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ON THE BRÜCK CONJECTURE

Part of: Differential equations in the complex domain Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  02 October 2015

TINGBIN CAO
TINGBIN CAO*
Affiliation:
Department of Mathematics, Nanchang University, Jiangxi 330031, PR China email [email protected]
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Abstract

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The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

Keywords

entire functionBrück conjectureNevanlinna theorycomplex differential equation

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 34M10: Oscillation, growth of solutions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 248 - 259
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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