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Fekete-Szegö inequality for a certain class of analytic functions. (English) Zbl 1174.30009
Let $\Cal{A}$ denote the class of analytic functions $f$ in the unit disc $\Delta$ of the form $$ f(z) = z + \sum_{k=2}^\infty a_kz^k\,, $$ and let $\Cal{S}$ denote the subclass of $\Cal{A}$ consisting of univalent functions. Let $\phi$ be an analytic function in $\Delta$ with positive real part, $\phi(0)=1$, $\phi'(0)>0$ which maps $\Delta$ onto a region starlike with respect to $1$ and symmetric with respect to the real axis. Then let $S^*(\phi)$ be the class of functions $f \in \Cal{S}$ with $\frac{zf'(z)}{f(z)} \prec \phi(z)$, and let $C(\phi)$ be the class of functions $f \in \Cal{S}$ with $1+\frac{zf''(z)}{f'(z)} \prec \phi(z)$, where $\prec$ denotes subordination between analytic functions. These classes were introduced and studied by {\it W. Ma} and {\it D. Minda} [Li, Zhong (ed.) et al., Proceedings of the conference on complex analysis, held June 19--23, 1992 at the Nankai Institute of Mathematics, Tianjin, China. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Anal. 1, 157--169 (1994; Zbl 0823.30007)]. In this paper the authors consider the more general class $M_\alpha(\phi)$ which is defined as follows. For $\alpha \geq 0$ let $M_\alpha(\phi)$ be the class of functions $f \in \Cal{A}$ with $\frac{zf'(z)}{f(z)} + \alpha z^2\,\frac{f''(z)}{f(z)} \prec \phi(z)$. For $f \in M_\alpha(\phi)$ the authors prove a sharp coefficient estimate for $\vert a_3-\mu a_2^2\vert $ in terms of $\alpha$, $\mu$ and the Taylor coefficients $B_1$ and $B_2$ of $\phi$. As an application of the main result, they prove a respective estimate for a class of analytic functions which are defined by convolution (Hadamard product), and as a special case they obtain such an estimate for a class of functions defined by fractional derivatives.

30C45Special classes of univalent and multivalent functions