On T-neighbourhoods of Harmonic Univalent Functions

Abstract

In this present paper, given a sequence $T=\{T_{n}\}_{n=2}^{\infty }$ consisting of positive numbers, we define the $T_{\delta }$-neighbourhood of the function $f=h+{{\overline{g}}}\in {{\mathcal {H}}}$ is defined as

$$\begin{aligned} N_{\delta }(f)= & {} \left\{ G(z)\;:\;G(z)=z+\sum _{n=2}^{\infty }\left( A_{n}z^{n}+\overline{B_{n}}\overline{z^{n}}\right) ,\right. \\&\;\left. ~\sum _{n=2}^{\infty }T_{n}(|a_{n}-A_{n}|+|b_{n}-B_{n}|)\le \delta ,\;\delta \ge 0\right\} . \end{aligned}$$

Furthermore, we investigate some problems concerning $T_{\delta }$-neighbourhoods of functions in various classes of analytic functions. The results obtained here are sharp.