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 Open Access
Sharp twoparameter bounds for the identric mean
 Omran Kouba^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366001819172
© The Author(s) 2018
 Received: 26 September 2018
 Accepted: 14 November 2018
 Published: 22 November 2018
Abstract
Keywords
 Arithmetic Mean
 Geometric Mean
 Harmonic Mean
 Identric Mean
MSC
 26E60
 26D07
1 Introduction
The study of inequalities involving means has become very popular in recent years because of their applications in various kinds of areas of mathematics. Finding sharp bounds for inequalities is an important task in order to have more accurate results in the aforementioned areas.
Inequalities relating means in two variables have attracted and continue to attract the attention of mathematicians. Many articles studying the properties of means of two variables have been published, and there is a large body of mathematical literature about comparing pairs of means. The interested reader may consult [1–3, 5–7, 9–11] and the references therein.
In this paper we continue the search for nontrivial bounds for the identric mean by studying a new family of two parameter means of two variables. The article is organized as follows. In Sect. 2 we present and discuss the main results. Section 3 is devoted to the proof of several technical lemmas that will be useful for the proof of the main theorems, and in Sect. 4 the main results are discussed and proved.
2 Results and discussion
The aim of this work is to produce a general result comparing the identric mean to members of the family \((Q_{t,s})_{t\in [0,1/2]\times [1,+\infty )}\), which generalizes the results of [4] and [13].
3 Preliminaries
The following lemmas pave the way to the main theorem. In Lemma 3.1 we study a family of functions using simple methods from classical analysis.
Lemma 3.1
 (a)
The necessary and sufficient condition to have \(f_{u,s}(x)>0\) for \(x\in (0,1)\) is that \(3su\leq 1\).
 (b)
The necessary and sufficient condition to have \(f_{u,s}(x)<0\) for \(x\in (0,1)\) is that \(u+(2/e)^{2/s}\geq 1\).
Proof

First, \(T_{u,s}(0)\geq 0\), or equivalently \(3su\leq 1\). Again, we distinguish two cases:
 ∘:

If \(s=1\), then clearly the zero of \(T_{u,1}\) does not belong to \((0,1)\) and \(T_{u,s}\) has a positive sign on \((0,1)\).
 ∘:

If \(s>1\), then the coefficient of \(X^{2}\) in \(T_{u,s}\) is negative, and the fact that both \(T_{u,s}(0)\) and \(T_{u,s}(1)\) are nonnegative implies that the zeros \(z_{0}< z_{1}\) of \(T_{u,s}\) satisfy the inequality \(z_{0}\leq 0<1\leq z_{1}\). Hence, \(T_{u,s}\) has a positive sign on \((0,1)\) in this case also.
It follows that in this case \(h_{u,s}\) is increasing on \([0,1)\). But \(h_{u,s}(0)=0\), so \(h_{u,s}\) is positive on \((0,1)\). Therefore \(f_{u,s}\) is increasing on \((0,1)\). Finally, the fact that \(\lim_{x\to 0^{+}}f_{u,s}(x)=0\) implies that \(f_{u,s}(x)>0\) for every \(x\in (0,1)\) in this case.

Second, \(T_{u,s}(0)<0\), or equivalently \(3su> 1\). This means that \(T_{u,s}\) has a unique zero \(z_{0}\) in the interval \((0,1]\) (because \(\deg (T_{u,s})\leq 2\)).
 ∘:

If \(u=1\), then \(z_{0}=1\) and \(h_{1,s}\) is decreasing on \([0,1]\). But \(h_{1,s}(0)=0\), so \(h_{1,s}\) is negative on \((0,1)\). This shows that \(f_{1,s}\) is decreasing on \((0,1)\). Finally, we have \(\lim_{x\to 0^{+}}f_{1,s}(x)=0\), and consequently \(f_{1,s}(x)<0\) for every \(x\in (0,1)\).
 ∘:

