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# Fekete-Szegö problem for a new subclass of analytic functions satisfying subordinate condition associated with Chebyshev polynomials

## 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

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