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T-NEIGHBORHOODS IN VARIOUS CLASSES OF ANALYTIC FUNCTIONS

  • Shams, Saeid (Department of Mathematics University of Urmia) ;
  • Ebadian, Ali (Department of Mathematics Payame Noor University) ;
  • Sayadiazar, Mahta (Department of Mathematics University of Urmia) ;
  • Sokol, Janusz (Department of Mathematics Rzeszow University of Technology)
  • Received : 2013.01.18
  • Published : 2014.05.31

Abstract

Let $\mathcal{A}$ be the class of analytic functions f in the open unit disk $\mathbb{U}$={z : ${\mid}z{\mid}$ < 1} with the normalization conditions $f(0)=f^{\prime}(0)-1=0$. If $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and ${\delta}$ > 0 are given, then the $T_{\delta}$-neighborhood of the function f is defined as $$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$, where $T=\{T_n\}_{n=2}^{\infty}$ is a sequence of positive numbers. In the present paper we investigate some problems concerning $T_{\delta}$-neighborhoods of function in various classes of analytic functions with $T=\{2^{-n}/n^2\}_{n=2}^{\infty}$. We also find bounds for $^{\delta}^*_T(A,B)$ defined by $$^{\delta}^*_T(A,B)=jnf\{{\delta}&gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$ 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

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