The Initial Coefficients for Bazilevic Functions Defined by q-Fractional Derivative Saadatul Fitri, Marjono, and Ratno Bagus Edy Wibowo
Department of Mathematics, Universitas Brawijaya, Malang, Indonesia
Abstract
Let -S- be the class of analytic functions -f- in -\mathbb{D}=\{z: |z|< 1\}- with -f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}-. We investigate the subclass of Bazilevi{\v c} functions defined by, --\Omega^q f(z)=\Gamma(2-q) z^q D_z ^q f(z),-- where -\Omega ^q- be operator on -S- and -D_z ^q f- is the -q--fractional derivative of -f-. For -\alpha\ge 0- and -0\le q<1-, let -\mathcal{B}_1^q (\alpha,\lambda)- denote the class of Bazilevi{\v c} functions satisfying
--\left| \dfrac {z^{1-\alpha}(\Omega^q f(z))^}{(\Omega^q f(z))^{1-\alpha}}-1\right| <\lambda. --
The class -\mathcal{B}_1^q (\alpha,\lambda)- generalizes the class -\mathcal{B}_1 (\alpha,\lambda)- which introduced by Ponnusamy and Singh in 1996\cite{singh}. Sharp estimates for the first few coefficients of function in -\mathcal{B}_1^q (\alpha,\lambda)- are given.
