Relations between growth of entire functions and behavior of its Taylor coefficients
aa r X i v : . [ m a t h . C V ] F e b Relations between growth of entire functionsand behavior of its Taylor coefficients.M.R.Formica, E.Ostrovsky, and L.Sirota.
Universit`a degli Studi di Napoli Parthenope, via Generale Parisi 13, PalazzoPacanowsky, 80132, Napoli, Italy.e-mail: [email protected] of Mathematics and Statistics, Bar-Ilan University, 59200, RamatGan, Israel.e-mail: [email protected] of Mathematics and Statistics, Bar-Ilan University,59200, Ramat Gan, Israel.e-mail: [email protected]
Abstract.
We derive in the closed and unimprovable form the bilateral non - asymptoticrelations between growth of entire functions and decay rate at infinity of its Tay-lor coefficients. We investigate the functions of one as well as of several complexvariables.We will apply the convex analysis: Young - Fenchel (Legendre) transform, Younginequality, saddle - point method etc.
Key words and phrases.
Entire functions, complex variables and functions, radii(radius) of convergence, convex functions, scalar product, upper and lower estimatesand limits, Taylor (power) series, maximal function, order and proximate order,type, saddle - point method, ordinary and multivariate Young - Fenchel (Legendre)transform, slowly varying functions, Young inequality, theorem of Fenchel - Moreau,vectors, factorizable function, examples, random variable, generating function.1
Notations, definitions, statement of problem,previous results.
Let f = f ( z ) be entire (analytical) complex valued function defined on the wholecomplex plane: f ( z ) = ∞ X k =0 c k z k , c k = c k [ f ] , (1)i.e. such that the radius of convergence of the Taylor (power) series (1) is equal toinfinity: lim n →∞ n q | c n | = 0 . Recall that the so - called maximal function, or equally maximal majorant M f ( r ) = M ( r ) , r ∈ (0 , ∞ ) for the source one is defined as follows M f ( r ) def = max z : | z |≤ r | f ( z ) | = max z : | z | = r | f ( z ) | . (2) It is the one of the classical problem from the theory of the entire functions:establish the relations between the asymptotical behavior as n → ∞ the coefficients c n = c n [ f ] and the asymptotical behavior for the maximal function M f ( r ) = M ( r ) as r → ∞ . See for example the classical monographs [3], [10]; where are described also someimportant applications, for instance in the theory of distributions of zeros of entirefunctions and in the functional analysis, in particular in the theory of operators.As regards for the modern work we mention an article [5].Let us bring some notions from the classical theory. The order ρ [ f ] of thefunction f = f ( z ) may be calculated by the following relations ρ [ f ] = lim n →∞ ( n ln n | ln | c n | | ) . (3)Correspondingly, the type β [ f ] is equal to β [ f ] = lim n →∞ (cid:26) n /ρ [ f ] n q | c n | (cid:27) . (4) Our target in this short report is to establish the non - asymptoti-cal exact bilateral estimations between the coefficients of the consideredfunction and its maximal majorant.They have a very simple, closed and very general form. e extend obtained results in the last section onto the entire functionsof several complex variables. We derive as a consequence the natural conditions for the possible coincidenceof these estimates: upper and lower ones.
Recall also one important for us auxiliary facts from the theory of convex func-tion. The so - called Young - Fenchel, or Legendre transform g ∗ ( y ) , y ∈ R for thegiven numerical valued function g = g ( x ) having the convex non - empty domainof definition Dom[g] is defined as follows g ∗ ( y ) def = sup x ∈ Dom[g] ( xy − g ( x )) . (5)The function g ∗ ( · ) named as customary Young conjugate, or simple conjugate to the source one g ( · ) . Many examples and properties of this transform with applications e.g. to thetheory of Orlicz spaces may be found in the classical monographs [8], [15], [16], [17],[19].The famous theorem of Fenchel - Moreau tell us that if the function g = g ( x )is convex and continuous defined on the convex set, then g ∗∗ ( x ) = g ( x ) , (6)see [17], chapters 2,3; [19].Recall also the following important Young’s inequality xy ≤ g ( x ) + g ∗ ( y ) , x, y ∈ R ; (7)or more generally xy ≤ g ( γx ) + g ∗ ( y/γ ) , γ = const > , x , y ∈ R . (8) Upper estimate.
