JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

doi:10.1007/s10473-021-0415-7

Abstract

Abstract: For entire or meromorphic function $f$, a value $\theta\in[0,2\pi)$ is called a Julia limiting direction if there is an unbounded sequence $\{z_n\}$ in the Julia set satisfying $\lim\limits_{n\rightarrow\infty}\arg z_n=\theta.$ Our main result is on the entire solution $f$ of $P(z,f)+F(z)f^s=0$, where $P(z,f)$ is a differential polynomial of $f$ with entire coefficients of growth smaller than that of the entire transcendental $F$, with the integer $s$ being no more than the minimum degree of all differential monomials in $P(z,f)$. We observe that Julia limiting directions of $f$ partly come from the directions in which $F$ grows quickly.

Key words: Julia set, meromorphic function, Julia limiting direction, complex differential equations

CLC Number:

34M05

Jun WANG, Xiao YAO, Chengchun ZHANG. JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS[J].Acta mathematica scientia,Series B, 2021, 41(4): 1275-1286.

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