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Applications of fractional calculus to parabolic starlike and uniformly convex functions. (English) Zbl 0948.30018
Let \({\mathcal A}\) be the class of analytic functions in the open unit disk \({\mathcal U}\). Given \(0\leq\lambda<1\), let \(\Omega^\lambda\) be the operator defined on \({\mathcal A}\) by \[ (\Omega^\lambda f)(z)=\Gamma(2-\lambda) z^\lambda D^\lambda_z f(z), \] where \(D^\lambda_zf\) is the fractional derivative of \(f\) of order \(\lambda\). A function \(f\) in \({\mathcal A}\) is said to be in the class \(SP_\lambda\) if \(\Omega^\lambda f\) is a parabolic starlike function. In this paper several basic properties and characteristics of the class \(SP_\lambda\) are investigated. These include subordination, inclusion, and growth theorems, class-preserving operators, Fekete-Szegő problems, and sharp estimates for the first few coefficients of the inverse function.
Reviewer: H.M.Srivastava

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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