$T$-neighborhoods in various classes of analytic functions

摘要

Let $mathcal{A}$ be the class of analytic functions $f$ in the open unit disk $mathbb U=left{z:|z|0$ are given, then the $T_{delta}$-neighborhood of the function $f$ is defined as $$TN_{delta}(f)=left{g(z)=z+sum^{infty}_{n=2} b_{n}z^{n} in mathcal{A}:sum^{infty}_{n=2} T_{n}|a_{n}-b_{n}|leq deltaight},$$ where $T={T_{n}}^{infty}_{n=2}$ is a sequence of positive numbers. In the present paper we investigate some problems concerning $T_{delta}$-neighborhoods of functions in various classes of analytic functions with $T=left{2^{-n}^{2}ight}^{infty}_{n=2}$. We also find bounds for $delta_T^{st}(A,B)$ defined by $$ delta_T^*(A,B)=inf left{ delta>0 : Bsubset TN_delta (f) {m for all} fin Aight}, $$ where $A$, $B$ are given subsets of $mathcal{A}$.