Let $\mathcal{ A}$ denote the class of analytic functions in the unit disk $\mathbb{ D}$ of the form $f(z)= z+\sum_{ n=2}^{ \infty}a_n z^n$ and $\mathcal{ S}$ denote the class of functions $f\in\mathcal{ A}$ which are univalent ({ \it i.e.}, one-to-one). In 1960s, L. Zalcman conjectured that $|a_n^2-a_{ 2n-1}|\le (n-1)^2$ for $n\ge 2$, which implies the famous Bieberbach conjecture $|a_n|\le n$ for $n\ge 2$. For $f\in \mathcal{ S}$, Ma \cite{ Ma-1999} proposed a generalized Zalcman conjecture $$|a_{ n}a_{ m}-a_{ n+m-1}|\le (n-1)(m-1) $$ for $n\ge 2, m\ge 2$. Let $\mathcal{ U}$ be the class of functions $f\in\mathcal{ A}$ satisfying $$ \left|f'(z)\left(\frac{ z}{ f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{ D}. $$ and $\mathcal{ F}$ denote the class of functions $f\in \mathcal{ A}$ satisfying ${ \rm Re\,}(1-z)^{ 2}f'(z)>0$ in $\mathbb{ D}$. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{ U}$ using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{ F}$ for the initial coefficients.
Categories: math.CV