Moduli Difference of Initial Inverse Coefficients of Bazilevič Functions

Abstract

Let $$f$$ be analytic in the unit disk $$\mathbb{D}=\{z\in\mathbb{C}\colon|z|<1\}$$,

and $$\mathcal{B}_1(\alpha,i\beta)$$ represent the class of Bazilevič functions

$$f$$ satisfying $$\left(zf'(z)/f(z)\right)(f(z)/z)^{\alpha+i\beta}=p(z)$$,

where $$\mathrm{Re}(p(z))>0$$, $$\alpha\ge 0$$, and $$\beta\in\mathbb{R}$$. The function $$f$$ has the series expansion $$f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$$ for $$z\in\mathbb{D}$$. Let $$F$$ denote the inverse function of $$f$$, defined in some disk $$|w|\le r_0(f)$$, with series expansion $$F(w)=w+\sum_{n=2}^{\infty}A_{n}w^{n}$$. In this article, we establish sharp bounds on $$||A_3|-|A_2||$$ for the class $$\mathcal{B}_1(\alpha,i\beta)$$.