$T$-neighborhoods in various classes of analytic functions
Bull. Korean Math. Soc. 2014 Vol. 51, No. 3, 659-666
https://doi.org/10.4134/BKMS.2014.51.3.659
Published online May 31, 2014
Saeid Shams, Ali Ebadian, Mahta Sayadiazar, and Janusz Sok\'{o}\l
University of Urmia, Payame Noor University, University of Urmia, Rzesz\'{o}w University of Technology
Abstract : Let $\mathcal{A}$ be the class of analytic functions $f$ in the open unit disk $\mathbb U=\left\{z:|z|<1\right\}$ with the normalization conditions $f(0)=f'(0)-1=0$. If $f(z)=z+\sum^{\infty}_{n=2} a_{n}z^{n}$ and ${\delta}>0$ are given, then the $T_{\delta}$-neighborhood of the function $f$ is defined as $$TN_{\delta}(f)=\left\{g(z)=z+\sum^{\infty}_{n=2} b_{n}z^{n} \in \mathcal{A}:\sum^{\infty}_{n=2} T_{n}|a_{n}-b_{n}|\leq \delta\right\},$$ 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 $T=\left\{2^{-n}/n^{2}\right\}^{\infty}_{n=2}$. We also find bounds for $\delta_T^{\ast}(A,B)$ defined by $$ \delta_T^*(A,B)=\inf \left\{ \delta>0 : B\subset TN_\delta (f) \ \ {\rm for\ all}\ \ f\in A\right\}, $$ where $A$, $B$ are given subsets of $\mathcal{A}$.
Keywords : analytic functions, univalent, starlike, convex, close-to-convex, concave functions, neighborhood, $T_{\delta}$-neighborhood, $T$-factor
MSC numbers : Primary 30C45; Secondary 30C50, 40A05
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