ABOUT TWO CLASSES OF ANALYTIC FUNCTIONS

Abstract

Let \(\mathcal{A}\) be the class of analytic functions in the unit disk \(\mathcal{U} = \{z : |z| < 1\}\) normalized such that \(f(z) = z + a_2z^2 + a_3z^3 + \cdots\). Classes

\[
M'_{\alpha,\rho} = \left\{f \in \mathcal{A} : \left|f'(z) - \alpha f(z)/z + \alpha - 1\right| < \rho, z \in \mathcal{U}\right\}
\]

and

\[
M''_{\alpha,\rho} = \left\{f \in \mathcal{A} : \left|zf''(z) - \alpha f'(z) + \alpha\right| < \rho, z \in \mathcal{U}\right\}
\]

were studied earlier by Fournier–Mocanu and Ponnusamy–Singh. In this paper, sharp upper bound of the Fekete–Szegö functional over these two classes is obtained. Also, sufficient conditions that embed this classes in the class of starlike functions of order \(\alpha\) and in the class of convex functions order \(\alpha\) are given.