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A certain subclass of analytic functions associated with operators of fractional calculus. (English) Zbl 0867.30015
Let \({\mathcal F} (n)\) denote the class of functions of the form: \(f(z)=z+ \sum_{k=n +1}^\infty a_kz^k\) \((a_k\geq 0; n\in\mathbb{N})\), which are analytic in the open unit disk. The authors consider the subclass \(S_{\lambda,\mu,\eta} (n,\sigma,\alpha)\) of functions in \({\mathcal F} (n)\) which also satisfy the following inequality: \[ \text{Re}\Bigl[\varphi_1 (\lambda,\mu,\eta) z^{\mu-1} \Bigl((1-\sigma) J_{0,z}^{\lambda, \mu,\eta} f(z)+\sigma zJ_{0,z}^{\lambda+1, \mu+1, \eta+1} f(z) \Bigr)\Bigr] >\alpha \]
\[ (0\leq\lambda <1;\;0\leq\alpha <1;\;0\leq\sigma \leq 1;\;\mu,\eta\in \mathbb{R};\;\mu<2;\;\lambda- \eta<2;\;\mu-\eta<2), \] where \(\varphi_m (\lambda,\mu,\eta) =\Gamma(1-\mu+m) \Gamma(1+\eta- \lambda+m)/ \Gamma(1+m ) \Gamma(1+\eta- \mu+m)\) and \(J_{0,z}^{\lambda, \mu,\eta}\) is a certain fractional derivative operator defined in terms of the Gauss hypergeometric function \(_2F_1\) [see H. M. Srivastava, M. Saigo and S. Owa, J. Math. Anal., Appl. 131, 412-420 (1988; Zbl 0628.30014)]. In this paper, some results connected with the new class \(S_{\lambda\mu,\eta} (n,\sigma,\alpha)\) of functions, including the characterization property, the radii of close-to-convexity, starlikeness, convexity and distortion inequalities, are obtained.

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals
Zbl 0628.30014
Full Text: DOI Link
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