Radially distributed values and normal families. II
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WALTER BERGWEILER AND ALEXANDRE EREMENKO∗ Dedicated to Larry Zalcman Abstract. We consider the family of all functions holomorphic in the unit disk for which the zeros lie on one ray while the 1-points lie on two different rays. We prove that for certain configurations of the rays this family is normal outside the origin.
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Introduction and results
There is an extensive literature on entire functions whose zeros and 1-points are distributed on finitely many rays. One of the first results of this type is the following theorem of Biernacki [5, p. 533] and Milloux [11]. Theorem A. There is no transcendental entire function for which all zeros lie on one ray and all 1-points lie on a different ray. Biernacki and Milloux proved this under the additional hypothesis that the function considered has finite order, but by a later result of Edrei [6] this is always the case if all zeros and 1-points lie on finitely many rays. A thorough discussion of the cases in which an entire function can have its zeros on one system of rays and its 1-points on another system of rays, intersecting the first one only at 0, was given in [4]. Special attention was paid to the case where the zeros are on one ray while the 1-points are on two rays. For this case the following result was obtained [4, Theorem 2]. Theorem B. Let f be a transcendental entire function whose zeros lie on a ray L0 and whose 1-points lie on two rays L1 and L−1 , each of which is distinct from L0 . Suppose that the numbers of zeros and 1-points are infinite. Then ∠(L0 , L1 ) = ∠(L0 , L−1 ) < π/2. ∗
Supported by NSF grant DMS-1665115.
99 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0116-5
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W. BERGWEILER AND A. EREMENKO
The hypothesis that f has infinitely many zeros excludes the example f (z) = ez in which case we have ∠(L1 , L−1 ) = π, and L0 can be taken arbitrarily. Without this hypothesis we have the following result. Theorem B . Let f be a transcendental entire function whose zeros lie on a ray L0 and whose 1-points lie on two rays L1 and L−1 , each of which is distinct from L0 . Then ∠(L1 , L−1 ) = π or ∠(L0 , L1 ) = ∠(L0 , L−1 ) < π/2. Bloch’s heuristic principle says that the family of all functions holomorphic in some domain which have a certain property is likely to be normal if there does not exist a non-constant entire function with this property. More generally, properties which are satisfied only by “few” entire functions often lead to normality. We refer to [2], [14] and [16] for a thorough discussion of Bloch’s principle. The following normal family analogue of Theorem A was proved in [3, Theorem 1.1]. Here D denotes the unit disk. Theorem C. Let L0 and L1 be two distinct rays emanating from the origin and let F be the family of all functions holomorphic in D for which all zeros lie on L0 and all 1-points lie on L1 . Then F is normal in D\{0}. The purpose of this paper is to prove a normal-family analogue of Theorem B . Theorem 1.1. Let L0 , L1 and L−1 be three distinct rays emanating from the origin and let F be
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