On the coefficients of $\mathbf {{\mathcal {B}}_1}(\varvec{\alpha })$ Bazilevič functions

Abstract

Denote by ${\mathcal {A}}$, the class of functions f, analytic in ${\mathbb {D}} =\{z:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ for $z\in {\mathbb {D}}$, and by ${\mathcal {S}}$ the subset of ${\mathcal {A}}$ whose elements are univalent in ${\mathbb {D}}$. The class ${\mathcal {B}}_{1}(\alpha )\subset {\mathcal {S}}$, of Bazilevič functions is defined by $Re\dfrac{zf^{\prime }(z)}{f(z)}\left( \dfrac{f(z)}{z}\right) ^{\alpha }>0$, for $\alpha \ge 0$ and $z\in {\mathbb {D}}$. We give sharp bounds for $|\gamma _{n}|$, where $\log \dfrac{f(z)}{z}=2\sum _{n=1}^{\infty }\gamma _{n}z^{n}$, when $n=1,2,3$, and $\alpha \ge 0$, and obtain the sharp bound for $|\gamma _4|$ when $0\le \alpha \le \alpha ^{*}(\alpha ^{*}\approx 1.5464),$ together with another bound for $|\gamma _{4}|$ when $\alpha \ge 0.$ Sharp bounds for some initial coefficients of the inverse function when $f\in {\mathcal {B}} _{1}(\alpha )$ are also found, which augment known results.