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# On the Keller limit and generalization

*Journal of Inequalities and Applications*
**volume 2016**, Article number: 97 (2016)

## Abstract

## Introduction motivation

The limit

is well known in the literature as the Keller’s limit, see [2]. Such a limit is very useful in many mathematical contexts and contributes as a tool for establishing some interesting inequalities [3–6].

In the recent paper [1], Mortici *et al.* have constructed a new proof of the limit and have discovered the following new results which generalize the Keller limit.

### Theorem 1

*Let*
*c*
*be any real number and let*

*Then*

The proof of Theorem 1 given in [1] is based on the following double inequality for every *x* in \(0< x\leq1\):

where

and

But, this proof has a major objection, namely, for the reader it is very difficult to observe the behavior of \(u_{n}(c)\) as \(n\rightarrow\infty\).

In this note, we will establish an integral expression of \(u_{n}(c)\), which tells us that Theorem 1 is a natural result.

## Main results

To establish an integral expression of \(u_{n}(c)\), we first recall the following result we obtained in [7].

### Theorem 2

*Let*
\(h(s)=\frac{\sin(\pi s)}{\pi}s^{s}(1-s)^{1-s}\), \(0\leq s\leq 1\). *Then for every*
\(x>0\), *we have*

*where*

In [8] (see also [9, 10]) Yang has proved that \(b_{2}=\frac{1}{24}\), \(b_{3}=\frac{1}{48}\).

Hence

Now, we establish an integral expression of \(u_{n}(c)\). Equation (2.1) implies the following results:

Hence by (2.2), (2.3), (2.6), and (2.7), we have

Note that

Therefore, from (2.8)-(2.11), we obtain the desired result:

where

From (2.12), we get immediately

## References

Mortici, C, Jang, X: Estimates of \((1+ 1/x)^{x}\) involved in Carleman’s inequality and Keller’s limit. Filomat

**7**, 1535-1539 (2015)Sandor, J, Debnath, L: On certain inequalities involving the constant

*e*and their applications. J. Math. Anal. Appl.**249**(2), 569-582 (2000)Polya, G, Szego, G: Problems and Theorems in Analysis, vol. I. Springer, New York (1972)

Hardy, GH, Littlewood, JE, Polya, G: Inequalities. Cambridge University Press, London (1952)

Mortici, C, Hu, Y: On some convergences to the constant

*e*and improvements of Carleman’s inequality. Carpath. J. Math.**31**, 249-254 (2015)Mortici, C, Hu, Y: On an infinite series for \((1+ 1/x)^{x}\). arXiv:1406.7814 [math.CA]

Hu, Y, Mortici, C: On the coefficients of an expansion of \((1+\frac{1}{x})^{x}\) related to Carleman’s inequality. arXiv:1401.2236 [math.CA]

Yang, X: Approximations for constant

*e*and their applications. J. Math. Anal. Appl.**262**, 651-659 (2001)Gyllenberg, M, Yan, P: On a conjecture by Yang. J. Math. Anal. Appl.

**264**, 687-690 (2001)Chen, H: On an infinite series for \((1+\frac{1}{x})^{x}\) and its application. Int. J. Math. Math. Sci.

**11**, 675-680 (2002)

## Acknowledgements

The work was supported by the National Natural Science Foundation of China, No. 11471103.

## Author information

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## Additional information

### Competing interests

The authors declare that there is no conflict of interests regarding the publication of this article.

### Authors’ contributions

The authors completed the paper together. They also read and approved the final manuscript.

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## About this article

### Cite this article

Hu, Y., Mortici, C. On the Keller limit and generalization.
*J Inequal Appl* **2016, **97 (2016). https://doi.org/10.1186/s13660-016-1042-z

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DOI: https://doi.org/10.1186/s13660-016-1042-z

### MSC

- 35B05
- 35B10

### Keywords

- Keller’s limit
- constant
*e* - integral expression