Zalcman Conjecture for certain analytic and univalent functions

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.