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Characterization properties for starlikeness and convexity of some subclasses of analytic functions involving a class of fractional derivative operators. (English) Zbl 0952.30011
The paper is devoted to the properties of starlike and convex functions of order $$\rho$$ defined on the unit disk $$U=\{z\in{\mathbb{C}}:\left|z\right|<1\}$$ [see P. L. Duren, Univalent functions (1983; Zbl 0514.30001)]. Some characterization properties are obtained and applied to fractional derivative operators on the mentioned function classes: $J_{0,z}^{\lambda,\mu,\eta}f\left(z\right)= {d^m\over dz^m}\left\{{z^{\lambda-\mu}\over\Gamma\left(m-\lambda\right)} \int_0^z\left(z-t\right)^{m-\lambda-1}_2F_1\left(\mu-\lambda,m-\eta; m-\lambda;1-{t\over z}\right)f\left(t\right) dt\right\},$ where $$m-1\leq\lambda<m$$, $$m\in{\mathbb{N}}$$, $$\mu,\eta\in{\mathbb{R}}$$, $${}_2F_1$$ is the Gaussian hypergeometric function. The operator $$J_{0,z}^{\lambda,\mu,\eta}$$ is a generalization of the Riemann-Liouville operator and is closely connected to the Erdélyi-Kober operator of the fractional calculus.
The characterization properties are formulated by using some conditions (inequalities) on Taylor’s coefficients of the function $$f$$. Some exactness of these conditions is illustrated by examples of functions for which the corresponding inequalities are attained.
Let $$f_i$$ ($$i=1,2$$) be given by $f_i\left(z\right)=z+\sum_{n=2}^\infty a_{n,i}z^n.$ The Hadamard product of these functions is defined by $\left(f_1*f_2\right)\left(z\right)=z+\sum_{n=2}^\infty a_{n,1}a_{n,2}z^n.$ The authors obtain several characterization properties of the Hadamard product on the classes of starlike and convex functions. The starlikeness and convexity properties of the following fractional derivative operator $P_{0,z}^{\lambda,\mu,\eta}f\left(z\right)= {\Gamma\left(2-\mu\right)\Gamma\left(2-\lambda+\eta\right)\over\Gamma\left(2 -\mu+\eta\right)} z^\mu J_{0,z}^{\lambda,\mu,\eta}f\left(z\right),$ with $$\lambda\geq 0$$, $$\mu<2$$, $$\eta>\max\left\{\lambda,\mu\right\}-2$$ are investigated too. These results can be applied to obtaining corresponding properties for Riemann-Liouville and Erdélyi-Kober operators.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 26A33 Fractional derivatives and integrals
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