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The Fekete-Szegö problem for a generalized subclass of analytic functions. (English) Zbl 1200.30010
Summary: The authors obtain the Fekete-Szegö inequality for a certain normalized analytic function $$f(z)$$ defined on the open unit disk, for which
$\frac{(1-\alpha)z\big(D^m_{\lambda,\mu}f(z)\big)'+\alpha z\big(D^{m+1}_{\lambda,\mu}f(z)\big)'}{(1-\alpha)D^m_{\lambda,\mu}f(z)+\alpha D^{m+1}_{\lambda,\mu}f(z)},\qquad \lambda\geq \mu\geq 0, m\in \mathbb N_0, \alpha\geq 0,$ lies in a region starlike with respect to $$1$$, and which is symmetric with respect to the real axis. Also certain applications of the main result to a class of functions defined by the Hadamard product (or convolution ) are given. As a special case of this result, the Fekete-Szegö inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to generalize the Fekete-Szegö inequalities obtained by H. M. Srivastava et al. [Complex Variables, Theory Appl. 44, No. 2, 145–163 (2001; Zbl 1021.30014)] , H. Orhan and E. Gunes [Gen. Math. 14, No. 1, 41–54 (2006; Zbl 1164.30345)] and T. N. Shanmugam and S. Sivasubramanian [JIPAM, J. Inequal. Pure Appl. Math. 6, No. 3, Paper No. 71, 6 p., electronic only (2005; Zbl 1080.30019)], by making use of the generalized differential operator $$D^m_{\lambda,\mu}$$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
Fekete-Szegö problem; Hadamard product; starlike
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