Remarks to properties of analytic in the unit disc functions P-valent along with their derivatives

Анотація

Let $K=\left(k_j\right)$ be an increasing sequence of  integers, $k_0=0$, and $E\left(K\right)$ be a class of analytic in the disk $\mathbb{D}=\left\{z:\left|z\right|<1\right\}$ functions $f$ such that for some $p\in \mathrm{\mathbb{N}}$ all derivatives $f^{\left(k_j\right)}$ are in the middle $p$-valent in $\mathbb{D}$. It is proved that  every function $f\in E\left(K\right)$ is entire if and only if

$\underset{j\to \infty }{\lim }\left\{\ln \left(k_j\right)-\frac{1}{k_j}\sum _{s=1}^j\left(k_s-k_{s-1}\right)\ln \left(k_s-k_{s-1}\right)\right\}=+\infty .$