷ Through the rest of this section, let k : (0 , ∞ ) → [1 , ∞ ) denote any non-decreasingfunction. To simplify the notation, we restrict to the growth rate k ( r ) = O ( r σ ) for some σ >
0, even though the results that will follow would allow even faster growth. Let f be anexponential polynomial in the normalized form (1.8), and let { z n } denote the sequence ofzeros of f and of all of its transcendental coefficients H j , listed according to multiplicitiesand ordered according to increasing moduli. Finally, let C ( f, k ) denote the collection ofdiscs | z − z n | ≤ /k ( | z n | ). For simplicity, we may also assume that a disc | z | < R for asuitably large R > C ( f, k ).The next auxiliary result is obtained by modifying the proof of [9, Theorem 2.4], seealso the remark following [9, Theorem 2.4]. Lemma 7.2
Let f be an exponential polynomial in the normalized form (1.8) , where wesuppose that ρ ( H j ) ≤ q − p for ≤ p ≤ q . Then there exist constants A > q and c > ero distribution theory for exponential polynomials with the following properties: If Λ denotes the domain in between any two consecutivezero-domains Λ p ( θ ∗ , c ) and Λ p ( θ ∗ , c ) of f , and if z ∈ Λ \ C ( f, k ) , then there exists a uniqueindex j such that f ( z ) = H j ( z ) e w j z q (1 + ε q − p ( r )) , (7.2) where | ε k ( r ) | ≤ B exp (cid:0) − Ar k log r (cid:1) , k ≥ , | z | = r , and B > is some constant. Denote G j ( z ) = H ′ j ( z ) + qw j z q − H j ( z ) . For the same values of z and j as above, we have f ′ ( z ) = (cid:26) G j ( z ) e w j z q (1 + ε q − p ( r )) , if j = 0 ,H ′ ( z ) + ε q − p ( r ) , if j = 0 . (7.3) Proof. (1) In order to prove (7.2), we first suppose that q ≥ ≤ p ≤ q − H j ( z ) are assumed to be transcendental. There existconstants A > q and r > M ( r, H j ) ≤ Ar q − p , j ∈ { , . . . , m } , (7.4)whenever r ≥ r . We conclude by Corollary 4.2 that there exists a constant A > q suchthat log | H j ( z ) | ≥ − Ar q − p log r, j ∈ { , . . . , m } , (7.5)whenever z
6∈ C ( f, k ). We may suppose that the constant A in (7.5) is the same as that in(7.4) by choosing the larger of the two. Noting that r q ψ k ( θ ) = ℜ ( w k z q ), we make use ofLemma 4.4 by choosing c > Aa : If z = re iθ ∈ Λ and if r is large enough, then there existsa unique index j such that ℜ ( w j z q ) ≥ ℜ ( w k z q ) + 4 Ar q − p log r (7.6)for all k = j . If in addition z
6∈ C ( f, k ), then the estimates (7.4)–(7.6) applied to (1.8)yield (cid:12)(cid:12) f ( z ) e − w j z q (cid:12)(cid:12) ≥ | H j ( z ) | − X k = j | H k ( z ) || e ( w k − w j ) z q |≥ | H j ( z ) | − exp (cid:0) log m + Ar q − p − Ar q − p log r (cid:1) , from which | f ( z ) || H j ( z ) e w j z q | ≥ − exp (cid:0) − Ar q − p log r (cid:1) . On the other hand, | f ( z ) || H j ( z ) e w j z q | ≤ X k = j | H k ( z ) || H j ( z ) | (cid:12)(cid:12) e ( w k − w j ) z q (cid:12)(cid:12) ≤ (cid:0) − Ar q − p log r (cid:1) , where z ∈ Λ \ C ( f, k ). This discussion covers the case w j = w = 0 also, that is, the casewhen f ( z ) − H ( z ) = ε p − q ( r ) in λ \ C ( f, k ).If some (but not all) coefficients are polynomials, the previous reasoning simplifies.Indeed, if a particular coefficient H j ( z ) is a polynomial, then the growth of | H j ( z ) | is ero distribution theory for exponential polynomials | z | deg( H j ) for r large enough. Consideration of the set C ( f, k ) is not neededfor this