Abstract
In the article, we prove that the double inequality
holds for all \(x\in(0, 1)\), we present the best possible constants λ and μ such that
for all \(x\in(0, 1)\), and we find the value of \(x^{\ast}\) in the interval \((0, 1)\) such that \(\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)\) for \(x\in(0, x^{\ast})\) and \(\Gamma(x+1)<(x^{2}+1/\gamma)/(x+1/\gamma )\) for \(x\in(x^{\ast}, 1)\), where \(\Gamma(x)\) is the classical gamma function, \(\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots\) is Euler-Mascheroni constant and \(p_{0}=\gamma/(1-\gamma )=1.365\ldots\) .
Similar content being viewed by others
1 Introduction
For \(x>0\), the classical Euler gamma function \(\Gamma(x)\) and its logarithmic derivative, the so-called psi function \(\psi(x)\) [1] are defined by
respectively.
A real-valued function f is said to be completely monotonic [2] on an interval I if f has derivatives of all orders on I and \((-1)^{n}f^{(n)}(x)\geq0\) for all \(n\geq0\) and \(x\in I\). The well-known Bernstein theorem [3] states that a function f on \([0,\infty)\) is completely monotonic if and only if there exists a bounded and non-decreasing function \(\omega(t)\) such that \(f(x)=\int_{0}^{\infty }e^{-xt}\,d\omega(t)\) converges for all \(x\in[0, \infty)\).
Recently, the gamma function have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for \(\Gamma(x)\) can be found in the literature [4–14].
Due to \(\Gamma(x+1)=x\Gamma(x)\) and \(\Gamma(n+1)=n!\), we will only need to focus our attention on \(\Gamma(x+1)\) with \(x\in(0, 1)\). Gautschi [15] proved that the double inequality
holds for all \(s\in(0, 1)\) and \(n\in\mathbb{N}\).
Inequality (1.1) was generalized and improved by Kershaw [16] as follows:
for all \(x>0\) and \(s\in(0, 1)\).
Elezović, Giordano and Pečarić [17] established the double inequality
for the gamma function being valid for all \(x\in(0, 1)\), and asked for ‘other bounds for the gamma function in terms of elementary functions’.
Ivády [18] provided the bounds for gamma function in terms of very simple rational functions as follows:
for all \(x\in(0, 1)\). Inequality (1.3) can be regarded as a simple estimation of the value of the gamma function.
In [19], Zhao, Guo and Qi proved that the function
is strictly increasing on \((0, 1)\). The monotonicity of \(Q(x)\) on the interval \((0, 1)\) and the facts that \(Q(0^{+})=\gamma\) and \(Q(1^{-})=2(1-\gamma)\) lead to the conclusion that
for all \(x\in(0, 1)\), where \(\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots\) is the Euler-Mascheroni constant.
Let
Then we clearly see that
for all \(x\in(0, 1)\), and numerical computations show that
Motivated by (1.3)-(1.8), it is natural to ask what the better parameters p and q on the interval \((1, 2)\) are such that the double inequality
holds for all \(x\in(0, 1)\). The main purpose of the article is to deal with this questions. Some complicated computations are carried out using the Mathematica computer algebra system.
2 Lemmas
In order to establish our main results we need several lemmas, which we present in this section.
Lemma 2.1
See [20, Theorem 1.25]
Let \(-\infty< a< b<\infty\), \(f,g:[a,b]\rightarrow{\mathbb{R}}\) be continuous on \([a,b]\) and differentiable on \((a,b)\), and \(g'(x)\neq0\) on \((a,b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a,b)\), then so are the functions
If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
See [21, Lemma 7]
Let \(n\in\mathbb{N}\) and \(m\in\mathbb{N}\cup\{0\}\) with \(n>m\), \(a_{i}\geq0\) for all \(0\leq i\leq n\), \(a_{n}a_{m}>0\) and
Then there exists \(t_{0}\in(0, \infty)\) such that \(P_{n}(t_{0})=0\), \(P_{n}(t)<0\) for \(t\in(0, t_{0})\) and \(P_{n}(t)>0\) for \(t\in(t_{0}, \infty)\).
Lemma 2.3
See [22, Corollary 3.1]
The inequality
holds for all \(x>0\).
Lemma 2.4
See [23, Corollary 3.3(ii)]
The double inequality
holds for all \(x>0\).
Lemma 2.5
The inequalities
hold for all \(x>-1/2\).
Proof
Let \(x>-1/2\), and \(R_{1}(x)\) and \(R_{2}(x)\) be defined by
respectively. Then making use of the well-known formulas
we get
where \(\sinh(t)=(e^{t}-e^{-t})/2\) is the hyperbolic sine function.
Note that
for \(t>0\).
It follows from (2.6)-(2.9) and the Bernstein theorem for complete monotonicity property that the two functions \(R_{1}(x)\) and \(R_{2}(x)\) are completely monotonic on the interval \((-1/2, \infty)\).
Therefore, Lemma 2.5 follows easily from (2.4), (2.5) and the complete monotonicity of \(R_{1}(x)\) and \(R_{2}(x)\) on the interval \((-1/2, \infty )\) together with the facts that
□
Lemma 2.6
The double inequality
holds for all \(x\in(0, 1)\) and \(q>1\).
Proof
Let \(x\in(0, 1)\), \(q>1\), and \(H_{1}(x)\) and \(H_{2}(x)\) be defined by
respectively. Then simple computations lead to
where
From (2.15) and (2.16) we clearly see that \(H_{1}^{\prime }(x)/H_{2}^{\prime}(x)\) is strictly increasing on \((0, \sqrt{2}-1)\cup (\sqrt{2}-1, 1)\). We assert that the function \(H_{1}(x)/H_{2}(x)\) is strictly increasing on \((0, 1)\). Indeed, if \(x\in(0, \sqrt{2}-1)\), then \(H_{2}^{\prime}(x)\neq0\), and Lemma 2.1 and (2.13) together with the monotonicity of \(H_{1}^{\prime}(x)/H_{2}^{\prime}(x)\) on \((0, \sqrt {2}-1)\) lead to the conclusion that \(H_{1}(x)/H_{2}(x)\) is strictly increasing on \((0, \sqrt{2}-1)\); if \(x\in(\sqrt{2}-1, 1)\), then \(H_{2}^{\prime}(x)\neq0\), and Lemma 2.1 and (2.14) together with the monotonicity of \(H_{1}^{\prime}(x)/H_{2}^{\prime}(x)\) on \((\sqrt {2}-1, 1)\) lead to the conclusion that \(H_{1}(x)/H_{2}(x)\) is strictly increasing on \((\sqrt{2}-1, 1)\).
Therefore, Lemma 2.6 follows easily from (2.11) and (2.12) together with the monotonicity of the function \(H_{1}(x)/H_{2}(x)\) on \((0, 1)\). □
Let \(p>0\), \(x\in(0, 1)\), and \(f(p; x)\), \(f_{1}(p; x)\), \(f_{2}(p; x)\) and \(f_{3}(p; x)\) be defined by
Lemma 2.7
Let \(f_{2}(p, x)\) be defined by (2.19). Then
for \(p\in[8/5, 9/5]\).
Proof
From (2.19) and the second inequality in Lemma 2.4 we have
Elaborated computations lead to
for \(p\in[8/5, 9/5]\).
□
Lemma 2.8
Let \(f_{2}(p, x)\) be defined by (2.19). Then
Proof
From (2.19) and the first inequality in Lemma 2.4 we have
□
Lemma 2.9
Let \(f_{1}(p, x)\) be defined by (2.18). Then
for \(x\in(7/50, 1/3)\).
Proof
It follows from Lemma 2.3 and (2.18) that
Elaborated computations lead to
where
From Lemma 2.2, (2.28) and (2.29) we know that \(h(x)\) is strictly decreasing on \((7/50, 1/3)\), then (2.27) leads to the conclusion that \(h(x)<0\) for \(x\in(7/50, 1/3)\).
Therefore,
for \(x\in(7/50, 1/3)\) follows from (2.25) and (2.26) together with \(h(x)<0\) for \(x\in (7/50, 1/3)\). □
Lemma 2.10
Let \(p\in[3/2, 2]\) and \(f_{3}(p, x)\) be defined by (2.20). Then there exists \(\eta(p)\in(0, 1)\) such that \(f_{3}(p, x)<0\) for \(x\in (0, \eta(p))\) and \(f_{3}(p, x)>0\) for \(x\in(\eta(p), 1)\).
