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Turkish Journal of Mathematics

DOI

10.3906/mat-2101-20

Abstract

In this paper,we define a class of analytic functions $F_{\left( \beta ,\lambda \right) }\left( H,\alpha ,\delta ,\mu \right) ,$ satisfying the following subordinate condition associated with Chebyshev polynomials \begin{equation*} \left\{ \alpha \left[ \frac{zG^{^{\prime }}\left( z\right) }{G\left( z\right) }\right] ^{\delta }+\left( 1-\alpha \right) \left[ \frac{% zG^{^{\prime }}\left( z\right) }{G\left( z\right) }\right] ^{\mu }\left[ 1+% \frac{zG^{^{\prime \prime }}\left( z\right) }{G^{^{\prime }}\left( z\right) }% \right] ^{1-\mu }\right\} \prec H\left( z,t\right) , \end{equation*}% where $G\left( z\right) =\lambda \beta z^{2}f^{^{\prime \prime }}\left( z\right) +\left( \lambda -\beta \right) zf^{^{\prime }}\left( z\right) +\left( 1-\lambda +\beta \right) f\left( z\right) ,$ $0\leq \alpha \leq 1,$ $% 1\leq \delta \leq 2,$ $0\leq \mu \leq 1,$ $0\leq \beta \leq \lambda \leq 1$ and $t\in \left( \frac{1}{2},1\right] $. We obtain initial coefficients $% \left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ for this subclass by means of Chebyshev polynomials expansions of analytic functions in $\mathcal{D}.$ Furthermore, we solve Fekete-Szegö problem for functions in this subclass.We also provide relevant connections of our results with those considered in earlier investigations. The results presented in this paper improve the earlier investigations.

Keywords

Analytic and univalent functions, subordination, Chebyshev polynomials, coefficient estimates, Fekete-Szegö inequality

First Page

1195

Last Page

1208

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