A property of the derivative of an entire function
aa r X i v : . [ m a t h . C V ] J u l A property of the derivative of an entirefunction
Walter Bergweiler ∗ and Alexandre Eremenko † November 15, 2018
Abstract
We prove that the derivative of a non-linear entire function is un-bounded on the preimage of an unbounded set.MSC 2010: 30D30. Keywords: entire function, normal family.
The main result of this paper is the following theorem conjectured by AllenWeitsman (private communication):
Theorem 1.
Let f be a non-linear entire function and M an unbounded setin C . Then f ′ ( f − ( M )) is unbounded. We note that there exist entire functions f such that f ′ ( f − ( M )) isbounded for every bounded set M , for example, f ( z ) = e z or f ( z ) = cos z .Theorem 1 is a consequence of the following stronger result: Theorem 2.
Let f be a transcendental entire function and ε > . Thenthere exists R > such that for every w ∈ C satisfying | w | > R there exists z ∈ C with f ( z ) = w and | f ′ ( z ) | ≥ | w | − ε . ∗ Supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-1, and the ESF Net-working Programme HCAA. † Supported by NSF grant DMS-1067886. f ( z ) = √ z sin √ z shows that that the exponent 1 − ε in thelast inequality cannot be replaced by 1. The function f ( z ) = cos √ z has theproperty that for every w ∈ C we have f ′ ( z ) → z → ∞ , z ∈ f − ( w ).We note that the Wiman–Valiron theory [20, 12, 4] says that there existsa set F ⊂ [1 , ∞ ) of finite logarithmic measure such that if | z r | = r / ∈ F and | f ( z r ) | = max | z | = r | f ( z ) | , then f ( z ) ∼ (cid:18) zz r (cid:19) ν ( r,f ) f ( z r ) and f ′ ( z ) ∼ ν ( r, f ) r f ( z )for | z − z r | ≤ rν ( r, f ) − / − δ as r → ∞ . Here ν ( r, f ) denotes the centralindex and δ >
0. This implies that the conclusion of Theorem 2 holds forall w satisfying | w | = M ( r, f ) for some sufficiently large r / ∈ F . However,in general the exceptional set in the Wiman–Valiron theory is non-empty(see, e.g., [3]) and thus it seems that our results cannot be proved usingWiman–Valiron theory. Acknowledgment.
We thank Allen Weitsman for helpful discussions.
One important tool in the proof is the following result known as the ZalcmanLemma [21]. Let g = | g ′ | | g | denote the spherical derivative of a meromorphic function g . Lemma 1.
Let F be a non-normal family of meromorphic functions in aregion D . Then there exist a sequence ( f n ) in F , a sequence ( z n ) in D , asequence ( ρ n ) of positive real numbers and a non-constant function g mero-morphic in C such that ρ n → and f n ( z n + ρ n z ) → g ( z ) locally uniformlyin C . Moreover, g ( z ) ≤ g (0) = 1 for z ∈ C . We say that a ∈ C is a totally ramified value of a meromorphic function f if all a -points of f are multiple. A classical result of Nevanlinna says thata non-constant function meromorphic in the plane can have at most 4 totallyramified values, and that a non-constant entire function can have at most2 finite totally ramified values. Together with Zalcman’s Lemma this yieldsthe following result [5, 13, 14]; cf. [22, p. 219].2 emma 2. Let F be a family of functions meromorphic in a domain D and M a subset of C with at least elements. Suppose that there exists K ≥ such that for all f ∈ F and z ∈ D the condition f ( z ) ∈ M implies | f ′ ( z ) | ≤ K . Then F is a normal family.If all functions in F are holomorphic, then the conclusion holds if M hasat least elements. Applying Lemma 2 to the family { f ( z + c ) : c ∈ C } where f is an entirefunction, we obtain the following result. Lemma 3.
Let f be an entire function and M a subset of C with at least elements. If f ′ is bounded on f − ( M ) , then f is bounded in C . It follows from Lemma 3 that the conclusion of Theorems 1 and 2 holdsfor all entire functions for which f is unbounded.We thus consider entire functions with bounded spherical derivative. Thefollowing result is due to Clunie and Hayman [6]. Let M ( r, f ) = max | z |≤ r | f ( z ) | and ρ ( f ) = lim sup r →∞ log log M ( r, f )log r denote the maximum modulus and the order of f . Lemma 4.
