Abstract
By using methods in the theory of majorization, a double inequality for the gamma function is extended to the k-gamma function and the k-Riemann zeta function.
MSC:33B15, 26D07, 26B25.
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1 Introduction
The Euler gamma function is defined [1] for by
In 2005, by using a geometrical method, Alsina and Tomás [2] proved the following double inequality:
In 2009, Nguyen and Ngo [3] obtained the following generalization of the double inequality (2):
where , , , .
For , the function is defined [4] by
where .
The above definition is a generalization of the gamma function. For with , the function is given by the integral [4]
It satisfies the following properties [4–6]:
-
(i)
;
-
(ii)
.
For , the k-Riemann zeta function is defined [5] by the integral
Note that when k tends to 1 we obtain the known Riemann zeta function .
In this note, by using methods on the theory of majorization, we extended the double inequality (3) to the function and the k-Riemann zeta function, namely, we established the following theorems.
Theorem 1
where , , , , .
Theorem 2
where , , , , .
Substituting and () into (8) and taking into account that and , we obtain the following.
Corollary 1
where , , .
Remark 1 is Apéry’s constant [7].
2 Definitions and lemmas
We need the following definitions and auxiliary lemmas.
Let and .
-
(i)
We say y majorizes x (x is said to be majorized by y), denoted by , if for and , where and are rearrangements of x and y in a descending order.
-
(ii)
Let , a function is said to be a Schur-convex function on Ω if on Ω implies . A function φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.
Let and , .
-
(i)
A set is said to be a convex set if implies .
-
(ii)
Let be a convex set. A function φ: is said to be a convex function on Ω if
for all . A function φ is said to be a concave function on Ω if and only if −φ is a convex function on Ω.
-
(iii)
Let . A function is said to be a log-convex function on Ω if the function logφ is convex.
Lemma 1 [[8], p.186]
Let , , and . If for some k, , , , for , then .
Lemma 2 Let f, g be a continuous nonnegative functions defined on an interval . Then
is log-convex on .
Proof Let , by the Hölder integral inequality [[10], p.140], we have
i.e.
this means that is log-convex on . □
Remark 2 When , the results of Lemma 2 presented previously hold true.
Lemma 3 [[8], p.105]
Let g be a continuous nonnegative function defined on an interval . Then
is Schur-convex on if and only if logg is convex on I.
Lemma 4 Let
and
where , , , , . Then .
Proof It is clear that .
Without loss of generality, we may assume that . So . The following discussion is divided into two cases:
Case 1. . Notice that , and , , and we have
and
Hence from Lemma 1, it follows that .
Case 2. . Let denote the components of u in a decreasing order. There exist such that
Notice that , , and , and if , then
If , then
Hence from Definition 1(i), it follows that . □
Lemma 5 Let
and
where , , , , . Then .
Proof It is clear that .
The following discussion is divided into two cases:
Case 1. . Notice that and , , we have
and
Hence from the Lemma 1, it follows that .
Case 2. . Let denote the components of w in a decreasing order. There exist such that
Now notice that , and , we have
and
Hence from the Lemma 1, it follows that . □
The Schur-convexity described the ordering of majorization, the order-preserving functions were first comprehensively studied by Issai Schur in 1923. It has important applications in analytic inequalities, combinatorial optimization, special functions, probabilistic, statistical, and so on. See [8, 11–13].
3 Proof of main result
Proof of Theorem 1 Taking , , , , then
By Lemma 2, is log-convex on , and then from Lemma 3, is Schur-convex on . Combining Lemma 4 and Lemma 5, respectively, we have
and
i.e.
and
Thus, we have proved the double inequality (7).
The proof of Theorem 1 is completed. □
Proof of Theorem 2 Let
i.e.
Taking , , , , then
By Lemma 2, is log-convex on , and then from Lemma 3, is Schur-convex on . Combining Lemma 4 and Lemma 5, respectively, we have
and
i.e.
and
notice that .
Further, we have
and
Rearranging (18) and (19) gives the double inequality (8).
The proof of Theorem 2 is completed. □
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Acknowledgements
The work was supported by the National Natural Science Foundation of China (Grant No. 11101034) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407).
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The main idea of this paper was proposed by JZ and H-NS. This work was carried out in collaboration between both authors. They read and approved the final manuscript.
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Zhang, J., Shi, HN. Two double inequalities for k-gamma and k-Riemann zeta functions. J Inequal Appl 2014, 191 (2014). https://doi.org/10.1186/1029-242X-2014-191
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DOI: https://doi.org/10.1186/1029-242X-2014-191