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Monatshefte für Mathematik

, Volume 185, Issue 3, pp 489–501

# Logarithmic coefficients and a coefficient conjecture for univalent functions

• Milutin Obradović
• Saminathan Ponnusamy
• Karl-Joachim Wirths
Article

## Abstract

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

## Keywords

Univalent Starlike Convex and close-to-convex functions Subordination Logarithmic coefficients and coefficient estimates

30C45

## Notes

### Acknowledgements

The work of the first author was supported by MNZZS Grant, No. ON174017, Serbia. The second author is on leave from the IIT Madras.

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## Copyright information

© Springer-Verlag Wien 2017

## Authors and Affiliations

• Milutin Obradović
• 1
• Saminathan Ponnusamy
• 2
• Karl-Joachim Wirths
• 3
1. 1.Department of Mathematics, Faculty of Civil EngineeringUniversity of BelgradeBelgradeSerbia
2. 2.Indian Statistical Institute (ISI)Chennai Centre, SETS (Society for Electronic Transactions and Security)ChennaiIndia
3. 3.Institut für Analysis und AlgebraTU BraunschweigBraunschweigGermany