Meromorphic functions that share four values

1983/01/01 by Gary G. Gundersen · 5 citations
Mathematics · Computer Science · #Meromorphic and Entire Functions #Polynomial and algebraic computation #Advanced Differential Equations and Dynamical Systems

paper · doi:10.1090/s0002-9947-1983-0694375-0

Abstract

An old theorem of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>R</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\text R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Möbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.

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• Questions on meromorphic functions and complex differential equations
• Remark on meromorphic functions that share five pairs
• Holomorphic curves with shift-invariant hyperplane preimages
• Reminiscence Of An Open Problem: Remarks On Nevanlinna's\n Four-Value-Theorem
• The difference analogues and delay-differential analogues of the Brück conjecture and their application

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