$\varphi$-normal criteria for meromorphic functions and its applications

Abstract

In 2011, Aulaskari and R$\overset{..}a$tty$\overset{..}a$ (J. Michigan. Math. Vol. 60) introduced the concept of $\varphi$-normal meromorphic functions on the unit disc $D,$ where
the function $\varphi (r): [0,1)\to (0,\infty)$ admits a sufficient regularity near 1 and exceeds $\frac{1}{1-r^2}$ in growth. They examined
the class of meromorphic functions $f$ on $D$ satisfying $f^{\#}(z)=O(\varphi(|z|),$ as $|z|\to 1^-.$
In this paper, using a result of Pang and Zalcman (Bull. London. Math. Soc. \textbf{32}(2000), 325-331), we prove some Zalcman's type result for $\varphi$-normal function. Using that results and Nevanlinna theory, we give some $\varphi$-normal criteria for meromorphic functions. In our best knowledge, they are first criteria for $\varphi$-normal functions sharing sets. Finally, we investigate the $\varphi$-normal function which is a solution of algebraic differential equations. Theorem \ref{th2a} shows that the solution is $\varphi$-normal without any comparision between $nk$ and the degree of $P[w](z)$ as previous results.