particular coefficient, as we only need to assume that | z | is large enough.Suppose then that q ≥ p = q , that is, all of the coefficients H j ( z )are polynomials. This covers the remaining case. There exist constants A > A > | H j ( z ) | ≤ A | z | d and | H j ( z ) | ≥ A | z | d j , where d j = deg( H j ) and d = max { d j } . Suppose that z = re iθ ∈ Λ, and that r is largeenough. Let A = max { d, } . We choose c > Aa in Lemma 4.4, and find that there existsa unique index j such that (cid:12)(cid:12) f ( z ) e − w j z (cid:12)(cid:12) ≥ | H j ( z ) | − A mr d e − A log r , from which | f ( z ) || H j ( z ) e w j z q | ≥ − exp ( − A log r ) , z ∈ Λ \ C . On the other hand, | f ( z ) || H j ( z ) e w j z q | ≤ X k = j | H k ( z ) || H j ( z ) | (cid:12)(cid:12) e ( w k − w j ) z q (cid:12)(cid:12) ≤ − A log r ) , where z ∈ Λ \ C ( f, k ). This completes the proof of (7.2).(2) In order to prove (7.3), we first notice that f ′ = G ( z ) e w z q + G ( z ) e w z q + · · · + G m ( z ) e w m z q , where G j ( z ) = H ′ j ( z ) + qw j z q − H j ( z ) for j ≥
0. If G j ( z ) ≡
0, then either H j ( z ) is aconstant and w j = 0 or ρ ( H j ) = q . The latter is clearly impossible, while the formeris possible only in the case when j = 0. Hence f ′ is an exponential polynomial withcoefficients being of order ≤ q − p , and shares the leading coefficients with f , exceptpossibly w . Thus, by considering separately the cases j = 0 and j = 0, the proof ofPart (1) applies to f ′ , and we obtain (7.3) for the same index j that appears in (7.2).This completes the proof. ✷ If any of the coefficients H j in (1.8) is transcendental, it also can be represented inan analogous normalized form as f , where the coefficients are either polynomials or expo-nential polynomials. Proceeding from one generation of transcendental coefficients to thenext, the order of growth decreases at least by one. Obviously there are at most q − { z n } denote the sequence of zeros of f , of the transcendental coefficients H j of f ,and of all transcendental descendants of these coefficients, listed according to the multiplic-ities and ordered according to increasing moduli. Let then C ( f, k ) denote the collectionof all discs | z − z n | ≤ /k ( | z n | ), and suppose that a suitably large disc | z | < R is alsoincluded in C ( f, k ). Clearly, C ( f, k ) is a sub-collection of C ( f, k ).Let Θ denote the set consisting of the critical angles of f in (1.8), the critical anglesof the transcendental coefficients H j of f , and the critical angles of all transcendentaldescendants of these coefficients. Some of the critical angles of f may coincide with thoseof its coefficients or descendant coefficients. Nevertheless, Θ is a finite subset of [0 , π ), sothe elements θ j of Θ can be ordered, say 0 ≤ θ < θ < · · · < θ l < π . Set θ l +1 = θ . ero distribution theory for exponential polynomials Lemma 7.3
Let f be an exponential polynomial in the normalized form (1.8) , where wesuppose that ρ ( H j ) ≤ q − p for ≤ p ≤ q . Let θ j , θ j +1 be two consecutive elements of Θ ,and let Λ be the domain in between the domains Λ p ( θ j , c ) and Λ p ( θ j +1 , c ) , where c > issufficiently large. If z ∈ Λ \ C ( f, k ) , then there exists a unique index j and constants C , . . . , C q − p − ∈ C such that f ′ ( z ) f ( z ) = qw j z q − + C q − p − z q − p − + · · · + C + O (cid:0) r − (cid:1) . (7.7) Proof.
