** 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

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