On the type of the meromorphic function of finite order

Abstract

Let $f(z)$ be a meromorphic function on the complex plane of finite order $\rho>0$. Let $\rho(r)$ be a proximate order in the sense of Boutroux such that $\limsup\limits_{r\to\infty}\rho(r)=\rho$, $\liminf\limits_{r\to\infty}\rho(r)=\alpha>0$. If $[\alpha]<\alpha\leqslant\rho<[\alpha]+1$ then the types of $T(r,f)$ and $|N|(r,f)$ coincide with respect to $\rho(r)$. If there are integers between $\alpha$ and $\rho$, then the resulting criterion is formulated in terms of the upper density of zeros and poles of the function $f$ and their argument symmetry.