Let f be an entire function for which f is bounded. Then log M ( r, f ) = O ( r ) as r → ∞ . In particular, ρ ( f ) ≤ . We will include a proof of Lemma 4 after Lemma 6.The following result is due to Valiron [20, III.10] and H. Selberg [17,Satz II].
Lemma 5.
Let f be a non-constant entire function of order at most forwhich and − are totally ramified. Then f ( z ) = cos( az + b ) , where a, b ∈ C , a = 0 . We sketch the proof of Lemma 5. Put h ( z ) = f ′ ( z ) / ( f ( z ) − h isentire and the lemma on the logarithmic derivative [9, p.94, (1.17)], togetherwith the hypothesis that ρ ( f ) ≤
1, yields that m ( r, h ) = o (log r ) and hencethat h is constant. This implies that f has the form given. Another proof isgiven in [10]The next lemma can be extracted from the work of Pommerenke [16,Sect. 5], see [8, Theorem 5.2]. 3 emma 6. Let f be an entire function and C > . If | f ′ ( z ) | ≤ C whenever | f ( z ) | = 1 , then | f ′ ( z ) | ≤ C | f ( z ) | whenever | f ( z ) | ≥ . Lemma 6 implies the theorem of Clunie and Hayman mentioned above(Lemma 4). For the convenience of the reader we include a proof of a slightlymore general statement, which is also more elementary than the proofs ofClunie, Hayman and Pommerenke; see also [1, Lemma 1].Let G = { z : | f ( z ) | > } and u = log | f | . Then | f ′ /f | = |∇ u | and ourstatement which implies Lemmas 4 and 6 is the following. Proposition.
Let G be a region in the plane, u a harmonic function in G ,positive in G , and such that for z ∈ ∂G we have u ( z ) = 0 and |∇ u ( z ) | ≤ .Then |∇ u ( z ) | ≤ for z ∈ G , and u ( z ) ≤ | z | + O (1) as z → ∞ .Proof. It is enough to consider the case of unbounded G with non-emptyboundary. For a ∈ G , consider the largest disc B centered at a and containedin G . The radius d = d ( a ) of this disc is the distance from a to ∂G . Thereis a point z ∈ ∂B such that u ( z ) = 0. Put z ( r ) = a + r ( z − a ) , where r ∈ (0 , u ( a ) d (1 + r ) ≤ u ( z ( r )) d (1 − r ) = u ( z ( r )) − u ( z ) d (1 − r ) . Passing to the limit as r → u ( a ) ≤ d ( a ) |∇ u ( z ) | ≤ d ( a ) . This holds for all a ∈ G . Now we take the gradient of both sides of thePoisson formula and, noting that u ( a + d ( a ) e it ) ≤ d ( a + d ( a ) e it ) ≤ d ( a ),obtain the estimate |∇ u ( a ) | ≤ πd ( a ) Z π − π | u ( a + d ( a ) e it ) | dt ≤ . So ∇ u is bounded in G . As the complex conjugate of ∇ u is holomorphicin G and |∇ u ( z ) | ≤ z of G , except infinity, thePhragm´en–Lindel¨of theorem [15, III, 335] gives that |∇ u ( z ) | ≤ z ∈ G .This completes the proof of the Proposition.We recall that for a non-constant entire function f the maximum modulus M ( r ) = M ( r, f ) is a continuous strictly increasing function of r . Denote by4 the inverse function of M . Clearly, for | w | > | f (0) | the equation f ( z ) = w has no solutions in the open disc of radius ϕ ( | w | ) around 0. The followingresult of Valiron ([18, 19], see also [7]) says that for functions of finite orderthis equation has solutions in a somewhat larger disc. Lemma 7.
Let f be a transcendental entire function of finite order and η > . Then there exists R > | f (0) | such that for all w ∈ C , | w | ≥ R , theequation f ( z ) = w has a solution z satisfying | z | < ϕ ( | w | ) η . We note that Hayman ([11], see also [2, Theorem 3]) has constructedexamples which show that the assumption about finite order is essential inthis lemma.
