New subclass of analytic functions defined by subordination

By using the subordination relation "\prec", we introduce an interesting subclass of analytic functions as follows: \begin{equation*} \mathcal{S}^*_{\alpha}:=\left\{f\in \mathcal{A}:\frac{zf'(z)}{f(z)}\prec \frac{۱}{(۱-z)^\alpha},\ \ |z|<۱\right\}, \end{equation*} where ۰<\alpha\leq۱ and \mathcal{A} denotes the class of analytic and normalized functions in the unit disk |z|<۱. In the present paper, by the class \mathcal{S}^*_{\alpha} and by the Nunokawa lemma we generalize a famous result connected to starlike functions of order ۱/۲. Also, coefficients inequality and logarithmic coefficients inequality for functions of the class \mathcal{S}^*_{\alpha} are obtained.

Keywords:

univalent , subordination , Starlike functions , coefficients estimates , logarithmic coefficients , Nunokawa's lemma