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An application of fractional calculus to a new class of multivalent functions with negative coefficients. (English) Zbl 0940.30006

From the summary: Motivated by some earlier works of M.-P. Chen and H. M. Srivastava [Comput. Math. Appl. 35, No. 5, 83-91 (1998; Zbl 0921.30012)]; H. M. Srivastava and M. K. Aouf [J. Math. Anal. Appl. 171, 1-13 (1992; Zbl 0760.30006); ibid. f92, 673-688 (1995; Zbl 0831.30008)], dealing with various applications of the operators of fractional calculus in analytic function theory, the authors introduce and study rather systematically a certain subclass of analytic and \(p\)-valent functions with negative coefficients. This subclass is defined by using a familiar fractional derivative operator. Coefficient estimates, growth and distortion theorems, and many other interesting and useful properties and characteristics of this class of analytic and \(p\)-valent functions are obtained; some of these properties involve, for example, linear combinations and modified Hadamard products (or convolution) of functions belonging to the class introduced here.

MSC:

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