Proof
Let
Then simple computations lead to
It follows from the first inequalities in (2.2) and (2.3) together with the identity \(\psi^{(n)}(x+1)=\psi^{(n)}(x)+(-1)^{n}n!/x^{n+1}\) that
where
with
From Lemma 2.2, (2.37) and (2.38) we clearly see that
for \(p\in[3/2, 1]\).
Making use of Lemma 2.2 again, and (2.36) and (2.39) together with the facts that \({b_{k}>0}\) for \(p\in[3/2, 1]\) and \(k=0, 1, 2, \ldots, 16\) we know that \(g_{1}(p, x)>0\) for \(p\in[3/2, 1]\) and \(x\in(0, 1)\). Then inequality (2.35) leads to the conclusion that the function \(x\rightarrow g(p, x)\) is strictly increasing on \((0, 1)\) for \(p\in [3/2, 2]\).
From (2.2) and the identity \(\psi^{(n)}(x)=\psi ^{(n)}(x+1)+(-1)^{n+1}n!/x^{n+1}\) we get
Taking \(x=1\) in the first inequality of (2.40) and \(x=0\) in the second inequality of (2.40), one has
It follows from (2.31) and (2.41) that
for \(p\in[3/2, 2]\).
Note that
Inequality (2.43) and equation (2.44) imply that
for \(p\in[3/2, 2]\).
Therefore, Lemma 2.10 follows easily from (2.30), (2.42), (2.45) and the monotonicity of the function \(x\rightarrow g(p, x)\) on the interval \((0, 1)\). □
Lemma 2.11
Let \(p\in[8/5, 9/5]\) and \(f_{2}(p, x)\) be defined by (2.19). Then there exist \(\eta_{1}(p), \eta_{2}(p)\in(0, 1)\) with \(\eta_{1}(p)<\eta _{2}(p)\) such that \(f_{2}(p, x)>0\) for \(x\in(0, \eta_{1}(p))\cup(\eta _{2}(p), 1)\) and \(f_{2}(p, x)<0\) for \(x\in(\eta_{1}(p), \eta_{2}(p))\).
Proof
It follows from (2.19) that
for \(p\in[8/5, 9/5]\).
From Lemma 2.10 and \([8/5, 9/5]\subset[3/2, 2]\) we know that there exists \(\eta(p)\in(0, 1)\) such that the function \(x\rightarrow f_{2}(p, x)\) is strictly decreasing on \((0, \eta(p))\) and strictly increasing on \((\eta(p), 1)\). Then Lemma 2.7 leads to the conclusion that
Therefore, there exist \(\eta_{1}(p)\in(0, \eta(p))\) and \(\eta _{2}(p)\in(\eta(p), 1)\) such that \(f_{2}(p, x)>0\) for \(x\in(0, \eta_{1}(p))\cup(\eta_{2}(p), 1)\) and \(f_{2}(p, x)<0\) for \(x\in(\eta_{1}(p), \eta_{2}(p))\) follow from (2.46)-(2.48) and the piecewise monotonicity of the function \(x\rightarrow f_{2}(p, x)\) on the interval \((0, 1)\). □
3 Main results
Theorem 3.1
Let \(p>0\) and \(p_{0}=\gamma/(1-\gamma)=1.365\ldots\) . Then the inequality
holds for all \(x\in(0, 1)\) if and only if \(p\leq p_{0}\), and the inequality
holds for all \(x\in(0, 1)\) if and only if \(\mu\geq\mu_{0}\), where
and \(x_{0}=0.346\ldots\) is the unique solution of the equation
on the interval \((0, 1)\).
Proof
If inequality (3.1) holds for all \(x\in(0, 1)\), then \(p\leq p_{0}\) follows easily from
Next, we prove that inequality (3.1) holds for all \(x\in(0, 1)\) and \(p=p_{0}\) and (3.2) holds for all \(x\in(0, 1)\) if and only if \(\mu\geq \mu_{0}\).
