On a conjecture for trigonometric sums and starlike functions. II.
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.
scientific article; zbMATH DE number 5722082
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