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arXiv · 2007.09716

On certain properties of the class $U(\lambda)$

N. M. Alarifi·M. Obradovic·N. Tuneski
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Abstract

Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda)$, $0<\lambda \leq1$, consisting of functions $f$ from ${\mathcal{A}}$ satisfying \[\left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right| < \lambda \quad (z\in {\mathbb D}).\] and give results regarding the Zalcman Conjecture, the generalised Zalcman conjecture, the Krushkal inequality and the second and third order Hankel determinant.

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N. M. Alarifi, M. Obradovic, N. Tuneski. 2020-07-19. On certain properties of the class $U(\lambda)$. https://arxiv.org/abs/2007.09716

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