Let \(f(p, x)\), \(f_{1}(p, x)\), \(f_{2}(p, x)\) be defined by (2.17)-(2.19) and
Then elaborated computations lead to
It follows from the second inequality in (2.1) and the first inequality in (2.2) together with (3.9) that
where
It is easy to verify that all the coefficients of the polynomial \(g_{1}(x)\) are positive, which implies that \(g(x)\) is strictly increasing on \((0, 1)\), then from (3.5) and (3.8) we know that there exists \(\eta\in(0, 1)\) such that the function \(f_{1}(p_{0}, x)\) is strictly decreasing on \((0, \eta)\) and strictly increasing on \((\eta, 1)\).
It follows from (2.18) and (3.7) together with the piecewise monotonicity of the function \(f_{1}(p_{0}, x)\) on the interval \((0, 1)\) that there exists \(x_{0}\in(0, 1)\) such that \(f(p_{0}, x)\) is strictly increasing on \((0, x_{0})\) and strictly decreasing on \((x_{0}, 1)\) and \(x_{0}\) is the unique solution of equation (3.4) on the interval \((0, 1)\).
Therefore, the desired results follow easily from (2.17), (3.3), (3.6) and the piecewise monotonicity of the function \(f(p_{0}, x)\) on the interval \((0, 1)\) together with the fact that the function \(p\rightarrow(x^{2}+p)/(x+p)\) is strictly increasing.
Numerical computations show that \(x_{0}=0.346\ldots\) and \(\mu_{0}=(x_{0}+p_{0})\Gamma(x_{0}+1)/(x^{2}_{0}+p_{0})=1.027\ldots\) . □
Theorem 3.2
The inequality
holds for all \(x\in(0, x^{\ast})\), and its reverse inequality
holds for all \(x\in(x^{\ast}, 1)\), where \(x^{\ast}=0.385\ldots\) is the unique solution of the equation
on the interval \((0, 1)\).
Proof
Let \(f(p, x)\), \(f_{1}(p, x)\) and \(f_{2}(p, x)\) be, respectively, defined by (2.17), (2.18) and (2.19). Then simple computations lead to
From Lemma 2.11 and \(1/\gamma=1.732\ldots\in[8/5, 9/5]\) we know that there exist \(\eta_{1}(1/\gamma), \eta_{2}(1/\gamma)\in(0, 1)\) with \(\eta_{1}(1/\gamma)<\eta_{2}(1/\gamma)\) such that \(f_{1}(1/\gamma, x)\) is strictly increasing on \((0, \eta_{1}(1/\gamma))\cup( \eta _{2}(1/\gamma), 1)\) and strictly decreasing on \((\eta_{1}(1/\gamma), \eta_{2}(1/\gamma))\). We claim that
Indeed, if \(f_{1}(1/\gamma, \eta_{2}(1/\gamma))\geq0\), then the piecewise monotonicity of the function \(f_{1}(1/\gamma, x)\) on the interval \((0, 1)\) and (3.11) lead to the conclusion that \(f(1/\gamma, x)\) is strictly increasing on \((0, 1)\), which contradicts (3.10).
It follows from (3.11) and (3.12) together with the piecewise monotonicity of the function \(f_{1}(1/\gamma, x)\) on the interval \((0, 1)\) that there exist \(\eta ^{\ast}_{1}(1/\gamma)\in(\eta_{1}(1/\gamma), \eta_{2}(1/\gamma))\) and \(\eta^{\ast}_{2}(1/\gamma)\in(\eta_{2}(1/\gamma), 1)\) such that \(f(1/\gamma, x)\) is strictly increasing on \((0, \eta^{\ast}_{1}(1/\gamma ))\cup(\eta^{\ast}_{2}(1/\gamma), 1)\) and strictly decreasing on \(( \eta ^{\ast}_{1}(1/\gamma), \eta^{\ast}_{2}(1/\gamma))\).
Therefore, Theorem 3.2 follows easily from (2.17) and (3.10) together with the piecewise monotonicity of \(f(1/\gamma, x)\) on \((0, 1)\). Numerical computations show that \(x^{\ast}=0.385\ldots\) . □
Theorem 3.3
The double inequality
holds for all \(x\in(0, 1)\) with the best possible constant
where \(\tau_{0}=0.719\ldots\) is the unique solution of the equation
on the interval \((0, 1)\).
