Skip to main content

Optimal bounds for the generalized Euler–Mascheroni constant

Abstract

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.

1 Introduction

Let \(a>0\). Then the generalized Euler–Mascheroni constant \(\gamma (a)\) [1] is given by

$$ \gamma (a)=\lim_{n\rightarrow \infty } \biggl[ \frac{1}{a}+ \frac{1}{a+1}+ \cdots +\frac{1}{a+n-1}-\log \biggl( \frac{a+n-1}{a} \biggr) \biggr] . $$

We clearly see that the generalized Euler–Mascheroni constant \(\gamma (a)\) is the natural generalization of the classical Euler–Mascheroni constant [25]

$$ \gamma =\gamma (1)=\lim_{n\rightarrow \infty } \biggl( 1+\frac{1}{2}+ \frac{1}{3}+\cdots +\frac{1}{n}-\log n \biggr) =0.577215664901\ldots \,. $$

Recently, the two bounds for γ and \(\gamma (a)\) have attracted the attention of many mathematicians. In particular, many remarkable inequalities and asymptotic formulas for γ and \(\gamma (a)\) can be found in the literature [610].

Let

$$\begin{aligned}& \gamma_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}-\log n, \\& R_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}-\log \biggl( n+ \frac{1}{2} \biggr) , \\& S_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n-1}+\frac{1}{2n}- \log n, \\& T_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}-\log \biggl( n+ \frac{1}{2}+\frac{1}{24n} \biggr) , \\& y_{n}(a)=\frac{1}{a}+\frac{1}{a+1}+\cdots + \frac{1}{a+n-1}-\log \biggl( \frac{a+n-1}{a} \biggr) , \\& \alpha_{n}(a)=\frac{1}{a}+\frac{1}{a+1}+\cdots + \frac{1}{a+n-2}+ \frac{1}{2(a+n-1)}-\log \biggl( \frac{a+n-1}{a} \biggr) , \end{aligned}$$
(1.1)
$$\begin{aligned}& \beta_{n}(a)=\frac{1}{a}+\frac{1}{a+1}+\frac{1}{a+n-1}- \log \biggl( \frac{a+n-1/2}{a} \biggr) , \end{aligned}$$
(1.2)
$$\begin{aligned}& \lambda_{n}(a)=\frac{1}{a}+\frac{1}{a+1}+\frac{1}{a+n-1}- \log \biggl( \frac{a+n-1/2}{a}+ \frac{1}{24a(a+n-1)} \biggr) , \end{aligned}$$
(1.3)
$$\begin{aligned}& \mu_{n}(a)=y_{n}(a)-\frac{1}{2(a+n-1)}+\frac{1}{12(a+n-1)^{2}}- \frac{1}{120(a+n-1)^{4}}. \end{aligned}$$
(1.4)

Negoi [11] proved that the two-sided inequality

$$ \frac{1}{48(n+1)^{3}}\leq \gamma -T_{n}\leq \frac{1}{48n^{3}} $$
(1.5)

is valid for \(n\geq 1\).

Qiu and Vuorinen [12] proved that the two-sided inequality

$$ \frac{1}{2n}-\frac{\lambda }{n^{2}}< \gamma_{n}-\gamma \leq \frac{1}{2n}-\frac{\mu }{n^{2}} $$
(1.6)

is valid for \(n\geq 1\) if and only if \(\lambda \geq 1/12\) and \(\mu \leq \gamma -1/2\).

In [13], DeTemple proved that the double inequality

$$ \frac{1}{24(n+1)^{2}}\leq R_{n}-\gamma \leq \frac{1}{24n^{2}} $$
(1.7)

holds for all \(n\geq 1\).

Chen [14] proved that \(\alpha =1/\sqrt{12\gamma -6}-1\) and \(\beta =0\) are the best possible constants such that the double inequality

$$ \frac{1}{12(n+\alpha )^{2}}\leq \gamma -S_{n}\leq \frac{1}{12(n+ \beta )^{2}} $$
(1.8)

holds for \(n\geq 1\).

