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Monatshefte für Mathematik

, Volume 185, Issue 3, pp 489–501 | Cite as

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 

Mathematics Subject Classification

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

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