We will make use of the proof of Lemma 7.2 not only for f but also for thecoefficients and the descendant coefficients of f . For each function the proof is used, newconstants A > q and c > A and c to be the maxima of all these corresponding coefficients.Independently on whether w j = 0 or w j = 0, it follows directly from (7.2) and (7.3)that f ′ ( z ) f ( z ) = (cid:18) qw j z q − + H ′ j ( z ) H j ( z ) (cid:19) (1 + ε q − p ( r )) . (7.8)Suppose that H j ( z ) is a polynomial. Then (7.8) yields f ′ ( z ) f ( z ) = qw j z q − + O (cid:16) r − (1 + ε q − p ( r )) (cid:17) + O (cid:16) exp (cid:16) ( q −
1) log r − Ar q − p log r (cid:17)(cid:17) = qw j z q − + O (cid:16) r − (cid:17) , z ∈ Λ \ C ( f, k ) , because of A > q . This proves (7.7) in the case when H j ( z ) is a polynomial.Suppose next that H j ( z ) is transcendental. Then q − p ≥
1, so we may write H j ( z )in the normalized form with coefficients being either exponential polynomials of order ≤ q − p − H j ( z ) instead of f , yields H ′ j ( z ) H j ( z ) = (cid:18) C q − p − z q − p − + K ′ ( z ) K ( z ) (cid:19) (1 + ε q − p − ( r )) , (7.9)where C q − p − ∈ C and where K ( z ) is either an exponential polynomial of order ≤ q − p − z .If K ( z ) in (7.9) is a polynomial, then we combine (7.8) and (7.9) for f ′ ( z ) f ( z ) = qw j z q − + C q − p − z q − p − + O (cid:16) r − (cid:17) , z ∈ Λ \ C ( f, k ) . If K ( z ) is transcendental, then we continue inductively in this fashion by reducing theorder on each step by one. Eventually the descendant coefficient must reduce down to apolynomial, and, as such, its logarithmic derivative is of growth O ( r − ). This gives us therepresentation (7.7). ✷ Finally, we remind the reader of the following well-known standard growth estimatefor logarithmic derivatives by Gundersen [6]. ero distribution theory for exponential polynomials Lemma 7.4
Let f be a meromorphic function, let k, j be integers such that k > j ≥ ,and let α > . Then there exists a set E ⊂ (1 , ∞ ) that has finite logarithmic measure,and there exists a constant A > depending only on α, k, j , such that for all z satisfying | z | 6∈ E ∪ [0 , , we have (cid:12)(cid:12)(cid:12)(cid:12) f ( k ) ( z ) f ( j ) ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ A (cid:18) T ( αr, f ) r + n j ( αr ) r log α r log n j ( αr ) (cid:19) k − j , (7.10) where r = | z | and n j ( r ) denotes the number of zeros and poles of f ( j ) in | z | < r . The first assertion n λ p ( r ) = O ( r q − p + log r ) is a simple consequence of (2.3) and thestandard estimate n ( r ) ≤ (log 2) − ( N (2 r ) − N ( r )) between any counting function n ( r )and its integrated counterpart N ( r ). Thus it suffices to prove (2.4), which is non-trivial.Let Γ denote any piecewise smooth positively oriented Jordan curve, and let n (Γ)denote the zeros of f in the domain bounded by Γ. By the argument principle, we have n (Γ) = 12 πi Z Γ f ′ ( z ) f ( z ) dz, (8.1)provided that f has no zeros on Γ. In addition, the curve Γ needs to be separated fromthe zeros of f so that we can use the minimum modulus estimate in Corollary 4.2 as wellas its further consequences in Section 7. Hence we need a smart choice for Γ.In order to simplify the situation, we may suppose that the positive real axis is a criticalray for f by appealing to the exponential polynomial g ( z ) = f ( e − iθ ∗ z ), if necessary. Thus,we assume that θ ∗ = 0 is a critical angle for f , determined by its leading coefficients w j , w k as in (2.1). It follows that w j and w k are the unique dominant leading coefficients of f on the sides of the positive real axis. Without loss of generality, we may assume that w j determines the dominant term of f below the x -axis and that w k determines the dominantterm of f above the x -axis. The uniqueness of the constants w j , w k carries over to theapplication of Lemma 7.3.Let k ( x ) = x q + p − log x and r n = 1 /k ( | z n | ) for x ≥ e and | z n | ≥ e . For this particular k , we let C ( f, k ) denote the set discussed in Section 7. Then (7.1) clearly holds, and so,by Lemma 7.1, we can find c > ∂ Λ p (0 , c ) meets at most finitely many discs | z − z n | ≤ r n . This constant c also takes into account the p -generalization (2.3) of theresult by Steinmetz. Denote by Λ + ( c ) and Λ − ( c ) the portions of Λ p (0 , c ) in the upperhalf-plane and the lower half-plane, respectively.Let α >
