Extremal problems in theory of central Wiman-Valiron index

Abstract

We consider some properties of central index in Wiman-Valiron index. We introduce the notion of a determining sequence of a central index $\nu(r)$ corresponding to a fixed transcendental function $f$ and the notion of a determining sequence for an arbitrary fixed central index $\nu(r)$. Let $\rho_1,\rho_2,\dots,\rho_s,\dots$ be the points of the jumps of the function $\nu(r)$ taken counting their multiplicities. This means that if at a point $\rho_s$ the jump is equal to $m_s$, then the quantity $\rho_s$ appears $m_s$ times in this sequence. Such sequence is called determining sequence of the function $\nu(r)$. We introduce the notion of the regularization of the function $\nu(r)$, which is employed for proving main statements. We study two extremal problems in the class of functions with a prescribed central index. We obtain the expression for the maximum of the modulus of the extremal function in terms of its central index. The main obtained results are as follows. Let $T_\nu$ be the set of all transcendental functions $f$ with a prescribed central index $\nu(r)$, $M(r,f)=\max\{|f(re^{i\theta})|:\, 0\leqslant\theta\leqslant2\pi\}$, and let $M(r,\nu)=\sup\{M(r,f):f\in T_\nu\}$. Then for each $r>0$, in the class of the functions $T_{\nu}$, the quantity $M(r,\nu)$ is attained at the same function for all $r>0$. We describe the form of such extremal function. We also prove that for each fixed $r_0>0$ and for each prescribed central index $\nu(r)$, in the class $T_\nu$ there exists a function $f_0(z)$ such that $M(r_0,f_0)=\inf\{M(r_0,f):f\in T_\nu\}$.