The Fifth and Sixth Coefficients for Bazilevič Functions $\mathcal B_{1}(\alpha )$

Abstract

Let f be analytic in ${\mathbb D}=\{z:|z|<1\}$, with $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ and belong to the set of Bazilevič functions of logarithmic growth ${\mathcal B}_{1}(\alpha )$ defined for $\alpha \ge 0$ by $\mathfrak {Re} \big \{ f'(z)\big (\frac{f(z)}{z}\big )^{\alpha -1}\big \}>0$. Sharp bounds for $|a_{5}|$, $|a_{6}|$ and the fifth coefficient of the inverse function are given when $0\le \alpha \le 1$.