On certain classes of series in systems of functions

Анотація

Let $f(z)=z^p-\sum_{k=1}^{\infty}f_k z^{k+p}$ with $f_k>0$ be an entire transcendental function and $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty$. Suppose that the series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ with $a_n>0$ regularly converges in ${\mathbb D}=\{z:\,|z|<1\}$. For $p\in {\mathbb N}$, $\alpha\ge 0$ and $0\le\beta<p$ by $\mathfrak{F}_p(\alpha,\beta)$ denote the class of an analytic in ${\mathbb D}$ functions $g(z)=z^p-\sum\limits_{n=1}^{\infty}g_n z^{n+p}$ with $g_n> 0$ such that $\text{Re}\left\{(1-\alpha)\dfrac{g(z)}{z^p}+\alpha\dfrac{g'(z)}{pz^{p-1}}\right\}>\dfrac{\beta}{p}$ for all $z\in \mathbb D$, and say that $g\in \mathfrak{G}_p(\alpha,\beta)$ if $\text{Re}\left\{(p+\alpha(1-p))\dfrac{g'(z)}{pz^{p-1}}+\alpha\dfrac{g''(z)}{pz^{p-2}}\right\}>\beta$ for all $z\in \mathbb D$. 
Conditions under which the function $A$ belongs either to $\mathfrak{F}_p(\alpha,\beta)$ or to $\mathfrak{G}_p(\alpha,\beta)$ are investigated.