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A subclass of uniformly convex functions with negative coefficients. (English) Zbl 1224.30026

Summary: Making use of the Sălăgean operator, we define the class \(T(n,\alpha,\beta)\). When \(n=1\) and \(n=0\), we obtain, respectively, a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients. In this paper, we obtain a distortion theorem and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class \(T(n,\alpha,\beta)\). We consider integral operators associated with functions belonging to \(T(n,\alpha,\beta)\). We also obtain several results for the modified Hadamard products of functions in \(T(n,\alpha,\beta)\). A distortion theorem for the fractional calculus (that is, fractional integral and fractional derivative) of functions in \(T(n,\alpha,\beta)\) is obtained.

MSC:

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