Logarithmic coefficients and a coefficient conjecture for univalent functions

Logarithmic coefficients and a coefficient conjecture for univalent functions 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 ) ) 2 f ′ ( z ) - 1 | < λ for some $$0<\lambda \le 1$$ 0 < λ ≤ 1 . The logarithmic coefficients $$\gamma _n$$ γ n 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 ∞ γ n z n . 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 $$|a_n|\le \sum _{k=0}^{n-1}\lambda ^k$$ | a n | ≤ ∑ k = 0 n - 1 λ k 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 + λ z ) ] . 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 ∞ | γ n | 2 ≤ 1 4 π 2 6 + 2 Li 2 ( λ ) + Li 2 ( λ 2 ) , where $$\mathrm{Li}_2$$ 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 $${\mathcal {S}}$$ S . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

Logarithmic coefficients and a coefficient conjecture for univalent functions

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Wien
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1024-3
Publisher site
See Article on Publisher Site

Abstract

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 ) ) 2 f ′ ( z ) - 1 | < λ for some $$0<\lambda \le 1$$ 0 < λ ≤ 1 . The logarithmic coefficients $$\gamma _n$$ γ n 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 ∞ γ n z n . 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 $$|a_n|\le \sum _{k=0}^{n-1}\lambda ^k$$ | a n | ≤ ∑ k = 0 n - 1 λ k 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 + λ z ) ] . 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 ∞ | γ n | 2 ≤ 1 4 π 2 6 + 2 Li 2 ( λ ) + Li 2 ( λ 2 ) , where $$\mathrm{Li}_2$$ 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 $${\mathcal {S}}$$ S .

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Feb 3, 2017

References

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