Logarithmic coefficients and a coefficient conjecture for univalent functions

By Milutin Obradovic, Saminathan Ponnusamy, Karl-Joachim Wirths. 2017

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Let $${{\mathcal {U}}}(\lambda )$$U(λ) denote the family of analytic functions f(z), $$f(0)=0=f'(0)-1$$f(0)=0=f′(0)-1, in the unit disk $${\mathbb {D}}$$D, which satisfy the condition $$\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $$|(z/f(z))2f′(z)-1|<λ for some $$0<\lambda \le 1$$0<λ≤1. The logarithmic coefficients $$\gamma _n$$γn of f are defined by the formula $$\log (f(z)/z)=2\sum _{n=1}^\infty \gamma _nz^n$$log(f(z)/z)=2∑n=1∞γnzn. In a recent paper, the present authors proposed a conjecture that if $$f\in {{\mathcal {U}}}(\lambda )$$f∈U(λ) for some $$0<\lambda \le 1$$0<λ≤1, then $$|a_n|\le \sum _{k=0}^{n-1}\lambda ^k$$|an|≤∑k=0n-1λk for $$n\ge 2$$n≥2 and provided a new proof for the case $$n=2$$n=2. One of the aims of this article is to present a proof of this conjecture for $$n=3, 4$$n=3,4 and an elegant proof of the inequality for $$n=2$$n=2, with equality for $$f(z)=z/[(1+z)(1+\lambda z)]$$f(z)=z/[(1+z)(1+λz)]. In addition, the authors prove the following sharp inequality for $$f\in {{\mathcal {U}}}(\lambda )$$f∈U(λ): $$\begin{aligned} \sum _{n=1}^{\infty }|\gamma _{n}|^{2} \le \frac{1}{4}\left( \frac{\pi ^{2}}{6}+2\mathrm{Li\,}_{2}(\lambda )+\mathrm{Li\,}_{2}(\lambda ^{2})\right) , \end{aligned}$$∑n=1∞|γn|2≤14π26+2Li2(λ)+Li2(λ2),where $$\mathrm{Li}_2$$Li2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of $${\mathcal {S}}$$S.

Subfields: Applied Mathematics, Geometry and Topology, Polymers and Plastics

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