Suppose that the conclusion is false. Then there exists ε >
0, a tran-scendental entire function f and a sequence ( w n ) tending to ∞ such that | f ′ ( z ) | ≤ | w n | − ε whenever f ( z ) = w n . By Lemma 3, the spherical derivativeof f is bounded, and we may assume without loss of generality that f ( z ) ≤ z ∈ C . (1)We may also assume that f (0) = 0. It follows from (1) that | f ′ ( z ) | ≤ | f ( z ) | = 1, and thus Lemma 6 yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | f ( z ) | ≥ . (2)It also follows from (1), together with Lemma 4, that ρ ( f ) ≤
1. We maythus apply Lemma 7 and find that if η > n is sufficiently large, thenthere exists ξ n satisfying | ξ n | ≤ ϕ ( | w n | ) η and f ( ξ n ) = w n . We put τ n = ϕ ( | w n | ) η and define Φ n ( z ) = w n − f ( τ n z ) w n = 1 − f ( τ n z ) w n . n (0) = 1 , Φ n ( ξ n /τ n ) = − , and ξ n /τ n → n → ∞ . Thus thesequence (Φ n ) is not normal at 0, and we may apply Zalcman’s Lemma(Lemma 1) to it. Replacing (Φ n ) by a subsequence if necessary, we thus findthat g n ( z ) = Φ n ( z n + ρ n z ) = 1 − w n f ( τ n z n + τ n ρ n z ) → g ( z )locally uniformly in C , where | z n | ≤ ρ n > ρ n →
0, and g is a non-constant entire function with bounded spherical derivative. With ζ n = τ n z n and µ n = τ n ρ n we have g n ( z ) = 1 − w n f ( ζ n + µ n z ) , (3)and g ′ n ( z ) = − µ n w n f ′ ( ζ n + µ n z ) . (4)We may assume that ρ n ≤ | ζ n | ≤ τ n and µ n ≤ τ n for all n .If g n ( z ) = 1, then f ( ζ n + µ n z ) = 0, hence | f ′ ( ζ n + µ n z ) | ≤ µ n ≤ τ n , we deduce that | g ′ n ( z ) | ≤ τ n w n if g n ( z ) = 1 . (5)If g n ( z ) = −
1, then f ( ζ n + µ n z ) = w n , and hence | f ′ ( ζ n + µ n z ) | ≤ | w n | − ε byour assumption. Thus | g ′ n ( z ) | ≤ µ n | w n | | w n | − ε ≤ τ n | w n | ε if g n ( z ) = − . (6)It follows from the definition of τ n that τ n = o ( | w n | ) δ ) as n → ∞ , (7)for any given δ > g ′ ( z ) = 0 whenever g ( z ) = 1 or g ( z ) = −
1. Since g has bounded spherical derivative, we conclude fromLemmas 3 and 4 that g ( z ) = cos( az + b ) . Without loss of generality, wemay assume that g ( z ) = cos z so that g ′ ( z ) = − sin z . In particular, thereexist sequences ( a n ) and ( b n ) both tending to 0, such that g n ( a n ) = 1 and g ′ n ( b n ) = 0. From (5) we deduce that | g ′ n ( a n ) | ≤ τ n | w n | . (8)6oting that g ′′ ( z ) = − cos z we find that g ′ n ( a n ) = g ′ n ( a n ) − g ′ n ( b n ) = Z a n b n g ′′ n ( z ) dz ∼ b n − a n (9)as n → ∞ , and thus | b n − a n | ≤ τ n | w n | (10)for large n , by (8). This implies that | g n ( b n ) − | = | g n ( b n ) − g n ( a n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b n a n g ′ n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | b n − a n | ≤ τ n | w n | (11)for large n .We put h n ( z ) = g n ( z + b n ) − g n ( b n )and note that h n (0) = 0, h ′ n (0) = g ′ n ( b n ) = 0 and h n ( z ) → cos z − n → ∞ . It follows that h n ( z ) z → cos z − z as n → ∞ , which implies that there exists r > ≤ | h n ( z ) || z | ≤
34 for | z | ≤ r. (12)and large n .Now we fix any γ ∈ (0 , /
