Some Geometric Results for a New subfamily of Regular Functions in the Generalized Janowski Domain

Abstract

Let $\mathcal{SK}\left[ P,Q;\alpha ,\beta \right] $ denote the subfamily of normalized regular functions $g(\xi )=\xi +d_{2}\xi ^{2}+d_{3}\xi ^{3}+d_{4}\xi ^{4}+...$ in the open unit disk $ \Delta $ holding the next subordination condition:
\begin{equation*}
\frac{\alpha g\left( \xi \right) +\beta \xi g^{\prime }\left( \xi \right) }{%
\alpha \xi +\beta g\left( \xi \right) }\prec \frac{1+\left[ Q+\left(
1-\gamma \right) \left( P-Q\right) \right] \xi }{1+M\xi },
\end{equation*}
where $0\leq \alpha ,\beta \leq 1$, $-1\leq Q<P\leq 1$, $0\leq \gamma <1$
and $\xi \in \Delta $. In this article, we study some geometric properties such as convolution results, coefficient estimates, the upper bounds for the initial coefficients of the first four coefficients and the Fekete-Szeg\"{o} inequalities for this subfamily.