We mean to derive the non - asymptotic upper estimate of the Taylors coefficientsfor the function f : c n [ f ] via its maximal function M f ( r ) . Introduce the function Λ( v ) def = ln M f ( e v ) , v ∈ R. (9)3owever, it is sufficient for us hereinafter to suppose at last instead (9) only anunilateral restriction M f ( r ) ≤ exp ( Λ(ln r ) ) , r > , (10)for certain function Λ = Λ( v ) , ∈ R. The restrictions on this function will beclarified below.
Proposition 2.1.
We propose under condition (10) | c n | ≤ exp ( − Λ ∗ ( n ) ) , n = 1 , , . . . . (11) Proof.
We start from the simple estimate | c n | ≤ M f ( r ) r n , r > . Therefore | c n | ≤ exp ( − n ln r + ln M f ( r ) ) = exp ( − nv + Λ( v ) ) =exp ( − ( nv − Λ( v )) ) , v = ln r ∈ ( −∞ , ∞ ) . Since the last inequality is true for arbitrary value v, one can take the minimumover v : | c n | ≤ exp( − sup v ∈ R ( nv − Λ( v )) ) = exp ( − Λ ∗ ( n ) ) , Q.E.D.
Lower estimate.
We intent to derive in this subsection the non - asymptotic lower estimate ofthe Taylors coefficients c n [ f ] = c n for the function f : c n [ f ] via its maximalfunction M f ( r ) . Equivalently: we want to obtain the upper estimate for the maximal function M f ( r ) through its series of Taylor coefficients.The lower estimate is more complicated. We will follow the authors on an article[7]. Suppose | c n | ≤ exp ( − Q ( n ) ) (12)for certain increasing to infinity function Q = Q ( z ) , z ≥ . We have for all thesufficient greatest values r ≥ e f ( r ) ≤ ∞ X n =0 exp ( nv − Q ( n ) ) =: R ( v ) R Q ( v ) , v = ln r. (13)It follows from the theory of the saddle - point method, see e.g. [4], chapters1,2, that for some finite positive constant C = C [ Q ] ∈ (1 , ∞ ) R Q ( v ) ≤ exp( sup n ( Cnv − Q ( n )) ) =exp( Q ∗ ( Cv ) ) = exp( Q ∗ ( C ln r ) ) , r ≥ e. (14)If in addition the function Q = Q ( v ) is in turn Young conjugate to somecontinuous and convex one, say G ( · ) : Q = G ∗ , then by virtue of Theorem ofFenchel - Moreau M f ( r ) ≤ C [ G ] exp ( G ( C ln r ) ) , r ≥ e. (15)In particular, the function G ( · ) may coincides with the introduced beforefunction Λ ∗ ( · ); and we conclude in this case M f ( r ) ≤ C [Λ] exp ( Λ( C ln r ) ) , r ≥ e. (16)Let us bring the strong proof of (15), of course, under appropriate conditions.Define the following function K Q ( ǫ ) := ∞ X n =0 exp ( − ǫQ ∗ ( n ) ) , ǫ ∈ (0 , . It is proved in particular in [7] that if ∃ ǫ ∈ (0 , ⇒ K ( ǫ ) < ∞ , then R ( v ) ≤ K Q ( ǫ ) exp (cid:18) (1 − ǫ ) Q ∗∗ (cid:18) v − ǫ (cid:19) (cid:19) . Further, as long as nv ≤ Q ∗ ((1 − ǫ ) v ) + Q ∗∗ ( n/ (1 − ǫ )) , ǫ ∈ (0 , . we have R ( v ) ≤ U Q ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) , where in addition to the (14) U ( ǫ ) = U Q ( ǫ ) := ∞ X n =0 exp( Q ∗ ((1 − ǫ ) n ) − Q ∗ ( n )) . To summarize:
Proposition 2.2.
Denote 5 Q ( ǫ ) = Y ( ǫ ) := min( U ( ǫ ) , K ( ǫ )) , ǫ ∈ (0 , . We assert R Q ( v ) ≤ Y Q ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) , ǫ ∈ (0 , . (17)Of course, R Q ( v ) ≤ inf ǫ ∈ (0 , { Y Q ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) } . (18)As a consequence: Proposition 2.3. If ∃ ǫ ∈ (0 , ⇒ Y Q ( ǫ ) < ∞ , (19)then the estimations (14) holds true. If in addition the function Q = Q ( v ) iscontinuous and convex, then (16) is valid, as well. To summarize. A. Recall that ln M f ( e v ) ≤ Λ( v ) , v ∈ R, therefore | c n [ f ] | ≤ exp ( − Λ ∗ ( n ) ) , n = 1 , , . . . . (20) B. Conversely, let the inequality (20) be given for certain non - negative con-tinuous convex function Λ = Λ( v ) , v ∈ R for which ∃ ǫ ∈ (0 , ⇒ S = S ( ǫ ) def = Y Λ ∗ ( ǫ ) < ∞ . Then M f ( r ) ≤ S ( ǫ ) exp ( Λ ln r − ǫ ! ) , r ≥ e. (21)Briefly: under formulated above conditions M f ( r ) ≤ e Λ(ln r ) ⇒ | c n | ≤ e − Λ ∗ ( n ) , r ≥ e ; (22) | c n | ≤ e − Λ ∗ ( n ) ⇒ M f ( r ) ≤ S e Λ(ln r/ ( 1 − ǫ )) . (23) C. ”Tauberian” theorem. additional restriction on the function Λ( · ) : ∃ γ = γ (Λ , ǫ ) = const < ∞ ⇒ Λ (cid:18) v1 − ǫ (cid:19) ≤ γ Λ(v) , v ≥ . (24)It follows immediately from the relations (22) and (23) the following assertion. Theorem 2.1.
We conclude under formulated above conditionslim r →∞ ln M f ( r )Λ(ln r ) = lim n →∞ | ln 1 / | c n || Λ ∗ ( n ) . (25)More precisely: if there exists the left - hand side of (25), then there exists alsothe right - hand one and they are equal; the converse proposition is also true: ifthere exists the right - hand side of (25), then there exists also the left - hand oneand they are equal. Auxiliary fact.
Introduce a following family of regular varying functions φ m,L ( λ ) def = 1 m λ m L ( λ ) , λ ≥ . (26)Here m = const > , and define as ordinary m ′ := m/ ( m −
1) and L = L ( λ ) , λ ≥ slowly varying as λ → ∞ function. It is known that as x → ∞ , x ≥ φ ∗ m,L ∼ ( m ′ ) − x m ′ L − / ( m − (cid:16) x / ( m − (cid:17) , (27)see [18], pp. 40 - 44; [7]. For instance, if φ m ( λ ) = m − | λ | m , λ ∈ R, then φ ∗ m ( x ) = ( m ′ ) − | x | m ′ , x ∈ R. More generally, if ψ m,q ( λ ) = C λ m [ln λ ] q , λ ≥ e, m = const > , q = const ≥ , then ψ ∗ m,q ( x ) ≍ C ( m, q ) x m ′ [ln x ] − q/ ( m − , x ≥ e. Example 3.1.
Suppose that for some entire function f = f ( z )ln M f ( r ) ≍ C ( m ) [ln r ] m , r ≥ e, m = const > . (28)Then 7 n [ f ] ≤ exp n − C ( m ) n m ′ o , n ≥
0; (29)and conversely proposition is also true: from the estimation (29) follows the inequal-ity (28).Note that the case m ≤ f ( z ) is polynomial, see [10],chapter 1, sections 2 - 4. This implies that ∃ N ∈ { , , . . . } ∀ n ≥ N ⇒ c n = 0 . Example 3.2.
Suppose that for some entire function f = f ( z )ln M f ( r ) ∼ C r ρ , r ≥ , ρ = const > , C = const ∈ (0 , ∞ ) , (30)a classical case, [10], chapter 1, sections 1 - 5. Then | c n | ≤ " nC ρ − n/ρ · e n/ρ , (31)and conversely proposition is also true: from the estimation (31) follows the inequal-ity (30).More generally, the relation of the formln M f ( r ) ∼ ρ − r ρ ln γ r, r → ∞ is quite equivalent to the following equality: as n → ∞ ln | c n | ! ∼ ρ − n ln n + γn ln ln n/ρ − nρ . Example 3.3.
We propose that for arbitrary entire function f = f ( z ) thefollowing relations are equivalent: ∃ C , C ∈ (0 , ∞ ) ⇒ ln M f ( r ) ≤ C e C r , r ≥ , (32)and ∃ C ∈ (0 , ∞ ) ⇒ | c n | ≤ C (ln n ) − n , n ≥ . (33)8 Generalization on the entire (holomorphic)functions of several complex variables.
It is no hard to generalize the obtained results on the case of the analyticalfunctions f = f ( z ) of several complex variables.Let us introduce first all some (ordinary) used notations. The (finite) dimensionof a considered problem will be denoted by d ; d = 2 , , . . . . Correspondingly themultivariate variable z consists on the d independent complex variables z = ~z = { z , z , . . . , z d } . Ordinary vector notations: k = ~k = { k , k , . . . , k d } , k j = 0 , , , . . . ; | k | := d X j =1 k j ; ~z ~k = z k = d Y j =1 z k j j ; ~r = r = { r , r , . . . , r d } ∈ R d + , r j ≥ v = ~v = ~v ( r ) := exp( ~r ) = { e r , e r , . . . , e r d } ∈ R d , ⇐⇒ ln ~v = ln v = { ln v j } , j = 1 , , . . . , d. The multivariate
Young - Fenchel transform g ∗ ( y ) , y ∈ R d for the givennumerical valued function g = g ( x ) , x ∈ R d having the convex non - emptydomain of definition Dom[g] is defined as usually g ∗ ( y ) def = sup x ∈ Dom[g] ( ( x, y ) − g ( x ) ) , (34)where ( x, y ) denotes the inner, or scalar product of the two d − dimensional vectors x, y. As above, if the function g = g ( x ) is continuous and convex, g ∗∗ = g. Thefamous Young inequality has a form( x, y ) ≤ g ( γx ) + g ∗ ( y/γ ) , γ = const > , x , y ∈ R d . (35)Further, the analytical function f = f ( z ) has a form f ( ~z ) = f ( z ) = X k c k z k = X ~k c ~k ~z ~k , (36)where the numbers { c k } = { c ~k } are the Taylor’s coefficients for f = f ( z ) andthe series in (36) converges for all the complex vectors ~z. M ( r ) = M f ( r ) = M f ( ~r ) for f = f ( z ) at the point r ∈ R d + is defined as before M f ( ~r ) def = max | z j |≤ r j | f ( ~z ) | = max | z j | = r j | f ( z ) | . (37)Define also the functionΛ( v ) = Λ( ~v ) def = ln M (cid:16) e ~v (cid:17) = ln M f ( e v ) , v ∈ R d . (38) Upper estimate.
We apply the following estimate | c ~k | ≤ M f ( ~r ) ~r ~k , ~r ∈ R d + , see, e.g., [6], page 15, formula 1.4.3. Following, | c k | ≤ exp ( − (( k, v ) − Λ( v )) ) ; | c ~k | ≤ exp( − sup v ∈ R (( k, v ) − Λ( v )) ) , and as before Proposition 4.1. | c ~k | ≤ exp (cid:16) − Λ ∗ ( ~k ) (cid:17) . (39) Lower estimate.
The lower estimate is quite alike ones obtained in the second section in the one- dimensional case, i.e. when d = 1; we will apply also the methods explained in[7]. Suppose | c k | = | c ~k | ≤ exp (cid:16) − Q ( ~k ) (cid:17) = exp ( − Q ( k ) ) (40)for certain increasing to infinity relative the all variables k j , j = 1 , , . . . , d function Q = Q ( z ) , z = ~z = { z j } , z j ≥ . We have for all the sufficient greatest values r j ≥ e M f ( ~r ) ≤ X ~k ≥ ~ exp ( ( k, v ) − Q ( k ) ) =: R ( v ) = R ( ~v ) , ~v = ~ ln r. (41)10efine the following function K ( ǫ ) := X ~k ≥ ~ exp (cid:16) − ǫQ ∗ ( ~k ) (cid:17) , ǫ ∈ (0 , . It is proved in particular in [7] that if ∃ ǫ ∈ (0 , ⇒ K ( ǫ ) < ∞ , then R ( v ) ≤ K ( ǫ ) exp (cid:18) (1 − ǫ ) Q ∗∗ (cid:18) v − ǫ (cid:19) (cid:19) . Further, as long as( k, v ) ≤ Q ∗ ((1 − ǫ ) v ) + Q ∗∗ ( k/ (1 − ǫ )) , ǫ ∈ (0 , , we have R ( v ) ≤ U ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) , where U ( ǫ ) := X ~k ≥ ~ exp( Q ∗ ((1 − ǫ ) ~k ) − Q ∗ ( ~k )) . To summarize:
Proposition 4.2.
Denote Y ( ǫ ) := min( U ( ǫ ) , K ( ǫ )) , ǫ ∈ (0 , . We assert R ( v ) ≤ Y ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) , ǫ ∈ (0 , . (42)Of course, R ( v ) ≤ inf ǫ ∈ (0 , { Y ( ǫ ) exp ( Q ∗∗ ( v/ (1 − ǫ )) ) } . (43)As a consequence: Proposition 4.3. If ∃ ǫ ∈ (0 , ⇒ Y ( ǫ ) < ∞ , (44)then the following estimation M f ( ~r ) ≤ exp( Q ∗∗ ( C ln r ) ) , r j ≥ e. (45)holds true.If in addition the function Q = Q ( ~v ) is continuous and convex, then11 f ( ~r ) ≤ exp( Q ( C ln r ) ) , r j ≥ e. (46)In particular, if the function Q ( · ) coincides with the introduced before functionΛ ∗ ( · ) , we conclude in this case M f ( r ) ≤ exp ( Λ( C ln r ) ) , r j ≥ e. (47) Multivariate examples.
It is no difficult to show the exactness of obtained estimates still in the mul-tidimensional case. It is sufficient to consider the case d = 2 and the so - calledfactorizable function f = f ( z ) = f ( z , z ) = f ( z ) · f ( z ) , z = ( z , z ) , where the functions f , f are function - examples considered in the third section.If f ( z ) = ∞ X k =0 a k z k , f ( z ) = ∞ X l =0 b l z l , then f ( z , z ) = X ∞ X k,l =0 a k b l z k z l = X ∞ X k,l =0 c k,l z k z l , where c k,l = a k b l . Obviously, M f ( r , r ) = M f ( r ) · M f ( r ) = exp { Λ f (ln r ) + Λ f (ln r ) } . As long as sup λ,µ ∈ R [ ( λx + µy ) − ( g ( λ ) + h ( µ )) ] = g ∗ ( x ) + h ∗ ( y ) , we conclude | c k,l | ≤ exp n − Λ ∗ f ( k ) − Λ ∗ f ( l ) o and so one; see theorem 3.1. 12 Concluding remarks. A. It is no hard to generalize obtained estimations on the derivatives of the sourcefunction f = f ( z ) , including the partial derivatives in the case of the function ofseveral complex variables. For instance, for the function from (1) we have f ′ ( z ) = ∞ X k =1 kc k [ f ] z k − , so that c k [ f ′ ] = ( k + 1) c k +1 [ f ] . B. The continuous version of our estimations, i.e. the Tauberian theoremsfor the Laplace transform are well known, see e.g. [20]. The non - asymptoticalestimates may be found, e.g. in [14], pp. 27 - 37. C. Let us list briefly a several works devoted to the applications of TauberianTheorems in the Probability Theory: [1], [2], [9], [11], [12], [13].Let us show some extension of obtained in these works results based on the ourestimations. Let ξ be an integer values non - negative random variable (r.v.): P ( ξ = k ) = c k , k = 0 , , , . . . . (48)Of course, c k ≥ , P k c k = 1 . The so - called generating function (g.f.) g [ ξ ] = g [ ξ ]( z ) for this r.v. is as ordinary defined as follows g [ ξ ]( z ) def = E z ξ = ∞ X k =0 c k z k . (49)This notion play a very important role in the probability theory, in particular, inthe reliability theory, in the grand deviation theory, in the theory of queue theoryetc. It is important especially for these applications the asymptotical behavior of g [ ξ ]( z ) as | z | → ∞ . One can for example apply our theorem 2.1 for the probability theory. Namely,we conclude under formulated in this theorem conditionslim r →∞ ln M g ( r )Λ P (ln r ) = lim n →∞ | ln 1 / | c n || Λ ∗ P ( n ) . (50)More precisely: if there exists the left - hand side of (50), then there exists alsothe right - hand one and they are equal; the converse proposition is also true: ifthere exists the right - hand side of (50), then there exists also the left - hand oneand they are equal.Here with accordance (9)Λ P ( v ) def = ln M g ( e v ) , v ∈ R. (51)13nalogous fact holds true still in the multidimensional case. Acknowledgement.
The first author has been partially supported by the Gruppo Nazionaleper l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionaledi Alta Matematica (INdAM) and by Universit`a degli Studi di Napoli Parthenope through theproject “sostegno alla Ricerca individuale”(triennio 2015 - 2017) . References [1]
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