On a conjecture for the fifth coefficients for the class $\mathcal{U}(\lambda )$

Abstract

Let $f$ be function that is analytic in the unit disk $\mathbb{D}=\{z:|z|<1\}$, normalized such that $f(0)=f^{\prime }(0)-1=0$, i.e., of type $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<λ\quad\quad (z\in{\mathbb D}), \] then $f$ belongs to the class $\mathcal{U}(\lambda )$, $0<\lambda \le 1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from $\mathcal{U}(\lambda )$ satisfying \[ \frac{f(z)}{z}\prec \frac{1}{(1+z)(1+λz)}, \] ("$\prec$" is the usual subordination) in the case when $0.400436\dots \le \lambda \le 1$.