1, and let E ⊂ (1 , ∞ ) denote the exceptional set of finite logarithmic measurein Lemma 7.4, where f is our exponential polynomial of order q , and k = 1, j = 0.Choose R ∈ ( e, ∞ ) \ E large enough such that the curves Λ + ( c ) and Λ − ( c ) do not meetany of the discs | z − z n | ≤ r n for r ≥ R . Then choose any r ∈ ( R, ∞ ) \ E . Finally, letΓ = Γ + Γ + Γ + Γ , whereΓ : z = te − ic log ttp , where t goes from R to r, Γ : z = re iθ , where θ goes from − c log rr p to c log rr p , Γ : z = te ic log ttp , where t goes from r to R, Γ : z = Re iθ , where θ goes from c log RR p to − c log RR p . ero distribution theory for exponential polynomials xy Γ : z = te − ic log ttp Γ : z = te − ic log ttp Γ : z = re θ Γ : z = Re θ Figure 3: The domain bounded by the curve Γ = Γ + Γ + Γ + Γ .The idea is to keep R fixed and eventually to let r increase. Due to the construction, it isclear that Γ has no zeros of f , even when r ∈ ( R, ∞ ) \ E is arbitrarily large. We find that I := Z Γ w j qz q − dz = Z rR w j qt q − e − iqc log ttp (cid:18) − − p log tt p +1 ci (cid:19) dt. If we set w j = a j + ib j , then ℑ I = Z rR b j qt q − (cid:26) cos (cid:18) qc log tt p (cid:19) − sin (cid:18) qc log tt p (cid:19) − p log tt p +1 c (cid:27) dt − Z rR a j qt q − (cid:26) sin (cid:18) qc log tt p (cid:19) + cos (cid:18) qc log tt p (cid:19) − p log tt p +1 c (cid:27) dt. Using the standard estimates 1 − x / ≤ cos x ≤ x − x / ≤ sin x ≤ x , we get ℑ I = b j r q + O (cid:0) r q − p log r (cid:1) . Using Lemma 7.3, it follows that ℑ Z Γ f ′ ( z ) f ( z ) dz = b j r q + O (cid:0) r q − p log r (cid:1) . Analogously, if w k = a k + ib k , then ℑ Z Γ f ′ ( z ) f ( z ) dz = − b k r q + O (cid:0) r q − p log r (cid:1) . The minus-sign here is a result of integrating in the opposite direction. Next we applyLemma 7.4 together with Corollary 3.3, and obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12)Z Γ f ′ ( z ) f ( z ) dz + Z Γ f ′ ( z ) f ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) r q − p log α r (cid:1) . ero distribution theory for exponential polynomials πi Z Γ f ′ ( z ) f ( z ) dz = b j − b k π r q + O (cid:0) r q − p log α r (cid:1) . (8.2)Keeping (8.1) in mind, the left-hand side of (8.2) is the number of zeros n ( r, Γ) of f ina domain bounded by the closed curve Γ that depends on r . Since counting functions arealways non-negative, it follows that b j − b k = | b j − b k | . The situation would be symmetricif we would have chosen the leading coefficients w j and w k the other way around in thereasoning above. Moreover, from (2.1), we get ℜ w j e iqθ ∗ = ℜ w k e iqθ ∗ , which can be writtenas a j cos qθ ∗ − b j sin qθ ∗ = a k cos qθ ∗ − b k sin qθ ∗ . Since we have chosen θ ∗ = 0, it follows that a j = a k . Therefore, we have | w j − w k | = | b j − b k | ,so that (8.2) now reads as n ( r, Γ) = | w j − w k | π r q + O (cid:0) r q − p log α r (cid:1) . The assertion follows from this via Lemma 4.3 by writing α = 1 + ε . References [1] Chuaqui M., J. Gr¨ohn, J. Heittokangas and J. R¨atty¨a,
Zero separation results forsolutions of second order linear differential equations . Adv. Math. (2013), 382–422.[2] Dickson D. G.,
Expansions in series of solutions of linear difference-differential andinfinite order equations with constant coefficients.
