About Borel type relation for some positive functional series

Abstract

Let $f$ be an entire transcendental function, $(\lambda_n)$ be a non-decreasing to $+\infty$ sequence, $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, and $\Gamma_f(r)/r=(\ln M_f(r))'_+$ be a right derivative, $r>0$. For a regularly convergent in ${\mathbb C}$ series of the form $F(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ is proved, in particular, the following statement (Corollary 1): If condition 
$$\sum\limits_{n=1}^{\infty}\dfrac{1}{n\Gamma_f(\lambda_n)}<+\infty$$
holds, then the relation $\ln M_F(r)=(1+o(1))\ln\mu_F(r)$ holds as $r\to +\infty$ outside a set of finite logarithmic measure, where $\mu_F(r)=\max\{|a_n|M_f(r\lambda_n)\colon\! n\geq 0\}, M_F(r)=\max\{|F(z)|\colon\! |z|=r\}$.