Document Type : Research Paper


Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan.


Let $f$ and $g$ be analytic in the open unit disc and, for $\alpha ,$ $\beta \geq 0$, let
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{.}
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.


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