A note on Hayman's conjecture

Some further q-shift difference results on Hayman conjecture

In this paper, we investigate the zero distributions of $q$-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential polynomials to q-shift difference. We also investigate the uniqueness problem of $q$-shift difference-differential polynomials sharing a polynomial value with finite weight.

On Hayman Conjecture for Paired Complex Delay-Differential Polynomials

We study Hayman conjecture for different paired complex polynomials under certain conditions. In 2021, the zeros distribution of $f^{n}(z)L(g)-a(z)$ and $g^{n}(z)L(f)-a(z)$ was studied by Gao and Liu for $n\geq 3$. In this paper, we work on the zeros distribution of $f^{2}(z)L(g)-a(z)$ and $g^{2}(z)L(f)-a(z)$, where $a(z)$ is a non-zero small function of both $f(z)$ and $g(z)$, and $L(h)$ takes the $k$th derivative $h^{(k)}(z)$ or shift $h(z+c)$ or difference $h(z+c)-h(z)$ or delay-difference $h^{(k)}(z+c)$, here $k\geq 1$ and $c$ is a non-zero constant. Moreover, we discuss Hayman conjecture …

非线性Loewy可分解代数常微分方程与Hayman猜想

Nonlinear Loewy Factorizable Algebraic ODEs and Hayman's Conjecture

本文首次引入了一类$n$ -阶非线性Loewy可分解代数常微分方程，并利用Nevanlinna特征函数研究了其亚纯解的增长性.证明了对于一般情形，它们的亚纯解都是椭圆函数或它们的退化，因此它们的增长阶至多为2。此外，对于二阶可分解代数常微分方程，还得到了除一种情形外的所有亚纯解。这使我们能够表明，海曼在1996年提出的猜想持有这些二阶常微分方程。

A note on Hayman's problem

In this note, it is shown that the differential polynomial of the form $Q(f)^{(k)}-p$ has infinitely many zeros, and particularly $Q(f)^{(k)}$ has infinitely many fixed points for any positive integer $k$, where $f$ is a transcendental meromorphic function, $p$ is a nonzero polynomial and $Q$ is a polynomial with coefficients in the field of small functions of $f$. The results are traced back to Problem 1.19 and Problem 1.20 in the book of research problems by Hayman and Lingham. As a consequence, we give an affirmative answer to an extended problem on the zero distribution of $(f^n)'-p$, prop…

关于Hayman-Wu定理的一个注记

A note on the Hayman-Wu theorem

Hayman-Wu定理指出，直线或圆L在从单位圆盘到单连通区域U的共形映射下的原像具有以绝对常数为界的总欧几里得长度。最好的可能常数已知位于区间[pi ^2，4 pi）中，这要归功于埃内马和罗德的工作。早些时候，布朗·弗林证明了在U包含L的特殊情况下，总长度至多为π ^2。设r是逐点固定L的反莫比乌斯映射。在这篇注记中，我们将尖界pi ^2推广到U与r（U）的交的每个连通分支都被U的一条弧及其在r下的像所包围的情况。我们还加强了边界略有取代欧几里德长度与严格更大的球面长度限制在单位盘。

一类非线性二阶代数常微分方程的Hayman猜想

Hayman's classical conjecture on some nonlinear second order algebraic ODEs

本文研究了三类二阶非线性代数常微分方程亚纯解的Nevanlinna特征函数的增长性。我们明确给出了它们的所有亚纯解，从而证明了所有这些常微分方程都满足Hayman在1996年提出的{ \it经典猜想}.