On some inequalities for the gamma and psi functions
Mathematics of Computation
Inequalities and monotonicity properties for the gamma function
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
On Gautschi's harmonic mean inequality for the gamma function
Journal of Computational and Applied Mathematics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
| Hi-index | 7.29 |
We prove that the double-inequality 1/x+1/x+x/(x+1/x)1/2 x)+1/Γ(1/x) ≤ 1/x+1/x+β/(x+1/x)1/2 holds for all x 0 with the best possible constants α=0 and β=3/√2. The right-hand side refines 1/Γ(x)+1/Γ(1/x)≤2 (x 0), which was proved in 1974 by W. Gautschi. Moreover, we present sharp inequalities for the product, the sum, the difference, and the ratio of Γ(x) and Γ(1/x).