Abstract
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1 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.
2 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
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Acknowledgements
The work was supported by the National Natural Science Foundation of China, No. 11471103.
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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