On a conjecture for trigonometric sums and starlike functions. II.
type: Item
instance of: scholarly article
title: On a conjecture for trigonometric sums and starlike functions. II.
zbMATH Open document ID: 1200.42002
author: Stamatis Koumandos
author: Martin Lamprecht
published in: Journal of Approximation Theory
publication date: 2010-06-17T00:00:00Z
review text: For \(\mu>0\) let \(s_n^\mu(z)=\sum_{k=0}^n (\mu)_k z^k/k!\), where \((\mu)_k=\mu(\mu+1)\dots (\mu+k-1)\). Denote by \(\mathcal{A}\) the class of analytic functions in the unit disk \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\). For \(f,g\in\mathcal{A}\) we write \(f\prec g\) (\(f\) is subordinate to \(g\) in \(\mathbb{D}\)) if there exists \(w\in\mathcal{A}\) with \(|w(z)|\leq z\) for \(z\in\mathbb{D}\), such that \(f=g\circ w\). Also, for \(\rho\in (0,1]\) we let \(\mu^*(\rho)\) be the unique solution of the integral equation \(\int_0^{(\rho+1)\pi}\sin(t-\rho\pi)t^{\mu-1}dt=0\). The paper under review is a continuation of the work of \textit{S. Koumandos} and \textit{S. Ruscheweyh} [J. Approximation Theory 149, No. 1, 42--58 (2007; Zbl 1135.42001)], where its authors guessed that: Conjecture. For \(\rho\in(0,1]\) the number \(\mu^*(\rho)\) is indeed maximal number \(\mu(\rho)\) such that for all \(n\in\mathbb{N}\) and \(\mu\in (0,\mu(\rho)]\) we have \[ (1-z)^\rho s_n^\mu(z)\prec \Big(\frac{1+z}{1-z}\Big)^\rho. \] In the present paper, the authors give a proof of this conjecture in case \(\rho=1/4\). To do this, they prove (based on the properties of completely monotonic functions) a sharp trigonometric inequality. Main result of paper has some applications concerning starlike functions, as more as a corollary, it gives an inequality concerning Gegenbauer polynomials.
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 26D05
Mathematics Subject Classification ID: 42A32
Mathematics Subject Classification ID: 26D20
zbMATH DE Number: 5722082
zbMATH Keywords: positive trigonometric inequality
zbMATH Keywords: completely monotonic function
zbMATH Keywords: starlike function
zbMATH Keywords: Gegenbauer polynomial
MaRDI profile type: Publication
full work available at URL: j.jat.2009.11.007
OpenAlex ID: W4254529661
Wikidata QID: Q123360997
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DOI: 10.1016/J.JAT.2009.11.007
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description: scientific article; zbMATH DE number 5722082