Sharp estimate of the fifth coefficients for the class U(lambda)

Document Type : Research Articles

Authors

1 Department of Mathematics, Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia

2 Department of Mathematics and Informatics, Faculty of Mechanical Engineering, Ss. Cyril and Methodius University in Skopje, Karpos II b.b., 1000 Skopje, Republic of North Macedonia

Abstract

Let $f$ be function that is analytic in the unit disk $D=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<\lambda \quad\quad (z\in D), \]
then $f$ belongs to the class $U(\lambda)$, $0<\lambda\le1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from $U(\lambda)$ in the case when $0.400436\ldots \le\lambda\le1$.

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Caspian Journal of Mathematical Sciences
Volume 11, Issue 2
2022
Pages 410-416

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