Inequalities for ultraspherical polynomials and the gamma function

Further Inequalities for the Gamma Function

“…Thus, if for some fixed a we can show that fk ultimately increases (or decreases) to 1 for k -» oo, we can conclude that fk < 1 (or fk > 1) for k > k0, and we obtain the inequalities which for integer k are the inequalities obtained by Lorch [4]. For the case 0 < X < 1 and real k the inequality (2.2) has also been proved independently by Kershaw [3].…”

“…Recently Lorch [4] has given some useful improvements of the bounds in (1.1) and has used his results to obtain a very interesting inequality for ultraspherical polynomials.…”

On some inequalities for the gamma and psi functions

“…In the past many articles were published providing different inequalities for the ratio Γ(x + 1)/Γ(x + s), where x > 0 and s ∈ (0, 1); see, e.g., [2], [13], [18], [25], [26], [29]- [31], [45], [50]. In this section we present upper and lower bounds for the difference ψ(x + 1) − ψ(x + s).…”

Uniform inequalities for Gegenbauer polynomials

“…In contrast to this, Lorch [5] proved the strict and uniform bound of holding for arbitrary ~ 6 R, but under the restriction of 0 < A < 1. In case of the Legendre-polynomials (A = ½) this was known before, see references in [1].…”