Sîntămărian [15], and Berinde and Mortici [16] proved that the double inequalities

$$\begin{aligned}& \frac{1}{2(n+a)}\leq y_{n}(a)-\gamma (a)\leq \frac{1}{2(n+a-1)}, \end{aligned}$$
(1.9)
$$\begin{aligned}& \frac{1}{24(n+a)^{2}}\leq \beta_{n}(a)-\gamma (a)\leq \frac{1}{24(n+a-1)^{2}} \end{aligned}$$
(1.10)

are valid for all \(a>0\) and \(n\geq 1\).

The main purpose of this article is to find the best possible constants \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\), \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\) and \(\beta_{4}\) such that the double inequalities

$$\begin{aligned}& \frac{1}{12(a+n-\alpha_{1})^{2}}\leq \gamma (a)-\alpha_{n}(a)< \frac{1}{12(a+n- \beta_{1})^{2}}, \\& \frac{1}{24(a+n-\alpha_{2})^{2}}\leq \beta_{n}(a)-\gamma (a)< \frac{1}{24(a+n- \beta_{2})^{2}}, \\& \frac{1}{48(a+n-\alpha_{3})^{3}}\leq \gamma (a)-\lambda_{n}(a)< \frac{1}{48(a+n- \beta_{3})^{3}}, \\& \frac{\alpha_{4}}{(a+n-1)^{6}}\leq \gamma (a)-\mu_{n}(a)< \frac{\beta _{4}}{(a+n-1)^{6}} \end{aligned}$$

hold for all \(a>0\) and \(n\geq n_{0}\) and improve the bounds for the Euler–Mascheroni constant.

2 Main results

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

For \(x>0\), the classical gamma function Γ and its logarithmic derivative, the so-called psi function ψ are defined [1724] as

$$ \Gamma (x)= \int_{0}^{\infty }t^{x-1}e^{-t}dt, \qquad \psi (x)=\frac{ \Gamma^{\prime }(x)}{\Gamma (x)}, $$

respectively.

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

$$\begin{aligned}& \psi (x+1)=\psi (x)+\frac{1}{x}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \psi (x)\sim \log x-\frac{1}{2x}-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}- \frac{1}{252x ^{6}}+\cdots \quad (x\rightarrow \infty ). \end{aligned}$$
(2.2)

Lemma 2.1

(See [14, Proof of Theorem 1])

The function

$$ f_{1}(x)=\frac{1}{ \sqrt{12 ( \log x-\psi (x+1)+\frac{1}{2x} ) }}-x $$
(2.3)

is strictly decreasing on \([2, \infty )\) with \(f_{1}(\infty )=0\).

Lemma 2.2

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

The function

$$ f_{2}(x)=\frac{1}{\sqrt{24 ( \psi (x+1)-\log (x+1/2) ) }}-x $$
(2.4)

is strictly decreasing on \([2, \infty )\) with \(f_{2}(\infty )=1/2\).

Lemma 2.3

(See [28, Proof of Theorem 2])

The function

$$ f_{3}(x)=\frac{1}{\sqrt[3]{48 [ \log ( x+\frac{1}{2}+ \frac{1}{24x} ) -\psi (x+1) ] }}-x $$
(2.5)

is strictly decreasing on \([5, \infty )\) with \(f_{3}(\infty )=83/360\).

Lemma 2.4

(See [29, Theorem 1.2(2)])

The function

$$ f_{4}(x)=\frac{x^{2}}{120}- \biggl( \psi (x)-\log x+ \frac{1}{2x}+\frac{1}{12x ^{2}} \biggr) x^{6} $$
(2.6)

is strictly increasing from \((0, \infty )\) onto \((0, 1/252)\).