Mem. Amer. Math. Soc., No. 23(1957), 72 pp.[3] Dickson D. G.,
Asymptotic distribution of zeros of exponential sums . Publ. Math. De-brecen (1964), 297–300.[4] Ding J., J. Heittokangas and Z.-T. Wen, From E -sets to R -sets and beyond .Manuscript in preparation.[5] Gackstatter F. and G. P. Meyer, Zur Wertverteilung der Quotienten von Exponen-tialpolynomen . Arch. Math. (Basel) (1981), no. 3, 255–274.[6] Gundersen G. G., Estimates for the logarithmic derivative of a meromorphic function,plus similar estimates.
J. London Math. Soc. (2) (1988), no. 1, 88–104.[7] Gundersen G. G., Finite order solutions of second order linear differential equations .Trans. Amer. Math. Soc. (1988), no. 1, 415–429.[8] Hayman W.,
Meromorphic Functions . Oxford Mathematical Monographs, ClarendonPress, Oxford, 1964.[9] Heittokangas J., K. Ishizaki, K. Tohge and Z.-T. Wen,
Zero distribution and divisionresults for exponential polynomials . Israel J. Math. (2018), 397–421. ero distribution theory for exponential polynomials
Completely regular growth solu-tions of second order complex linear differential equations . Ann. Acad. Sci. Fenn. (2015), no. 2, 985–1003.[11] Helmrath W. and J. Nikolaus, Ein elementarer Beweis bei der Anwendung derZentralindexmethode auf Differentialgleichungen . Complex Variables Theory Appl. (1984), no. 4, 387–396.[12] Laine I., Nevanlinna Theory and Complex Differential Equations . De Gruyter Studiesin Mathematics, 15. Walter de Gruyter & Co., Berlin, 1993.[13] Langer R. E.,
On the zeros of exponential sums and integrals .Bull. Amer. Math. Soc. (1931), no. 4, 213–239.[14] Lax P. D., The quotient of exponential polynomials . Duke Math. J. (1948), 967–970.[15] Levin B. Ja., Distribution of Zeros of Entire Functions . Translated from the Rus-sian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shieldsand H. P. Thielman. Revised edition. Translations of Mathematical Monographs, 5.American Mathematical Society, Providence, R.I., 1980.[16] MacColl L. A.,
On the distributions of the zeros of sums of exponentials of polynomials .Trans. Amer. Math. Soc. (1934), no. 2, 341–360.[17] P´olya G., Geometrisches ¨uber die Verteilung der Nullstellen spezieller ganzer Funk-tionen . Sitz.-Ber. Bayer. Akad. Wiss. (1920), 285–290.[18] P´olya G.,
Untersuchungen ¨uber L¨ucken und Singularit¨aten von Potenzreihen .Math. Z. (1929), no. 1, 549–640.[19] Ritt J. F., On the zeros of exponential polynomials . Trans. Amer. Math. Soc. (1929), no. 4, 680–686.[20] Schwengeler E., Geometrisches ¨uber die Verteilung der Nullstellen spezieller ganzerFunktionen (Exponentialsummen) . Diss. Z¨urich, 1925.[21] Shapiro H. S.,
The expansion of mean-periodic functions in series of exponentials .Comm. Pure Appl. Math. (1958), 1–21.[22] Steinmetz N., Zur Wertverteilung von Exponentialpolynomen . Manuscripta Math. (1978/79), no. 1–2, 155–167.[23] Tsuji M., Potential Theory in Modern Function Theory . Reprinting of the 1959 orig-inal. Chelsea Publishing Co., New York, 1975.[24] Voorhoeve M., A. J. van der Poorten and R. Tijdeman,
On the number of zeros ofcertain functions . Ned. Akad. Wer. Proc. Ser. A (1975), 407–416.[25] Wittich H., Bemerkung zur Wertverteilun von Exponentialsummen . Arch. Math.(Basel) (1953), 202–209. ero distribution theory for exponential polynomials J. M. Heittokangas
University of Eastern Finland, Department of Physics and Mathematics,P.O. Box 111, 80101 Joensuu, FinlandTaiyuan University of Technology, Department of Mathematics, YingzeWest Street No. 79, Taiyuan 030024, China e-mail:[email protected]
Z.-T. Wen