On Bernstein's inequality for ultraspherical polynomials (Q1383604)

The author offers a proof for \[ (\sin t)^{s}| P_{n}^{(s)}(\cos t)| < \frac{ 2^{1-s}}{\Gamma(s)} \frac{ \Gamma(n+(3s/2))}{\Gamma(n+1+(s/2))}, \] where \(0<t<\pi\), \(0<s<1\), and \(P_{n}^{(s)}\) is an ultraspherical polynomial. This improves several previous results, including that of \textit{A. Laforgia} [Boll. Unione Mat. Ital., VII. Ser., A 6, 267-269 (1992; Zbl 0582.33009)], who had \(\Gamma(n+2) / \Gamma(n+1)\) as second factor on the right and whose ``elegant methods motivated'' the present paper.