Proof
Let \(f(p, x)\), \(f_{1}(p, x)\) and \(f_{2}(p, x)\) be, respectively, defined by (2.17), (2.18) and (2.19). Then simple computations lead to
It follows from Lemma 2.11 that there exist \(\eta_{1}(9/5), \eta _{2}(9/5)\in(0, 1)\) with \(\eta_{1}(9/5)<\eta_{2}(9/5)\) such that \(f_{2}(9/5, x)>0\) for \(x\in(0, \eta_{1}(9/5))\cup(\eta_{2}(9/5), 1)\) and \(f_{2}(9/5, x)<0\) for \(x\in(\eta_{1}(9/5), \eta_{2}(9/5))\), and \(f_{1}(9/5, x)\) is strictly increasing on \((0, \eta_{1}(9/5))\cup(\eta_{2}(9/5), 1)\) and strictly decreasing on \((\eta_{1}(9/5), \eta_{2}(9/5))\). Then Lemmas 2.7-2.9 lead to the conclusion that \(\eta_{1}(9/5)\in(7/50, 1/3)\) and
From (2.18), (3.17), (3.18) and the piecewise monotonicity of \(f_{1}(9/5, x)\) on \((0, 1)\) we clearly see that there exists \(\tau_{0}\) such that \(\tau_{0}\) is the unique solution of equation (3.15) on the interval \((0, 1)\), and \(f(9/5, x)\) is strictly decreasing on \((0, \tau _{0})\) and strictly increasing on \((\tau_{0}, 1)\).
Equation (3.16) and the piecewise monotonicity of the function \(f(9/5, x)\) on the interval \((0, 1)\) lead to the conclusion that
for all \(x\in(0, 1)\).
Therefore, inequality (3.13) holds for all \(x\in(0, 1)\) follows from (2.17) and (3.19). We clearly see that the parameter λ given by (3.14) is the best possible constant such that the first inequality in (3.13) holds for all \(x\in(0, 1)\). Numerical computations show that \(\tau _{0}=0.719\ldots\) and \(\lambda=0.991\ldots\) . □
Remark 3.4
From Theorems 3.1 and 3.3 we clearly see that the double inequality
holds for all \(x\in(0, 1)\) with \(p_{0}=\gamma/(1-\gamma)=1.365\ldots\) and \(p_{1}=9/5\), the constant \(p_{0}\) appears to be the best possible, but this is not true for \(p_{1}\), and a slightly smaller value for \(p_{1}\) is possible. Unfortunately, we cannot find the best possible constant \(p_{1}\) in the article; we leave this as an open problem for the reader.
Remark 3.5
From the monotonicity of the function \(p\mapsto(x^{2}+p)/(x+p)\) we clearly see that both the upper and lower bounds for \(\Gamma(x+1)\) given in (3.20) are better than that given in (1.3), and the first (second) inequality in Theorem 3.2 is the improvement of the first (second) inequality in (1.3) for \(x\in(0, x^{\ast})\) \((x\in(x^{\ast}, 1))\), where \(x^{\ast}=0.385\ldots\) is given by Theorem 3.2.
Remark 3.6
From Lemma 2.6, \(\gamma+\gamma^{2}<1\), \(1/\gamma>1\), \(\gamma/(1-\gamma )>1\) and \((x^{2}+1)/(x+1)<1\) for \(x\in(0, 1)\) one has
Therefore, the lower bound for \(\Gamma(x+1)\) given in (3.20) is better than that given in (1.4), the first inequality in Theorem 3.2 is an improvement of the first inequality in (1.4) for \(x\in(0, x^{\ast})\) and the second inequality in Theorem 3.2 is an improvement of the second inequality in (1.4) for \(x\in(x^{\ast}, 1)\), where \(x^{\ast }=0.385\ldots\) is given by Theorem 3.2.
