## 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

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 [3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science: Gordon and Breach Science Reading, PA · Zbl 0818.26003 [4] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002 [5] (Srivastava, H. M.; Owa, S., Univalent Functions, Fractional Calculus, and Their Applications (1989), Halsted Press/John Wiley and Sons: Halsted Press/John Wiley and Sons Chichester/New York) · Zbl 0683.00012 [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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