A Big Picard Type Theorem Concerning Derivative and Its Application

Abstract

In this paper, the authors prove a big Picard type theorem concerning derivative: Let f(z) be meromorphic in D = {z: 0 < ∣z − z₀∣ < δ} for each δ > 0, if z₀ is an essential singularity of f(z), then either f(z) assumes every finite value infinitely often or f′(z) assumes every finite value except possibly zero infinitely often. As an application of this result, they extend Nevo, Pang and Zalcman’s quasinormal criterion: Let {f_n(z)} be a sequence of meromorphic functions on the plane domain D, all of whose zeros are multiple such that f′_n(z) − 1 has zeros with multiplicity at least n for all n on D, then {f_n(z)} is quasinormal of order 1 on D. Then they obtain a corresponding result in value distribution theory: Let f(z) be a meromorphic function on ℂ, all but finitely many of whose zeros are multiple such that

\underset{r\to +\mathrm{\infty }}{\overline{lim}}\frac{T(r,f)}{{(\log r)}^2}=+\mathrm{\infty },

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathop {\overline {\lim}}\limits_{r \to + \infty} {{T({r,f})} \over {{{({\log r})}^2}}} = + \infty,$$\end{document}

then there exist a positive integer M and R₀ > 0 such that for each r > R₀, there exists z₀ ∈ ℂ satisfying ∣z₀∣ > r such that z₀ is a zero of f′(z) − 1 with multiplicity at most M.

Keywords

Essential singularity / Quasinormal families / Value distribution theory / 30D30 / 30D35 / 30D45

References

[1] Hayman W K. Picard values of meromorphic functions and their derivatives. Ann. Math., 1959, 70(1): 9-42

[2] Hayman W K. Meromorphic Functions, 1964, Oxford, Oxford Univ. Press

[3] Lehto O. The spherical derivative of a meromorphic function in the neighborhood of an isolated singularity. Comment. Math. Helv., 1959, 33: 169-205

[4] Lehto O, Virtanen K I. On the behaviour of meromorphic functions in the neighbourhood of an isolated singularity. Ann. Acad. Sci. Fenn., 1957, 240: 1-9

[5] Minda D. A heuristic principle for a nonessential isolated singularity. Proc. Am. Math. Soc., 1985, 93(3): 443-447

[6] Nevo S, Pang X C, Zalcman L. Quasinormality and meromorphic functions with multiple zeros. J. Anal. Math., 2007, 101: 1-23

[7] Pang X C, Nevo S, Zalcman L. Derivatives of meromorphic functions with multiple zeros and rational functions. Comput. Methods Funct. Theory, 2008, 8(2): 483-491

[8] Pang X C, Zalcman L. Normal families and shared values. Bull. Lond. Math. Soc., 2000, 32(3): 325-331

[9] Picard E. Sur les fonctions analytiques uniformes dans le voisinage d–un point singular essentiel. C. R. Acad. Sci., 1879, 89: 745-747

[10] Wang Y F, Fang M L. Picard values and normal families of meromorphic functions with multiple zeros. Acta Math. Sinica (N. S.), 1998, 14(1): 17-26

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