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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.
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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