On rational bounds for the gamma function

In the article, we prove that the double inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$\end{document}x2+p0x+p0<Γ(x+1)<x2+9/5x+9/5 holds for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in(0, 1)$\end{document}x∈(0,1), we present the best possible constants λ and μ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\lambda(x^{2}+9/5)}{x+9/5}\leq\Gamma(x+1)\leq\frac{\mu (x^{2}+p_{0})}{x+p_{0}} $$\end{document}λ(x2+9/5)x+9/5≤Γ(x+1)≤μ(x2+p0)x+p0 for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in(0, 1)$\end{document}x∈(0,1), and we find the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\ast}$\end{document}x∗ in the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0, 1)$\end{document}(0,1) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$\end{document}Γ(x+1)>(x2+1/γ)/(x+1/γ) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in(0, x^{\ast})$\end{document}x∈(0,x∗) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma(x+1)<(x^{2}+1/\gamma)/(x+1/\gamma )$\end{document}Γ(x+1)<(x2+1/γ)/(x+1/γ) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in(x^{\ast}, 1)$\end{document}x∈(x∗,1), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma(x)$\end{document}Γ(x) is the classical gamma function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots$\end{document}γ=limn→∞(∑k=1n1/k−logn)=0.577… is Euler-Mascheroni constant and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{0}=\gamma/(1-\gamma )=1.365\ldots$\end{document}p0=γ/(1−γ)=1.365… .


Introduction
For x > , the classical Euler gamma function (x) and its logarithmic derivative, the so- Recently, the gamma function have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for (x) can be found in the literature [-].
Due to (x + ) = x (x) and (n + ) = n!, we will only need to focus our attention on (x + ) with x ∈ (, ). Gautschi [] proved that the double inequality n -s < (n + ) (n + s) < e (-s)ψ(n+) (.) holds for all s ∈ (, ) and n ∈ N. Inequality (.) was generalized and improved by Kershaw [] as follows: x + s  for the gamma function being valid for all x ∈ (, ), and asked for 'other bounds for the gamma function in terms of elementary functions' . Ivády [] provided the bounds for gamma function in terms of very simple rational functions as follows: for all x ∈ (, ). Inequality (.) can be regarded as a simple estimation of the value of the gamma function. In [], Zhao, Guo and Qi proved that the function is strictly increasing on (, ). The monotonicity of Q(x) on the interval (, ) and the facts that Q( + ) = γ and Q( -) = (γ ) lead to the conclusion that for all x ∈ (, ), where γ = lim n→∞ ( n k= /k -log n) = . . . . is the Euler-Mascheroni constant. Let Then we clearly see that for all x ∈ (, ), and numerical computations show that Motivated by (.)-(.), it is natural to ask what the better parameters p and q on the interval (, ) are such that the double inequality holds for all x ∈ (, ). The main purpose of the article is to deal with this questions. Some complicated computations are carried out using the Mathematica computer algebra system.

Lemmas
In order to establish our main results we need several lemmas, which we present in this section. .
is strictly monotone, then the monotonicity in the conclusion is also strict.
a n a m >  and Then there exists t  ∈ (, ∞) such that P n (t  ) = , P n (t) <  for t ∈ (, t  ) and P n (t) >  for t ∈ (t  , ∞).
holds for all x > .

Lemma . The inequalities
hold for all x > -/.
Proof Let x > -/, and R  (x) and R  (x) be defined by respectively. Then making use of the well-known formulas where sinh(t) = (e te -t )/ is the hyperbolic sine function.
Note that for t > . It follows from (.)-(.) and the Bernstein theorem for complete monotonicity property that the two functions R  (x) and R  (x) are completely monotonic on the interval (-/, ∞).
Therefore, Lemma . follows easily from (.), (.) and the complete monotonicity of R  (x) and R  (x) on the interval (-/, ∞) together with the facts that Lemma . The double inequality holds for all x ∈ (, ) and q > .
Proof Let x ∈ (, ), q > , and H  (x) and H  (x) be defined by respectively. Then simple computations lead to

From (.) and (.) we clearly see that H  (x)/H  (x) is strictly increasing on
, then H  (x) = , and Lemma . and (.) together with the monotonicity of , then H  (x) = , and Lemma . and (.) together with the monotonicity of . Therefore, Lemma . follows easily from (.) and (.) together with the monotonicity of the function Proof From (.) and the second inequality in Lemma . we have From (.) and (.) we get Proof From (.) and the first inequality in Lemma . we have Proof It follows from Lemma . and (.) that Then simple computations lead to It follows from the first inequalities in (.) and (.) together with the identity ψ (n) (x + ) = ψ (n) (x) + (-) n n!/x n+ that  From (.) and the identity ψ (n) (x) = ψ (n) (x + ) + (-) n+ n!/x n+ we get Taking x =  in the first inequality of (.) and x =  in the second inequality of (.), one has It follows from (.) and (.) that

Main results
Theorem . Let p >  and p  = γ /(γ ) = . . . . . Then the inequality holds for all x ∈ (, ) if and only if p ≤ p  , and the inequality and x  = . . . . is the unique solution of the equation on the interval (, ).
Proof If inequality (.) holds for all x ∈ (, ), then p ≤ p  follows easily from Next, we prove that inequality (.) holds for all x ∈ (, ) and p = p  and (.) holds for all x ∈ (, ) if and only if μ ≥ μ  .
Let f (p, x), f  (p, x), f  (p, x) be defined by (.)-(.) and Then elaborated computations lead to It follows from the second inequality in (.) and the first inequality in (.) together with (.) that It is easy to verify that all the coefficients of the polynomial g  (x) are positive, which implies that g(x) is strictly increasing on (, ), then from (.) and (.) we know that there exists η ∈ (, ) such that the function f  (p  , x) is strictly decreasing on (, η) and strictly increasing on (η, ).
It follows from (.) and (.) together with the piecewise monotonicity of the function f  (p  , x) on the interval (, ) that there exists x  ∈ (, ) such that f (p  , x) is strictly increasing on (, x  ) and strictly decreasing on (x  , ) and x  is the unique solution of equation (.) on the interval (, ). Therefore, the desired results follow easily from (.), (.), (.) and the piecewise monotonicity of the function f (p  , x) on the interval (, ) together with the fact that the function p → (x  + p)/(x + p) is strictly increasing.

Theorem . The double inequality
holds for all x ∈ (, ) with p  = γ /(γ ) = . . . . and p  = /, the constant p  appears to be the best possible, but this is not true for p  , and a slightly smaller value for p  is possible. Unfortunately, we cannot find the best possible constant p  in the article; we leave this as an open problem for the reader.
Remark . From the monotonicity of the function p → (x  + p)/(x + p) we clearly see that both the upper and lower bounds for (x + ) given in (.) are better than that given in (.), and the first (second) inequality in Theorem . is the improvement of the first (second) inequality in (.) for x ∈ (, x * ) (x ∈ (x * , )), where x * = . . . . is given by Theorem ..

Results and discussion
In this paper, we provide the accurate bounds for the classical gamma function in terms of very simple rational functions, which can be used to estimate the value of the gamma function in the area of engineering and technology.

Conclusion
In the article, we present several very simple and practical rational bounds for the gamma function, which can be regarded as a simple estimation of the value of the gamma function. The given results are improvements of some well-known results.