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 $f\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots$ In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies

$$|arg\left[{\left(\frac{z}{f\left(z\right)}\right)}^{1+\alpha }{f}^{\prime }\left(z\right)\right]\parallel <\gamma \frac{1}{2}\pi \left(z\in \mathbb{𝔻}\right)$$

for $0<\alpha <1$ and $0<\gamma \le 1$.