Theorem 2.5

Let \(\alpha_{n}(a)\) and \(f_{1}(x)\) be, respectively, defined by (1.1) and (2.3). Then \(\alpha_{1}=1-f_{1}(a+2)\) and \(\beta_{1}=1\) are the best possible constants such that the double inequality

$$ \frac{1}{12(a+n-\alpha_{1})^{2}}\leq \gamma (a)-\alpha_{n}(a)< \frac{1}{12(a+n- \beta_{1})^{2}} $$
(2.7)

holds for all \(a>0\) and \(n\geq 3\).

Proof

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

$$\begin{aligned} \gamma (a)-\alpha_{n}(a)&=\lim_{n\rightarrow \infty } \biggl[ \psi (n+a)- \psi (a)-\log \biggl( \frac{a+n-1}{a} \biggr) \biggr] \\ &\quad {}- \biggl[ \psi (n+a)-\psi (a)-\frac{1}{2(a+n-1)}-\log \biggl( \frac{a+n-1}{a} \biggr) \biggr] \\ &=\lim_{n\rightarrow \infty }\bigl[\psi (n+a)-\log (a+n-1)\bigr] \\ &\quad {}-\psi (n+a)+\frac{1}{2(a+n-1)}+\log (a+n-1) \\ &=\log (a+n-1)-\psi (n+a)+\frac{1}{2(a+n-1)}. \end{aligned}$$
(2.8)

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

$$ \alpha_{1}\leq 1-f_{1}(n+a-1)< \beta_{1}. $$
(2.9)

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

Theorem 2.6

Let \(\beta_{n}(a)\) and \(f_{2}(x)\) be, respectively, defined by (1.2) and (2.4). Then \(\alpha_{2}=1-f_{2}(a+2)\) and \(\beta_{2}=1/2\) are the best possible constants such that the double inequality

$$ \frac{1}{24(a+n-\alpha_{2})^{2}}\leq \beta_{n}(a)-\gamma (a)< \frac{1}{24(a+n- \beta_{2})^{2}} $$
(2.10)

holds for all \(a>0\) and \(n\geq 3\).

Proof

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

$$ \beta_{n}(a)-\gamma (a)=\psi (n+a)-\log \biggl( a+n-\frac{1}{2} \biggr) . $$
(2.11)

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

$$ \alpha_{2}\leq 1-f_{2}(n+a-1)< \beta_{2}. $$
(2.12)

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 \(\beta_{n}(a)-\gamma (a)\) are better than that given in (1.10) for \(n\geq 3\) due to \(1-f_{2}(2)=3-1/\sqrt{36-24( \gamma +\log 5-\log 2)}=0.466904841516\ldots \) .

Theorem 2.7

Let \(\lambda_{n}(a)\) and \(f_{3}(x)\) be, respectively, defined by (1.3) and (2.5). Then \(\alpha_{3}=1-f_{3}(a+5)\) and \(\beta_{3}=277/360\) are the best possible constants such that the double inequality

$$ \frac{1}{48(a+n-\alpha_{3})^{3}}\leq \gamma (a)-\lambda_{n}(a)< \frac{1}{48(a+n- \beta_{3})^{3}} $$
(2.13)

holds for all \(a>0\) and \(n\geq 6\).

Proof

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

$$ \gamma (a)-\lambda_{n}(a)=\log \biggl( a+n- \frac{1}{2}+ \frac{1}{24(a+n-1)} \biggr) -\psi (a+n). $$
(2.14)

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

$$ \alpha_{3}\leq 1-f_{3}(a+n-1)< \beta_{3}. $$
(2.15)

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

Theorem 2.8

Let \(\mu_{n}(a)\) and \(f_{4}(x)\) be, respectively, defined by (1.4) and (2.6). Then \(\alpha_{4}=f_{4}(a)\) and \(\beta_{4}=1/252\) are the best possible constants such that the double inequality

$$ \frac{\alpha_{4}}{(a+n-1)^{6}}\leq \gamma (a)-\mu_{n}(a)< \frac{\beta _{4}}{(a+n-1)^{6}} $$
(2.16)

holds for all \(a>0\) and \(n\geq 1\).

