On some properties of the maximal term of series in systems of functions - M. M. Sheremeta; Yu. M. Gal' (Математичні Студії, 2024)
For an entire transcendental function $f$ and a sequence $(λ_n)$ of positive numbers increa\-sing to $+∞$ a series $A(z)=\sum_{n=1}^{∞}a_nf(λ_n z)$ in the system $\{f(λ_nz)\}$ is said to be regularly convergent in ${\mathbb C}$ if $\mathfrak{M}(r,A)=\sum_{n=1}^{∞} |a_n|M_f(rλ_n)<+∞$ for all $r\in (0,+∞)$, where $ M_f(r)=\max\{|f(z)|\colon |z|=r\}$. Conditions are found on $(λ_n)$ and $f$, under which $\ln\mathfrak{M}(r,A)\sim \ln μ(r,A)$ as $r→+∞$, where $μ(r,A)= \max\{|a_n|M_f(rλ_n)\colon n\ge 1\}$ is the maximal term of the series. A~formula for finding the lower generalized order $$λ_{α,β}[A]=\varliminf\limits_{r→+∞}\dfrac{α(\ln \mathfrak{M}(r,A))}{β(r)}$$ is obtained, where the functions $α$ and $β$ are positive, continuous and increasing to $+∞$.