2) and put c n = b n + 1 | w n | γ . Then g n ( c n ) − h n ( | w n | − γ ) + g ( b n ) − n : | g n ( c n ) − | ≤ (cid:12)(cid:12)(cid:12) h n ( | w n | − γ ) (cid:12)(cid:12)(cid:12) + | g ( b n ) − | ≤ | w n | γ + 6 τ n | w n | ≤ | w n | γ . (13)7imilarly | g n ( c n ) − | ≥ (cid:12)(cid:12)(cid:12) h n ( | w n | − γ ) (cid:12)(cid:12)(cid:12) − | g ( b n ) − | ≥ | w n | γ . (14)On the other hand, arguing as in (9), we have g ′ n ( c n ) = g ′ n ( c n ) − g ′ n ( b n ) = Z c n b n g ′′ n ( z ) dz ∼ b n − c n = − | w n | γ , and thus | g ′ n ( c n ) | ≥ | w n | γ (15)for large n . Put v n = ζ n + µ n c n . Then f ( v n ) = w n − g n ( c n )) and f ′ ( v n ) = w n µ n g ′ n ( c n ) , by (3) and (4). Hence110 | w n | − γ ≤ | f ( v n ) | ≤ | w n | − γ , (16)by (13) and (14) while | f ′ ( v n ) | ≥ | w n | γ µ n . Since | f ( v n ) | ≥ n , by (16), this contradicts (2) and (7). References [1] M. Barrett and A. Eremenko, A generalization of a theorem of Clunieand Hayman, to appear in Proc. Amer. Math. Soc., arXiv: 1011.3907.[2] W. Bergweiler, Order and lower order of composite meromorphic func-tions, Michigan Math. J., 36 (1989) 135–146.[3] W. Bergweiler, On meromorphic functions that share three values andon the exceptional set in Wiman–Valiron theory, Kodai Math. J. 13(1990) 1–9. 84] W. Bergweiler, P. J. Rippon and G. M. Stallard, Dynamics of meromor-phic functions with direct or logarithmic singularities, Proc. LondonMath. Soc., 97 (2008) 368–400.[5] H. H. Chen and X. H. Hua, Normality criterion and singular direc-tions, in “Proceedings of the Conference on Complex Analysis (Tianjin,1992)”, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA,1994, pp. 34–40.[6] J. Clunie and W. Hayman, On the spherical derivative of integral andmeromorphic functions, Comm. Math. Helv., 40 (1966) 117–148.[7] A. Edrei and W. Fuchs, On the zeros of f ( g ( z )) where f and g are entirefunctions, J. Anal. Math., 2 (1964) 243–255.[8] A. Eremenko, Normal holomorphic curves from parabolic regions to pro-jective spaces, arXiv: 0710.1281.[9] A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromor-phic functions. Transl. Math. Monographs 236, American Math. Soc.,Providence, R. I., 2008.[10] A. Gol~dberg, V. Tairova, O elyh funk i(cid:31)h s dvum(cid:31) koneqnymivpolne razvetvlennymi znaqeni(cid:31)mi, Zapiski meh-mat fakultetaHar~kovskogo universiteta i Har~kovskogo mat. obw., 29 (1963) 67{78 (Russian).[11] W. Hayman, Some integral functions of infinite order, Math. Notae, 20(1965) 1–5.[12] W. Hayman, The local growth of power series: a survey of the Wiman–Valiron method, Canad. Math. Bull., 17 (1974) 317–358.[13] A. Hinkkanen, Normal families and Ahlfors’s five islands theorem, NewZealand J. Math., 22 (1993) 2, 39–41.[14] P. Lappan, A uniform approach to normal families, Rev. RoumaineMath. Pures Appl., 39 (1994) 691–702.[15] G. P´olya and G. Szeg¨o, Problems and theorems in analysis. I, Springer,New York, 1998. 916] Ch. Pommerenke, Estimates for normal meromorphic functions, Ann.Acad. Sci. Fenn., Ser. A I, 476 (1970) 10pp.[17] H. Selberg, ¨Uber einige Eigenschaften bei der Werteverteilung der mero-morphen Funktionen endlicher Ordnung, Avh. Det Norske Videnskaps-Akademi i Oslo I. Matem.-Naturvid. Kl., 7 (1928) 17pp.[18] G. Valiron, Sur les fonctions enti`eres et leurs fonctions inverses, ComptesRendus, 173 (1921) 1059–1061.[19] G. Valiron, Sur un th´eor`eme de M. Fatou, Bull. Sci. Math., 46 (1922)200–208.[20] G. Valiron, Lectures on the general theory of entire functions, Chelsea,New York, 1949.[21] L. Zalcman, A heuristic principle in complex function theory, Amer.Math. Monthly, 82 (1975) 813–817.[22] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc.(N. S.), 35 (1998) 215–230.