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

Authors: MUHAMMET KAMALİ, MURAT ÇAĞLAR, ERHAN DENİZ, MIRZAOLIM TURABAEV

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

Full Text: PDF