Proof

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

$$\begin{aligned} &\gamma (a)-\mu_{n}(a) \\ &\quad =\frac{1}{120(n+a-1)^{4}} \\ &\quad \quad {}- \biggl[ \psi (n+a-1)-\log (n+a-1)+ \frac{1}{2(n+a-1)}+ \frac{1}{12(n+a-1)^{2}} \biggr] . \end{aligned}$$
(2.17)

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

$$ \alpha_{4}\leq f_{4}(n+a-1)< \beta_{4}. $$
(2.18)

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

Remark 2.2

Note that

$$ \alpha_{n}(a)=y_{n}(a)-\frac{1}{2(a+n-1)}. $$
(2.19)

It follows from (1.4), Theorem 2.5, Theorem 2.8 and (2.19) that \(\alpha_{1}=1-f_{1}(a+2)\), \(\beta_{1}=1\), \(\alpha_{4}=f_{4}(a)\) and \(\beta_{4}=1/252\) are the best possible constants such that the double inequalities

$$\begin{aligned}& \frac{1}{2(a+n-1)}-\frac{1}{12(a+n-\beta_{1})^{2}}< y_{n}(a)-\gamma (a) \\& \hphantom{\frac{1}{2(a+n-1)}-\frac{1}{12(a+n-\beta_{1})^{2}}}\leq \frac{1}{2(a+n-1)}-\frac{1}{12(a+n-\alpha_{1})^{2}}, \end{aligned}$$
(2.20)
$$\begin{aligned}& \frac{1}{2(a+n-1)}-\frac{1}{12(a+n-1)^{2}}+\frac{1}{120(a+n-1)^{4}}-\frac{ \beta_{4}}{(a+n-1)^{6}} \\& \quad < y_{n}(a)-\gamma (a) \\& \quad \leq \frac{1}{2(a+n-1)}-\frac{1}{12(a+n-1)^{2}}+ \frac{1}{120(a+n-1)^{4}}-\frac{\alpha_{4}}{(a+n-1)^{6}}, \end{aligned}$$
(2.21)

hold for all \(a>0\) and \(n\geq 3\).

We clearly see that the two inequalities (2.20) and (2.21) are the improvements of the inequality (1.9) for \(n\geq 3\).

Let \(a=1\) and

$$\begin{aligned}& c_{1}=f_{1}(3)=1/\sqrt{12(\gamma +\log 3)-20}-3=0.015998 \ldots \,, \\& c_{2}=f_{2}(3)=1/ \sqrt{44-24(\gamma +\log 7-\log 2)}-3=0.5242567\ldots \,, \\& c_{3}=f_{3}(6)=-6+1/\sqrt[3]{48(\gamma -49/20+\log 937-\log 144)}=0.242347\ldots \end{aligned}$$

and

$$\begin{aligned}& c_{4}=f_{4}(1)=\gamma -23/40=0.00221566\ldots \,. \end{aligned}$$

Then

$$\begin{aligned}& \gamma (1)=\gamma , \qquad \alpha_{n}(1)=\gamma_{n}- \frac{1}{2n}=S_{n}, \qquad \beta_{n}(1)=R_{n}, \\& \lambda_{n}(1)=T_{n}, \qquad \mu_{n}(1)= \gamma_{n}-\frac{1}{2n}+\frac{1}{12n ^{2}}-\frac{1}{120n^{4}}. \end{aligned}$$

Therefore, Theorems 2.52.8 lead to Corollaries 2.12.5 immediately.

Corollary 2.1

The double inequality

$$ \frac{1}{2n}-\frac{1}{12n^{2}}< \gamma_{n}-\gamma \leq \frac{1}{2n}-\frac{1}{12(n+c _{1})^{2}} $$
(2.22)

holds for all \(n\geq 3\).

Corollary 2.2

The double inequality

$$ \frac{1}{12(n+c_{1})^{2}}\leq \gamma -S_{n}< \frac{1}{12n^{2}} $$
(2.23)

holds for all \(n\geq 3\).