If \(u<1\), then \(z_{0}\in (0,1)\). So \(h_{u,s}\) is decreasing on \([0,z_{0}]\) and increasing on \([z_{0},1]\). But \(h_{u,s}(0)=0\) so \(h_{u,s}(z_{0})<0\). On the other hand \(\lim_{x\to 1^{}}h_{u,s}(x)=+\infty \). So there exists a unique real number \(y_{0}\in (z_{0},1)\) such that \(h_{u,s}(y_{0})=0\). Thus \(h_{u,s}(x)<0\) for \(x\in (0,y_{0})\) and \(h_{u,s}(x)>0\) for \(x\in (y_{0},1)\). This implies that \(f_{u,s}\) is decreasing on \((0,y_{0})\) and increasing on \((y_{0},1)\). Finally we have \(\lim_{x\to 0^{+}}f_{u,s}(x)=0\) and \(\lim_{x\to 1^{}}f_{u,s}(x)= \ln ( e(1u)^{s/2}/2 ) \).
This shows that the necessary and sufficient condition for \(f_{u,s}\) to be negative on \((0,1)\) is that either \(u=1\) or \(u<1\) and \(\ln ( e(1u)^{s/2}/2 ) \leq 0\) which is equivalent to the condition \(1\leq u+(2/e)^{2/s}\).
This achieves the proof of Lemma 3.1. □
Lemma 3.2
 (a)For \(s\geq 1\) and \(t\in [0,1/2]\), we have$$ \ln \biggl( \frac{Q_{t,s}(a,b)}{A(a,b)} \biggr) =\frac{s}{2}\ln \bigl( 1(12t)^{2} v^{2} \bigr) . $$
 (b)Also, for the identric mean, we have$$ \ln \biggl( \frac{I(a,b)}{A(a,b)} \biggr) =1+\frac{1}{2}\ln \bigl(1v^{2}\bigr)+ \frac{1}{2v}\ln \biggl( \frac{1+v}{1v} \biggr) . $$
Proof
The next corollary is an immediate consequence of part (a) of Lemma 3.2.
Corollary 3.1
For distinct positive real numbers a and b, and for given \(s\ge 1\), the function \(t\mapsto Q_{t,s}(a,b) \) is continuous and increasing on the interval \([0,1/2]\).
Remark 3.1
4 The main theorem
Further, this result is used to obtain in Corollary 4.1 an upper bound counterpart of the inequality due to Seiffert [11] about the ratio \(A(a,b)/I(a,b)\).
Theorem 4.1
Proof
When \(s=2\), the definition of \(Q_{t,s}(a,b)\) given by (1) is reduced to the harmonic mean of \(ta+(1t)b\) and \((1t)a+tb\). So Theorem 4.1 yields in this case the following theorem from [4].
Theorem 4.2
([4])
Similarly, when \(s=1\), the definition of \(Q_{t,s}(a,b)\) given by (1) is reduced to the geometric mean of \(ta+(1t)b\) and \((1t)a+tb\). So Theorem 4.1 yields in this case the following theorem from [13].
Theorem 4.3
([13])
In the next corollary, the lower bound is an inequality due to Seiffert [11], it appears also in [10], while the upper bound is new and to be compared with the results of Sándor and Trif in [10].
Corollary 4.1
Proof
Theorem 4.4
5 Conclusion
In this work, we have considered a new twoparameter family of means, and we have compared them to the identric mean giving sharp upper and lower bounds.
Declarations
Acknowledgements
The author would like to thank the anonymous referees for reading this article carefully and providing valuable suggestions.
Availability of data and materials
Not applicable.
Funding
The author declares that he received no funding for doing this research.
Authors’ contributions
The author declares that he carried out this work by himself. The author read and approved the final manuscript.
Competing interests
The author declares that there are no competing interests with any individual or institution, and that he has not received any financial support to do this research.
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.
Authors’ Affiliations
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