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A Note on the Conjectures of Hayman, Mues and Gol’dberg

A Note on the Conjectures of Hayman, Mues and Gol’dberg As consequences of Yamanoi’s (Proc Lond Math Soc 106(3):703–780, 2013) important paper, we prove the following results. (1) Let $$f$$ f be a transcendental meromorphic function on $${\mathbb{C }}$$ C . Then for any $$k\ge 1$$ k ≥ 1 , $$\begin{aligned} \underset{a\in {\mathbb{C }}}{\sum }\delta (a,f^{(k)})+\sum ^{\infty }_{j=k+1}\underset{b\in {\mathbb{C }}/\{0\}}{\sum }\delta (b, f^{(j)}) \le 1. \end{aligned}$$ ∑ a ∈ C δ ( a , f ( k ) ) + ∑ j = k + 1 ∞ ∑ b ∈ C / { 0 } δ ( b , f ( j ) ) ≤ 1 . (2) Let $$f$$ f be a transcendental meromorphic function on $$\mathbb{C }$$ C having at most finitely many simple zeros. Then $$f^{(k)}$$ f ( k ) takes on every non-zero complex value infinitely often for $$k=1,2,3,\ldots $$ k = 1 , 2 , 3 , … . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Note on the Conjectures of Hayman, Mues and Gol’dberg

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Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-013-0036-9
Publisher site
See Article on Publisher Site

Abstract

As consequences of Yamanoi’s (Proc Lond Math Soc 106(3):703–780, 2013) important paper, we prove the following results. (1) Let $$f$$ f be a transcendental meromorphic function on $${\mathbb{C }}$$ C . Then for any $$k\ge 1$$ k ≥ 1 , $$\begin{aligned} \underset{a\in {\mathbb{C }}}{\sum }\delta (a,f^{(k)})+\sum ^{\infty }_{j=k+1}\underset{b\in {\mathbb{C }}/\{0\}}{\sum }\delta (b, f^{(j)}) \le 1. \end{aligned}$$ ∑ a ∈ C δ ( a , f ( k ) ) + ∑ j = k + 1 ∞ ∑ b ∈ C / { 0 } δ ( b , f ( j ) ) ≤ 1 . (2) Let $$f$$ f be a transcendental meromorphic function on $$\mathbb{C }$$ C having at most finitely many simple zeros. Then $$f^{(k)}$$ f ( k ) takes on every non-zero complex value infinitely often for $$k=1,2,3,\ldots $$ k = 1 , 2 , 3 , … .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 26, 2013

References