Sharp Smith’s bounds for the gamma function

Among various approximation formulas for the gamma function, Smith showed that Γ(x+12)∼S(x)=2π(xe)x(2xtanh12x)x/2,x→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Gamma \biggl( x+\frac{1}{2} \biggr) \thicksim S ( x ) =\sqrt{2 \pi } \biggl( \frac{x}{e} \biggr) ^{x} \biggl( 2x\tanh \frac{1}{2x} \biggr) ^{x/2}, \quad x\rightarrow \infty , $$\end{document} which is a little-known but accurate and simple one. In this note, we prove that the function x↦lnΓ(x+1/2)−lnS(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto \ln \Gamma ( x+1/2 ) - \ln S ( x ) $\end{document} is strictly increasing and concave on (0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( 0,\infty ) $\end{document}, which shows that Smith’s approximation is just an upper one.


Introduction
The Stirling formula n! ∼ √ 2πnn n e -n (1.1) has many important applications in statistical physics, probability theory and number theory. Due to its practical importance, it has attracted much interest of many mathematicians and has also motivated a large number of research papers concerning various generalizations and improvements; see for example, Burnside's [1], Gosper [2], Batir [3], Mortici [4]. The gamma function (x) = ∞ 0 t x-1 e -t dt for x > 0 is closely related to the Stirling formula, since (n + 1) = n! for all n ∈ N. This inspired some authors to also pay attention to find various better approximations for the gamma function; see, for instance, Ramanujan [5, p. 339], Windschitl (see Nemes [6,Corollary 4.1]), Yang and Chu [7], Chen [8].
In this note, we are interested in Smith's approximation formula (see [24, equation (42)]): x x e x 2x tanh 1 2x x/2 It is easy to check that which shows that the rate of S(x) converging to (x + 1/2) as x → ∞ is like x -5 . According to the comment in [8, (3.5)-(3.10)], it is well known that Smith's approximation is an accurate but simple one for gamma function. The aim of this short note is to further prove the Smith approximation S(x) is an upper one. Our main result is stated as follows.

Proof of Theorem 1
To prove Theorem 1 we need the following two lemmas.

Lemma 1 The inequality
holds for all x > 0.
Proof Let .

Lemma 2 The inequality
holds for all t > 0.
Proof It is obvious that the inequality what we consider is equivalent to Simplifying and expanding it in power series lead us to where a n = 62n 2 -31n + 120 2 2n-1 -24n(2n -1) 14n 2 -35n + 31 .
It is easy to check that a 2 = a 3 = 0 and a 4 = 49 184 > 0. It remains to prove a n > 0 for n ≥ 5.
Now we are in a position to prove Theorem 1.
Then it is deduced that which in turn implies that This completes the proof.

Corollaries and remarks
Using the increasing property of f (x + 1/2) given in Theorem 1 and noting that π (tanh 1) 1/4 and f we have the corollaries.

Corollary 3 For x > 0, the inequalities
hold, where the constants 1/2 and are the best possible.
Finally, as a by-product of Lemma 1, we draw the following conclusion.
where the inequality holds due to Lemma 1. This completes the proof.
Remark 1 Theorem 2 gives a new approximation for the gamma function

Conclusions
In this note, we mainly presented an upper bound of Smith's approximation in accordance with the fact that the function x → ln (x + 1/2) -ln S(x) is strictly increasing and concave on (0, ∞). As a consequence, we get some new sharp estimates to various classical inequalities concerning the gamma function and hyperbolic functions.