About Borel type relation for some positive functional series - A.Yu. Bodnarchuk; O.B. Skaskiv; O.M. Trusevych (Математичні Студії, 2025)
Let $f$ be an entire transcendental function, $(λ_n)$ be a non-decreasing to $+∞$ sequence, $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, and $Γ_f(r)/r=(\ln M_f(r))'_+$ be a right derivative, $r>0$. For a regularly convergent in ${\mathbb C}$ series of the form $F(z)=\sum_{n=1}^{∞}a_nf(λ_n z)$ is proved, in particular, the following statement (Corollary 1): If condition $$ \sum\limits_{n=1}^{∞}\dfrac{1}{nΓ_f(λ_n)}<+∞ $$ holds, then the relation $\ln M_F(r)=(1+o(1))\lnμ_F(r)$ holds as $r→ +∞$ outside a set of finite logarithmic measure, where $μ_F(r)=\max\{|a_n|M_f(rλ_n)\colon\! n≥ 0\}, M_F(r)=\max\{|F(z)|\colon\! |z|=r\}$.