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

G.D. Anderson and S. L. Qiu (A monotonicity property of the gamma function, Proc. Amer. Math. Soc. 125 (11) (1997), 3355–3362) ob-tained a double inequality for the function 􀀀(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 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 such that it will be possible to compare lower bounds in the intervals (1, ξ) and (ξ,+∞)