On the main relation of the Wiman-Valiron theory and asymptotic $h$-density of an exceptional sets

Abstract

Let $L$ be the class of the positive increasing on $[0;+\infty)$ functions, and $L_{1}$ be the subclass of the functions $h\in L$ such that $h\Big(x+\frac{1}{h(x)}\Big)=O(h(x)),(x\to +\infty).$ For measurable by Lebesgue set $E\subset [0;+\infty),$ of finite Lebesgue measure $\mathop{\rm meas } E=\int_{E}dx < +\infty,$ we define the lower asymptotic $h$-density of $E$ on $+\infty$ $\displaystyle {d}_{h}(E)= \varliminf_{R \rightarrow +\infty} h(R)\cdot \mathop{\rm meas }(E \cap [R;+\infty)).$   Let $S(a,b),$\ $-\infty\le a<b\le +\infty,$\  be the class of an analytic in  $\Pi(a,b)=\{z: a<{\rm Re}\, z<b\}$  functions such that $(\forall x\in(a,b))\colon\ M(x,F):=\sup\{|F(t+iy)|: 0<t\le x, y\in{\mathbb R}\}<+\infty,$ and $L(x,F)=(\ln M(x,F))'_+$ is right-hand derivative. In the paper it is proved the following Theorem: Let  $\Phi, h \in L$\ be the functions such that $ h(2r)=o(\Phi(r)) (r\to\infty).$ If $F \in S_{\infty}(0,\infty)$ and $\displaystyle (\exists x_{n}\nearrow + \infty\ (n\to +\infty))\colon\quad L(x_{n},F)\geq \Phi(x_{n})\ (n\geq 1),$ the relation $F'(z)=(1+o(1))L(x,F)F(z)$ holds as $ x\to +\infty$\ $(x\notin E,\ d_{h} (E) = 0)$ for all $z$\ such that $Re z = x$ and $ |F(z)| = (1+o(1))M(x,F) $ as $ x\to +\infty.$