On the initial coefficients for certain class of functions analytic in the unit disc

Abstract

Let function $f$ be analytic in the unit disk $\mathbb{D}$ and be normalized so that $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$. In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies \[ \left|\arg \left[\left(\frac{z}{f(z)}\right)^{1+α}f'(z) \right] \right|<γ\fracπ{2} \quad\quad (z\in {\mathbb D}),\] for $0<\alpha <1$ and $0<\gamma \le 1$.