On Certain Generalized Bazilevic type Functions Associated with Conic Regions

Let f and g be analytic in the open unit disc and, for \alpha , \beta \geq ۰, let\begin{align*}J\left( \alpha ,\beta ,f,g\right) & =\frac{zf^{\prime }(z)}{f^{۱-\alpha}(z)g^{\alpha }(z)}+\beta \left( ۱+\frac{zf^{\prime \prime }(z)}{f^{\prime}(z)}\right) -\beta \left( ۱-\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