On rational bounds for the gamma function.

In the article, we prove that the double inequality [Formula: see text] holds for all [Formula: see text], we present the best possible constants λ and μ such that [Formula: see text] for all [Formula: see text], and we find the value of [Formula: see text] in the interval [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is the classical gamma function, [Formula: see text] is Euler-Mascheroni constant and [Formula: see text] .

Introduction

For , the classical Euler gamma function and its logarithmic derivative, the so-called psi function [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 for all and . The well-known Bernstein theorem [3] states that a function f on is completely monotonic if and only if there exists a bounded and non-decreasing function such that converges for all .

Recently, the gamma function have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for can be found in the literature [4-14].

Due to and , we will only need to focus our attention on with . Gautschi [15] proved that the double inequality holds for all and .

Inequality (1.1) was generalized and improved by Kershaw [16] as follows: for all and .

Elezović, Giordano and Pečarić [17] established the double inequality for the gamma function being valid for all , 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 . 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 . The monotonicity of on the interval and the facts that and lead to the conclusion that for all , where is the Euler-Mascheroni constant.

Let Then we clearly see that for all , 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 are such that the double inequality holds for all . 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.

Lemma 2.1

See [20, Theorem 1.25]

Let , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are the functions If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

See [21, Lemma 7]

Let and with , for all , and Then there exists such that , for and for .

Lemma 2.3

See [22, Corollary 3.1]

The inequality holds for all .

Lemma 2.4

See [23, Corollary 3.3(ii)]

The double inequality holds for all .

Lemma 2.5

The inequalities hold for all .

Proof

Let , and and be defined by respectively. Then making use of the well-known formulas we get where is the hyperbolic sine function.

Note that for .

It follows from (2.6)-(2.9) and the Bernstein theorem for complete monotonicity property that the two functions and are completely monotonic on the interval .

Therefore, Lemma 2.5 follows easily from (2.4), (2.5) and the complete monotonicity of and on the interval together with the facts that  □

Lemma 2.6

The double inequality holds for all and .

Let , , and and be defined by respectively. Then simple computations lead to where

From (2.15) and (2.16) we clearly see that is strictly increasing on . We assert that the function is strictly increasing on . Indeed, if , then , and Lemma 2.1 and (2.13) together with the monotonicity of on lead to the conclusion that is strictly increasing on ; if , then , and Lemma 2.1 and (2.14) together with the monotonicity of on lead to the conclusion that is strictly increasing on .

Therefore, Lemma 2.6 follows easily from (2.11) and (2.12) together with the monotonicity of the function on . □

Let , , and , , and be defined by

Lemma 2.7

Let be defined by (2.19). Then for .

From (2.19) and the second inequality in Lemma 2.4 we have

Elaborated computations lead to for .

From (2.22) and (2.23) we get  □

Lemma 2.8

Let be defined by (2.19). Then

From (2.19) and the first inequality in Lemma 2.4 we have  □

Lemma 2.9

Let be defined by (2.18). Then for .

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 is strictly decreasing on , then (2.27) leads to the conclusion that for .

Therefore, for follows from (2.25) and (2.26) together with for . □

Lemma 2.10

Let and be defined by (2.20). Then there exists such that for and for .

Let Then simple computations lead to

It follows from the first inequalities in (2.2) and (2.3) together with the identity that

From (2.32)-(2.34) we have where with

From Lemma 2.2, (2.37) and (2.38) we clearly see that for .

Making use of Lemma 2.2 again, and (2.36) and (2.39) together with the facts that for and we know that for and . Then inequality (2.35) leads to the conclusion that the function is strictly increasing on for .

From (2.2) and the identity we get

Taking in the first inequality of (2.40) and in the second inequality of (2.40), one has

It follows from (2.31) and (2.41) that for .

Note that

Inequality (2.43) and equation (2.44) imply that for .

Therefore, Lemma 2.10 follows easily from (2.30), (2.42), (2.45) and the monotonicity of the function on the interval . □

Lemma 2.11

Let and be defined by (2.19). Then there exist with such that for and for .

It follows from (2.19) that for .

From Lemma 2.10 and we know that there exists such that the function is strictly decreasing on and strictly increasing on . Then Lemma 2.7 leads to the conclusion that

Therefore, there exist and such that for and for follow from (2.46)-(2.48) and the piecewise monotonicity of the function on the interval . □

Main results

Theorem 3.1

Let and  . Then the inequality holds for all if and only if , and the inequality holds for all if and only if , where and is the unique solution of the equation on the interval .

If inequality (3.1) holds for all , then follows easily from

Next, we prove that inequality (3.1) holds for all and and (3.2) holds for all if and only if .

Let , , 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 are positive, which implies that is strictly increasing on , then from (3.5) and (3.8) we know that there exists such that the function is strictly decreasing on and strictly increasing on .

It follows from (2.18) and (3.7) together with the piecewise monotonicity of the function on the interval that there exists such that is strictly increasing on and strictly decreasing on and is the unique solution of equation (3.4) on the interval .

Therefore, the desired results follow easily from (2.17), (3.3), (3.6) and the piecewise monotonicity of the function on the interval together with the fact that the function is strictly increasing.

Numerical computations show that and  . □

Theorem 3.2

The inequality holds for all , and its reverse inequality holds for all , where is the unique solution of the equation on the interval .

Let , and be, respectively, defined by (2.17), (2.18) and (2.19). Then simple computations lead to

From Lemma 2.11 and we know that there exist with such that is strictly increasing on and strictly decreasing on . We claim that

Indeed, if , then the piecewise monotonicity of the function on the interval and (3.11) lead to the conclusion that is strictly increasing on , which contradicts (3.10).

It follows from (3.11) and (3.12) together with the piecewise monotonicity of the function on the interval that there exist and such that is strictly increasing on and strictly decreasing on .

Therefore, Theorem 3.2 follows easily from (2.17) and (3.10) together with the piecewise monotonicity of on . Numerical computations show that  . □

Theorem 3.3

The double inequality holds for all with the best possible constant where is the unique solution of the equation on the interval .

Let , and 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 with such that for and for , and is strictly increasing on and strictly decreasing on . Then Lemmas 2.7-2.9 lead to the conclusion that and

From (2.18), (3.17), (3.18) and the piecewise monotonicity of on we clearly see that there exists such that is the unique solution of equation (3.15) on the interval , and is strictly decreasing on and strictly increasing on .

Equation (3.16) and the piecewise monotonicity of the function on the interval lead to the conclusion that for all .

Therefore, inequality (3.13) holds for all 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 . Numerical computations show that and  . □

Remark 3.4

From Theorems 3.1 and 3.3 we clearly see that the double inequality holds for all with and , the constant appears to be the best possible, but this is not true for , and a slightly smaller value for is possible. Unfortunately, we cannot find the best possible constant in the article; we leave this as an open problem for the reader.

Remark 3.5

From the monotonicity of the function we clearly see that both the upper and lower bounds for 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 , where is given by Theorem 3.2.

Remark 3.6

From Lemma 2.6, , , and for one has

Therefore, the lower bound for 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 and the second inequality in Theorem 3.2 is an improvement of the second inequality in (1.4) for , where is given by Theorem 3.2.

Remark 3.7

It is not difficult to verify that for and for , where and  . Therefore, the upper bound for given in (3.20) is better than that given in (1.4) for , and it is also better than that given in (1.2) for .

Remark 3.8

Let Then numerical computations show that

Therefore, there exists such that the lower bound for given in (3.20) is better than that given in (1.2) for .

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.