Remark 3.7
It is not difficult to verify that
for \(x\in(0.44, 0.45)\) and
for \(x\in(\theta_{0}, \theta_{1})\), where \(\theta_{0}=(0.95-\sqrt {0.5425})/2=0.106\ldots\) and \(\theta_{1}=(0.95+\sqrt {0.5425})/2=0.843\ldots\) . Therefore, the upper bound \((x^{2}+9/5)/(x+9/5)\) for \(\Gamma(x+1)\) given in (3.20) is better than that given in (1.4) for \(x\in(0.44, 0.45)\), and it is also better than that given in (1.2) for \(x\in(\theta_{0}, \theta_{1})\).
Remark 3.8
Let
Then numerical computations show that
Therefore, there exists \(\delta\in(0, 1/8)\) such that the lower bound for \(\Gamma(x+1)\) given in (3.20) is better than that given in (1.2) for \(x\in(\delta, 1/8+\delta)\cup(1/4-\delta, 1/4+\delta)\cup (3/8-\delta, 3/8+\delta)\cup(1/2-\delta, 1/2+\delta)\cup(5/8-\delta, 5/8+\delta)\cup(3/4-\delta, 3/4+\delta)\cup(7/8-\delta, 7/8+\delta)\).
4 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.
5 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.
References
Anderson, GD, Qiu, S-L: A monotoneity property of the gamma function. Proc. Am. Math. Soc. 125(11), 3355-3362 (1997)
Mitrinović, DS, Pečarić, JE, Fink, AM: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993)
Widder, DV: The Laplace Transform. Princeton University Press, Princeton (1941)
Selliah, JB: An inequality satisfied the gamma function. Can. Math. Bull. 19(1), 85-87 (1990)
Alzer, H: Some gamma function inequalities. Math. Comput. 60(201), 337-346 (1993)
Alzer, H: On a gamma function inequality of Gautschi. Proc. Edinb. Math. Soc. (2) 45(3), 589-600 (2002)
Zhang, X-M, Chu, Y-M: A double inequality for gamma function. J. Inequal. Appl. 2009, Article ID 503782 (2009)
Zhao, T-H, Chu, Y-M, Jiang, Y-P: Monotonic and logarithmically convex properties of a function involving gamma functions. J. Inequal. Appl. 2009, Article ID 728612 (2009)
Qi, F: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Article ID 493058 (2010)
Zhao, T-H, Chu, Y-M: A class of logarithmically completely monotonic functions associated with a gamma function. J. Inequal. Appl. 2010, Article ID 392431 (2010)
Zhao, T-H, Chu, Y-M, Wang, H: Logarithmically complete monotonicity properties related to the gamma function. Abstr. Appl. Anal. 2011, Article ID 896483 (2011)
Laforgia, A, Natalini, P: On an inequality for the ratio of gamma functions. Math. Inequal. Appl. 17(4), 1591-1599 (2014)
Wang, M-K, Chu, Y-M: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017)
Wang, M-K, Li, Y-M, Chu, Y-M: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. (2017). doi:10.1007/s11139-017-9888-3
Gautschi, W: Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38(1-4), 77-81 (1959)
Kershaw, D: Some extensions of W. Gautschi’s inequalities for the gamma function. Math. Comput. 41(164), 607-611 (1983)
Elezović, N, Giordano, C, Pečarić, J: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 3(2), 239-252 (2000)
Ivády, P: A note on a gamma function inequality. J. Math. Inequal. 3(2), 227-236 (2009)
Zhao, J-L, Guo, B-N, Qi, F: A refinement of a double inequality for the gamma function. Publ. Math. (Debr.) 80(3-4), 333-342 (2012)
Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
Yang, Z-H, Chu, Y-M, Tao, X-J: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 2014, Article ID 702718 (2014)
Yang, Z-H, Chu, Y-M, Zhang, X-H: Sharp bounds for psi function. Appl. Math. Comput. 268, 1055-1063 (2015)
Zhao, T-H, Yang, Z-H, Chu, Y-M: Monotonicity properties of a function involving the psi function with applications. J. Inequal. Appl. 2015, Article ID 193 (2015)
Acknowledgements
The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191) and the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yang, ZH., Qian, WM., Chu, YM. et al. On rational bounds for the gamma function. J Inequal Appl 2017, 210 (2017). https://doi.org/10.1186/s13660-017-1484-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1484-y