Generalized Zalcman conjecture for some classes of analytic functions

Abstract: For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m -a_{n+m-1}|$. For all real parameters $\lambda$ and $ \beta<1$, we provide the sharp upper bound of $\phi(f,n,m;\lambda)$ for functions $f$ satisfying $\operatorname{Re}f'(z) > \beta$ and hence settles the open problem of estimating $\phi(f,n,m;\lambda)$ recently proposed by Agrawal and Sahoo [S. Agrawal and S. k. Sahoo, On coefficient functionals associated with the Zalcman conjecture, arXiv preprint, 2016]. It is worth mentioning that the sharp estimations of $\phi(f,n,m;\lambda)$ follow for starlike and convex functions of order $\alpha$ $(\alpha <1)$ when $\lambda \leq 0$. Moreover, for certain positive $\lambda$, the sharp estimation of $\phi(f,n,m;\lambda)$ is given when $f$ is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions.