NOTES ON UPPER AND LOWER BOUNDS OF TWO INEQUALITIES FOR THE GAMMA FUNCTION

Year 2010, Volume: 39 Issue: 1, 11 - 15, 03.07.2019

Armend Sh. Shabani

Abstract

G. D. Anderson and S. L. Qiu (A monotonicity property of the gamma
function, Proc. Amer. Math. Soc. 125 (11) (1997), 3355–3362) obtained a double inequality for the function Γ(x). Their result was improved by H. Alzer (Inequalities for the gamma function, Proc. Amer.
Math. Soc. 128 (1), 141–147, 1999), and by X. Li, Ch. P. Chen (Inequalities for the gamma function, J. Ineq. Pure Appl. Math. 8 (1),
Art.28, 2007). Li and Chen remarked that their bounds could not be
compared with those of Alzer. In this note, we will show that there
exist a constant γ such that, in the intervals (1, γ) and (γ, +∞), the
upper bounds can be compared to each other. We will also show that
there exist a constant ξ such that it will be possible to compare lower
bounds in the intervals (1, ξ) and (ξ, +∞). 

Keywords

Gamma function, Inequalities

References

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