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 $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ 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$n\ge 2$,whichimpliesthefamousBieberbachconjecture$|a_n|\le n$for$n\ge 2$\mathrm{.}For$f\in $\mathcal{S}$$, Ma \cite{Ma-1999} proposed a generalized Zalcman conjecture $$\mathrm{\mid }{a}_n{a}_m-a_{n+m-1}\mathrm{\mid }\le (n-1)(m-1)$$ for $n\ge 2,m\ge 2$. Let $\mathcal{U}$ be the class of functions $f\in \mathcal{A}$ satisfying $$\mid f^{\mathrm{\prime }}(z){\left(\frac{z}{f(z)}\right)}^2-1\mid <1forz\in \mathbb{D}\mathrm{.}$$ and $\mathcal{F}$ denote the class of functions $f\in $\mathcal{A}$$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.