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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.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals

Citations:

Zbl 0628.30014
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References:

[1] Srivastava, H. M.; Saigo, M.; Owa, S., A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl., 131, 412-420 (1988) · Zbl 0628.30014
[2] Raina, R. K.; Saigo, M., A note on fractional calculus operators involving Fox’s \(H\)-function on space \(F_{p,μ}\), (Kalia, R. N., Recent Advances in Fractional Calculus (1993), Global Publishing: Global Publishing Sauk Rapids, MN), 219-229 · Zbl 0789.33006
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[6] (Srivastava, H. M.; Owa, S., Current Topics in Analytic Function Theory (1992), World Scientific: World Scientific Singapore) · Zbl 0976.00007
[7] Altintaş, O.; Irmak, H.; Srivastava, H. M., A subclass of analytic functions defined by using certain operators of fractional calculus, Compututers Math. Applic., 30, 1, 1-9 (1995) · Zbl 0838.30012
[8] Duren, P. L., Univalent Functions, (Grundlehren der Mathematischen Wissenschaften, Volume 259 (1983), Springer-Verlag: Springer-Verlag New York) · Zbl 0398.30010
[9] Srivastava, H. M.; Karlsson, P. W., Multiple Gaussian Hypergeometric Series (1985), Halsted Press/John Wiley and Sons: Halsted Press/John Wiley and Sons Chichester/New York · Zbl 0552.33001
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