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On the Fekete-Szegő inequality for a class of analytic functions defined by using generalized differential operator. (English) Zbl 1274.30027

Summary: In this present investigation, the Fekete-Szegő inequality for certain normalized analytic functions \(f\) defined on the open unit disk for which \[ \frac {z(D^k_{\alpha,\beta,\lambda,\delta} f(z))'}{D^k_{\alpha,\beta,\lambda,\delta} f(z)},\quad k\in\mathbb N_0,\alpha,\beta,\lambda,\delta\geq 0, \] lies in a region starlike with respect to 1 and is symmetric with respect to the real axis is obtained. In addition, certain applications of the main result for a class of functions defined by convolution are given. As a special case of this result, the Fekete-Szegő inequality for a class of functions defined by fractional derivatives is obtained. The motivation for this paper is due to the work given by H. M. Srivastava and A. K. Mishra in [Comput. Math. Appl. 39, No. 3–4, 57–69 (2000; Zbl 0948.30018)].

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable

Citations:

Zbl 0948.30018
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