A study of the Fekete-Szegö functional and coefficient estimates for subclasses of analytic functions satisfying a certain subordination condition and associated with the Gegenbauer polynomials
Authors: Srivastava H. M. ,   Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada, Kamalı Muhammet , Urdaletova Anarkül ,   Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan,   Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan,   Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy,   Department of Mathematics, Faculty of Science, Kyrgyz-Turkish Manas University, Chyngz Aitmatov Avenue, 720038 Biskek, Chuy Province, Kyrgyz Republic,   Department of Mathematics, Faculty of Science, Kyrgyz-Turkish Manas University, Chyngz Aitmatov Avenue, 720038 Biskek, Chuy Province, Kyrgyz Republic
Published: November 2021

<abstract><p>In this paper, we introduce and study a new subclass of normalized analytic functions, denoted by</p>

<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal F_{\left(\beta,\gamma\right)} \bigg(\alpha,\delta,\mu,H\big(z,C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg), $\end{document} </tex-math></disp-formula></p>

<p>satisfying the following subordination condition and associated with the Gegenbauer (or ultraspherical) polynomials $ C_{n}^{\left(\lambda\right)}(t) $ of order $ \lambda $ and degree $ n $ in $ t $:</p>

<p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha \left(\frac{zG^{'}\left(z\right)}{G\left(z\right)} \right)^{\delta}+\left(1-\alpha\right)\left(\frac{zG^{'} \left(z\right)}{G\left(z\right)}\right)^{\mu} \left(1+\frac{zG^{''}\left(z\right)}{G^{'} \left(z\right)} \right)^{1-\mu} \prec H\big(z,C_{n}^{\left(\lambda\right)} \left(t\right)\big), $\end{document} </tex-math></disp-formula></p>

<p>where</p>

<p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ H\big(z,C_{n}^{\left(\lambda\right)}\left(t\right)\big) = \sum\limits_{n = 0}^{\infty} C_n^{(\lambda)}(t)\;z^n = \left(1-2tz+z^2\right)^{-\lambda}, $\end{document} </tex-math></disp-formula></p>

<p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ G\left(z\right) = \gamma \beta z^{2} f^{''} \left(z\right)+\left(\gamma-\beta \right)zf^{'} \left(z\right)+\left(1-\gamma+\beta\right)f\left(z\right), $\end{document} </tex-math></disp-formula></p>

<p>$ 0\leqq \alpha \leqq 1, $ $ 1\leqq \delta \leqq 2, $ $ 0\leqq \mu \leqq 1, $ $ 0\leqq \beta \leqq \gamma \leqq 1 $, $ \lambda \geqq 0 $ and $ t\in \left(\frac{1}{\sqrt{2}}, 1\right] $. For functions in this function class, we first derive the estimates for the initial Taylor-Maclaurin coefficients $ \left|a_{2}\right| $ and $ \left|a_{3}\right| $ and then examine the Fekete-Szegö functional. Finally, the results obtained are applied to subclasses of normalized analytic functions satisfying the subordination condition and associated with the Legendre and Chebyshev polynomials. The basic or quantum (or $ q $-) calculus and its so-called trivially inconsequential $ (p, q) $-variations have also been considered as one of the concluding remarks.</p></abstract>