Optimal bounds for the generalized Euler-Mascheroni constant.

We provide several sharp upper and lower bounds for the generalized Euler-Mascheroni constant. As consequences, some previous bounds for the Euler-Mascheroni constant are improved.

Introduction

Let . Then the generalized Euler–Mascheroni constant [1] is given by

We clearly see that the generalized Euler–Mascheroni constant is the natural generalization of the classical Euler–Mascheroni constant [2-5]

Recently, the two bounds for γ and have attracted the attention of many mathematicians. In particular, many remarkable inequalities and asymptotic formulas for γ and can be found in the literature [6-10].

Let

Negoi [11] proved that the two-sided inequality is valid for .

Qiu and Vuorinen [12] proved that the two-sided inequality is valid for if and only if and .

In [13], DeTemple proved that the double inequality holds for all .

Chen [14] proved that and are the best possible constants such that the double inequality holds for .

Sîntămărian [15], and Berinde and Mortici [16] proved that the double inequalities are valid for all and .

The main purpose of this article is to find the best possible constants , , , , , , and such that the double inequalities hold for all and and improve the bounds for the Euler–Mascheroni constant.

Main results

In order to prove our main results, we need several formulas and lemmas which we present in this section.

For , the classical gamma function Γ and its logarithmic derivative, the so-called psi function ψ are defined [17-24] as respectively.

The psi function ψ has the recurrence and asymptotic formulas [25] as follows:

Lemma 2.1

(See [14, Proof of Theorem 1])

The function is strictly decreasing on with .

Lemma 2.2

(See [26, Proof of Theorem 1], [27, Remark 4])

The function is strictly decreasing on with .

Lemma 2.3

(See [28, Proof of Theorem 2])

The function is strictly decreasing on with .

Lemma 2.4

(See [29, Theorem 1.2(2)])

The function is strictly increasing from onto .

Theorem 2.5

Let and be, respectively, defined by (1.1) and (2.3). Then and are the best possible constants such that the double inequality holds for all and .

Proof

It follows from (1.1), (2.1) and (2.2) that

From (2.3) and (2.8) we clearly see that inequality (2.7) is equivalent to

Therefore, Theorem 2.5 follows easily from Lemma 2.1 and (2.19). □

Theorem 2.6

Let and be, respectively, defined by (1.2) and (2.4). Then and are the best possible constants such that the double inequality holds for all and .

It follows from (1.2), (2.1) and (2.2) that

From (2.4) and (2.11) we clearly see that inequality (2.10) can be rewritten as

Therefore, Theorem 2.6 follows easily from Lemma 2.2 and (2.12). □

Remark 2.1

We clearly see that both the upper and the lower bounds given in (2.10) for are better than that given in (1.10) for due to  .

Theorem 2.7

Let and be, respectively, defined by (1.3) and (2.5). Then and are the best possible constants such that the double inequality holds for all and .

From (1.3), (2.1) and (2.2) we have

It follows from (2.5) and (2.14) that inequality (2.13) can be rewritten as

Therefore, Theorem 2.7 follows easily from Lemma 2.3 and (2.15). □

Theorem 2.8

Let and be, respectively, defined by (1.4) and (2.6). Then and are the best possible constants such that the double inequality holds for all and .

It follows from (1.4), (2.1) and (2.2) that

From (2.6) and (2.17) we clearly see that inequality (2.16) is equivalent to

Therefore, Theorem 2.8 follows easily from Lemma 2.4 and (2.18). □

Remark 2.2

Note that

It follows from (1.4), Theorem 2.5, Theorem 2.8 and (2.19) that , , and are the best possible constants such that the double inequalities hold for all and .

We clearly see that the two inequalities (2.20) and (2.21) are the improvements of the inequality (1.9) for .

Let and and Then

Therefore, Theorems 2.5–2.8 lead to Corollaries 2.1–2.5 immediately.

Corollary 2.1

The double inequality holds for all .

Corollary 2.2

The double inequality holds for all .

Corollary 2.3

The double inequality holds for all .

Corollary 2.4

The double inequality holds for all .

Corollary 2.5

The double inequality holds for all .

Remark 2.3

We clearly see that the upper bound given in (2.22) is better than that given in (1.6) for due to is the solution of the inequality , the lower bound given in (2.23) is better than that given in (1.8) for due to  , both the upper and the lower bounds given in (2.24) are improvements of that given in (1.7) for , inequality (2.25) is stronger than inequality (1.5) for , the lower bound given in (2.26) is better than that given in (1.6) for , and the upper bound given in (2.26) is stronger than that given in (1.6) for due to being the solution of the inequality

Results and discussion

As the natural generalization of the Euler–Mascheroni constant the generalized Euler–Mascheroni constant is defined by for .

Recently, the evaluations for γ and have been the subject of intensive research. In the article, we provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant . As applications, we improve some previously results on the Euler–Mascheroni constant γ. The idea presented may stimulate further research in the theory of special function.

Conclusion

In this paper, we present several best possible approximations for the generalized Euler–Mascheroni constant and improve some well-known bounds for the Euler–Mascheroni constant,