On Certain Generalized Bazilevic type Functions Associated with Conic Regions

Abstract

Let $f$ and $g$ be analytic in the open unit disc and, for $alpha ,$ $beta geq 0$, letbegin{align*}Jleft( alpha ,beta ,f,gright) & =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 $Jleft( alpha ,beta ,f,gright) $ 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.