On the Bernstein-type inequalities for ultraspherical polynomials

We present a survey of the most recent results and inequalities for the gamma function and the ratio of the gamma functions and study, among other things, the relation between these results and known inequalities for ultraspherical polynomials. In particular, we discuss the inequality(sin θ)λ|Pn(λ)(cos θ)|<21−λΓ(λ) Γ(n+3/2λ)Γ(n+1+1/2λ), 0⩽θ⩽π,where Pn(λ)(cos θ) denotes the ultraspherical polynomial of degree n, established by Alzer (Arch. Math. 69 (1997) 487) and the one established by Durand (In: R.A. Askey (Ed.), Theory and Application of Special Functions, Proceedings of the Advanced Seminar on Mathematical Research Center, University of Wisconsin, Madison, Vol. 35, Academic Press, New York, 1975, p. 353)(sin θ)λ|Pn(λ)(cos θ)|⩽Γ(n/2+λ)Γ(λ)Γ(n/2+1), 0⩽θ⩽π.