On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series

We consider the class $S(\lambda,\beta,\tau)$ of convergent for all  $x\ge0$ Taylor-Dirichlet type series of the form $$F(x) =\sum_{n=0}^{+\infty}{b_ne^{x\lambda_n+\tau(x)\beta_n}},\  b_n\geq 0\ (n\geq 0),$$  where  $\tau\colon [0,+\infty)\to (0,+\infty)$\ is a continuously differentiable non-decreasing function, $\lambda=(\lambda_n)$ and $\beta=(\beta_n)$ are such that $\lambda_n\geq 0, \beta_n\geq 0$ $(n\geq 0)$. In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference  ``Complex Analysis and Related Topics'' (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function  $h(x)\colon [0,+\infty)\to (0,+\infty)$, $h'(x)\nearrow +\infty$ $ (x\to +\infty)$, every sequence  $\lambda=(\lambda_n)$ such that  $\displaystyle\sum_{n=0}^{+\infty}\frac1{\lambda_{n+1}-\lambda_n}<+\infty$ and for any non-decreasing sequence  $\beta=(\beta_n)$ such that $\beta_{n+1}-\beta_n\le\lambda_{n+1}-\lambda_n$ $(n\geq 0)$  there exist a function  $\tau(x)$ such that $\tau'(x)\ge 1$ $(x\geq x_0)$, a function  $F\in S(\alpha, \beta, \tau)$, a set  $E$ and  a constant $d>0$ such that $h-\mathop{meas} E:=\int_E dh(x)=+\infty$ and $(\forall x\in E)\colon\ F(x)>(1+d)\mu(x,F),$ where $\mu(x,F)=\max\{|a_n|e^{x\lambda_n+\tau(x)\beta_n}\colon n\ge 0\}$ is the maximal term of the series.   At the same time, we also pose some open questions and formulate one conjecture.