Analog of the Wiman inequality for Taylor-Dirichlet type series

Abstract

There are presented sufficient conditions for the Taylor-Dirichlet series  of the form $$ F(x)=\sum_{n=0}^{+\infty}a_ne^{x\lambda_n+\tau(x)\beta_n},  $$ and  $\lambda=(\lambda_n)$, $\beta=(\beta_n)$ be positive sequences, $\tau\colon\mathbb{R}_+\to\mathbb{R}_+$ be a differentiable function  such that $\tau'(x)\geq 0$ $(x\geq x_0)$ providing validity of Wiman-type inequality outside some set $E$ of finite Lebesgue measure. We prove the following statement: If for the sequences $\lambda=(\lambda_n)$, $\beta=(\beta_n)$ and a twice differentiable function $\tau\colon\mathbb{R}_+\to\mathbb{R}_+$ such that $\tau'(x)\geq 0$ $(x\geq x_0)$, $\tau''(x)\geq 0$ $(x\geq x_0)$ there exists a function $\psi\in\mathcal{L}_1$ that the condition  $$ \sup\limits_{x>0}\varlimsup\limits_{u\to+\infty}\frac{\ln \nu\big\{n\geq 0\colon u-\sqrt{\psi(u)}<\lambda_n+\tau'(x)\beta_n\leq u+\sqrt{\psi(u)}\big\}}{\ln u}<+\infty $$ holds, where $\nu(D)=\#\{n\geq 0\colon \lambda_n+\tau'(x)\beta_n\in D\}$, then  there exists a set $E$ of finite Lebesgue measure such that for arbitrary $\delta>0$ and $x\in \mathbb{R}_+\setminus E$ we have $$ F(x)\leq \mu(x,F)(\ln\mu(x,F))^{\alpha_0+\delta}. $$