Corollary 2.3

The double inequality

$$ \frac{1}{24(n+c_{2})^{2}}\leq R_{n}-\gamma < \frac{1}{24(n+1/2)^{2}} $$
(2.24)

holds for all \(n\geq 3\).

Corollary 2.4

The double inequality

$$ \frac{1}{48(n+c_{3})^{2}}\leq \gamma -T_{n}< \frac{1}{48(n+83/360)^{2}} $$
(2.25)

holds for all \(n\geq 6\).

Corollary 2.5

The double inequality

$$ \frac{1}{2n}-\frac{1}{12n^{2}}+\frac{1}{120n^{4}}-\frac{1}{252n^{6}} < \gamma_{n}-\gamma \leq \frac{1}{2n}-\frac{1}{12n^{2}}+ \frac{1}{120n ^{4}}-\frac{c_{4}}{n^{6}} $$
(2.26)

holds for all \(n\geq 1\).

Remark 2.3

We clearly see that the upper bound given in (2.22) is better than that given in (1.6) for \(n\geq 3\) due to \(n>\sqrt{12(\gamma -1/2)}c_{1}/(1-\sqrt{12(\gamma -1/2)})=0.4117\ldots \) is the solution of the inequality \(1/[12(n+c_{1})^{2}]>( \gamma -1/2)/n^{2}\), the lower bound given in (2.23) is better than that given in (1.8) for \(n\geq 3\) due to \(c_{1}<1\sqrt{12\gamma -6}-1=0.03885914\ldots\) , both the upper and the lower bounds given in (2.24) are improvements of that given in (1.7) for \(n\geq 3\), inequality (2.25) is stronger than inequality (1.5) for \(n\geq 6\), the lower bound given in (2.26) is better than that given in (1.6) for \(n\geq 1\), and the upper bound given in (2.26) is stronger than that given in (1.6) for \(n\geq 2\) due to

$$ n> \biggl( \frac{1+\sqrt{1-4800[1-12(\gamma -1/2)]c_{4}}}{20[1-12( \gamma -1/2)]} \biggr) ^{1/2}=1.00000000006823\ldots $$

being the solution of the inequality

$$ \frac{1}{2n}-\frac{1}{12n^{2}}+\frac{1}{120n^{4}}-\frac{c_{4}}{n^{6}}< \frac{1}{2n}-\frac{\gamma -1/2}{n^{2}}. $$

3 Results and discussion

As the natural generalization of the Euler–Mascheroni constant

$$ \gamma =\lim_{n\rightarrow \infty } \biggl( 1+\frac{1}{2}+ \frac{1}{3}+ \cdots +\frac{1}{n}-\log n \biggr) =0.5772156649\ldots\, , $$

the generalized Euler–Mascheroni constant is defined by

$$ \gamma (a)=\lim_{n\rightarrow \infty } \biggl[ \frac{1}{a}+ \frac{1}{a+1}+ \cdots +\frac{1}{a+n-1}-\log \biggl( \frac{a+n-1}{a} \biggr) \biggr] $$

for \(a>0\).

Recently, the evaluations for γ and \(\gamma (a)\) have been the subject of intensive research. In the article, we provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant \(\gamma (a)\). 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.

4 Conclusion

In this paper, we present several best possible approximations for the generalized Euler–Mascheroni constant

$$ \gamma (a)=\lim_{n\rightarrow \infty } \biggl[ \frac{1}{a}+ \frac{1}{a+1}+ \cdots +\frac{1}{a+n-1}-\log \biggl( \frac{a+n-1}{a} \biggr) \biggr] $$

and improve some well-known bounds for the Euler–Mascheroni constant,

$$ \gamma =\lim_{n\rightarrow \infty } \biggl( 1+\frac{1}{2}+ \frac{1}{3}+ \cdots +\frac{1}{n}-\log n \biggr) =0.5772156649\ldots\,. $$

