Cyclic mean-value inequalities for the gamma function

• Autorzy: Horst Alzer
• Języki publikacji: EN

Abstrakty

EN

We present two cyclic inequalities involving the classical Γ-function of Euler and the (unweighted) power mean <br> $M_t(a,b) = ((a^t + b^t)/2)^{1/t}$ (t ≠ 0), M₀(a,b) = √(ab) (a,b>0).<br>(I) Let 2 ≤ n ∈ ℕ and r ∈ ℝ. The inequality <br> $∏_{j=1}^{n} Γ(1/(1 + M_r(x_j,x_{j+1}))) ≤ ∏_{j=1}^{n} Γ(1/(1 + x_j)) (x_{n+1} = x₁)$ <br> holds for all $x_j > 0$ (j = 1,..., n) if and only if r ≤ 0. (II) Let 2 ≤ n ∈ ℕ and s ∈ ℝ. The inequality <br> $∏_{j=1}^{n} Γ(1/(1 + x_j)) ≤ ∏_{j=1}^{n} Γ(1/(1 + M_s(x_j,x_{j+1}))) (x_{n+1} = x₁)$ <br> is valid for all $x_j > 0$ (j = 1,...,n) if and only if <br> $s ≥ max_{0<x<1} P(x) = 1.0309...$. <br> Here, <br> P(x) = 2x - 1 + x(x-1) ψ'(x)/ψ(x) and ψ = Γ'/Γ.

Kategorie tematyczne

• 33B15: Gamma, beta and polygamma functions
• 26D07: Inequalities involving other types of functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 132
• Numer: 1
• Strony: 27-34

Daty

wydano

2013

Twórcy

autor

Horst Alzer

• Morsbacher Str. 10, D-51545 Waldbröl, Germany

Identyfikatory

DOI

10.4064/cm132-1-3