New subclass of analytic functions defined by subordination

Document Type : Research Paper

Authors

Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

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{1}{(1-z)^\alpha},\ \ |z|<1\right\},
\end{equation*}
where $0<\alpha\leq1$ and $\mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. 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 $1/2$. Also, coefficients inequality and logarithmic coefficients inequality for functions of the class $\mathcal{S}^*_{\alpha}$ are obtained.

Keywords