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Fekete-Szegö inequality for a certain class of analytic functions. (English) Zbl 1174.30009

Let $𝒜$ denote the class of analytic functions $f$ in the unit disc ${\Delta }$ of the form

$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_{k}{z}^{k}\phantom{\rule{0.166667em}{0ex}},$

and let $𝒮$ denote the subclass of $𝒜$ consisting of univalent functions. Let $\phi$ be an analytic function in ${\Delta }$ with positive real part, $\phi \left(0\right)=1$, ${\phi }^{\text{'}}\left(0\right)>0$ which maps ${\Delta }$ onto a region starlike with respect to 1 and symmetric with respect to the real axis. Then let ${S}^{*}\left(\phi \right)$ be the class of functions $f\in 𝒮$ with $\frac{z{f}^{\text{'}}\left(z\right)}{f\left(z\right)}\prec \phi \left(z\right)$, and let $C\left(\phi \right)$ be the class of functions $f\in 𝒮$ with $1+\frac{z{f}^{\text{'}\text{'}}\left(z\right)}{{f}^{\text{'}}\left(z\right)}\prec \phi \left(z\right)$, where $\prec$ denotes subordination between analytic functions. These classes were introduced and studied by W. Ma and 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 ${M}_{\alpha }\left(\phi \right)$ which is defined as follows. For $\alpha \ge 0$ let ${M}_{\alpha }\left(\phi \right)$ be the class of functions $f\in 𝒜$ with $\frac{z{f}^{\text{'}}\left(z\right)}{f\left(z\right)}+\alpha {z}^{2}\phantom{\rule{0.166667em}{0ex}}\frac{{f}^{\text{'}\text{'}}\left(z\right)}{f\left(z\right)}\prec \phi \left(z\right)$. For $f\in {M}_{\alpha }\left(\phi \right)$ the authors prove a sharp coefficient estimate for $|{a}_{3}-\mu {a}_{2}^{2}|$ 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.

##### MSC:
 30C45 Special classes of univalent and multivalent functions