References

  1. Knopp, K.: Theory and Applications of Infinite Series. Dover Publications, New York (1990)

    Google Scholar 

  2. Chen, C.-P., Qi, F.: The best bounds of the n-th harmonic number. Glob. J. Appl. Math. Math. Sci. 1(1), 41–49 (2008)

    Google Scholar 

  3. Niu, D.-W., Zhang, Y.-J., Qi, F.: A double inequality for the harmonic number in terms of the hyperbolic cosine. Turk. J. Anal. Number Theory 2(6), 223–225 (2014)

    Article  Google Scholar 

  4. Wang, M.-K., Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 521–537 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch–Pfluger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)

    Google Scholar 

  6. Alzer, H.: Inequalities for the gamma and polygamma functions. Abh. Math. Semin. Univ. Hamb. 68, 363–372 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23(1), 97–100 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mortici, C.: Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant. Anal. Appl. 8(1), 99–107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mortici, C.: Improved convergence towards generalized Euler–Mascheroni constant. Appl. Math. Comput. 215(9), 3443–3448 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler–Mascheroni constants. Appl. Math. Lett. 24(7), 1239–1244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Negoi, T.: A faster convergence to the constant of Euler. Gaz. Mat., Ser. A 15, 111–113 (1997)

    Google Scholar 

  12. Qiu, S.-L., Vuorinen, M.: Some properties of the gamma and psi functions, with applications. Math. Comput. 74(250), 723–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. DeTemple, D.W.: A geometric look at sequences that converge to Euler’s constant. Coll. Math. J. 37(2), 128–131 (2006)

    Article  MathSciNet  Google Scholar 

  14. Chen, C.-P.: The best bounds in Vernescu’s inequalities for the Euler’s constant. RGMIA Res. Rep. Collect. 12(3), Article ID 11 (2009)

    Google Scholar 

  15. Sîntămărian, A.: A generalization of Euler’s constant. Numer. Algorithms 46(2), 141–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Berinde, V., Mortici, C.: New sharp estimates of the generalized Euler–Mascheroni constant. Math. Inequal. Appl. 16(1), 279–288 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012). https://doi.org/10.1007/s00025-010-0090-9

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, B.-N., Qi, F.: Sharp inequalities for the psi function and harmonic numbers. Analysis 34(2), 201–208 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Yang, Z.-H., Chu, Y.-M., Zhang, X.-H.: Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind. J. Nonlinear Sci. Appl. 10(3), 929–936 (2017)

    Article  MathSciNet  Google Scholar 

  21. Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautsch inequality for parameter \(0< p<1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Guo, B.-N., Qi, F.: Sharp bounds for harmonic numbers. Appl. Math. Comput. 218(3), 991–995 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21(2), 469–479 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U. S. Government Printing Office, Washington (1964)

    MATH  Google Scholar 

  26. Chen, C.-P.: Inequalities for the Euler–Mascheroni constant. Appl. Math. Lett. 23(2), 161–164 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, C.-P.: Inequalities and monotonicity properties for some special functions. J. Math. Inequal. 3(1), 79–91 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chen, C.-P., Mortici, C.: Limits and inequalities associated with the Euler–Mascheroni constant. Appl. Math. Comput. 219(18), 9755–9761 (2013). https://doi.org/10.1016/j.amc.2013.03.089

    MathSciNet  MATH  Google Scholar 

  29. Qiu, S.-L., Zhao, X.: Some properties of the psi function and evaluations of γ. Appl. Math. J. Chin. Univ. Ser. A 31(1), 103–111 (2016). https://doi.org/10.1007/s11766-016-3272-8

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 11401531, 11601485), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101), the Natural Science Foundation of Zhejiang Province (Grant No. LQ17A010010) and the Science Foundation of Zhejiang Sci-Tech University (Grant No. 14062093-Y).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yu-Ming Chu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, TR., Han, BW., Ma, XY. et al. Optimal bounds for the generalized Euler–Mascheroni constant. J Inequal Appl 2018, 118 (2018). https://doi.org/10.1186/s13660-018-1711-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-018-1711-1

MSC

Keywords