On Certain Generalized Bazilevic type Functions Associated with Conic Regions

Abstract

Let $f$ and $g$ be analytic in the open unit disc and, for $\alpha ,$ $\beta \geq 0$, let\begin{align*}J\left( \alpha ,\beta ,f,g\right) & =\frac{zf^{\prime }(z)}{f^{1-\alpha}(z)g^{\alpha }(z)}+\beta \left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime}(z)}\right) -\beta \left( 1-\alpha \right) \frac{zf^{\prime }(z)}{f(z)} \\& \quad -\alpha \beta \frac{zg^{\prime }(z)}{g(z)}\text{.}\end{align*}The main aim of this paper is to study the class of analytic functions which map $J\left( \alpha ,\beta ,f,g\right) $ onto conic regions. Several interesting problems such as arc length, inclusion relationship, rate of growth of coefficient and Growth rate of Hankel determinant will be discussed.

Keywords

• conic regions
• bazilevic function
• bounded boundary rotation
• hankel determinant
• univalent functions