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Sharp twoparameter bounds for the identric mean
Journal of Inequalities and Applications volume 2018, Article number: 322 (2018)
Abstract
For \(t\in [0,1/2]\) and \(s\ge 1\), we consider the twoparameter family of means
where A and G denote the arithmetic and geometric means. Sharp bounds for the identric mean in terms of \(Q_{t,s}\) are obtained. Our results generalize and extend bounds due to Chu et al. (Abstr. Appl. Anal. 2011:657935, 2011) and to Wang et al. (Appl. Math. Lett. 25:471–475, 2012).
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.
Let us fix some notation in order to describe our results. For distinct positive real numbers a and b, we recall that the arithmetic mean \(A(a,b)\), the geometric mean \(G(a,b)\), the harmonic mean \(H(a,b)\), and the identric mean \(I(a,b)\) are respectively defined by
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 [1,2,3, 5,6,7, 9,10,11] and the references therein.
For example, Alzer and Qui considered in [3] the following inequality relating the identric, geometric, and arithmetic means:
They proved that it holds, for every distinct positive numbers a and b, if and only if \(\alpha \leq 2/3\) and \(\beta \geq 2/e\).
This was later complemented by Trif [12] who proved that, for \(p\geq 2\) and every distinct positive number a and b, we have
if and only if \(\alpha \leq (2/e)^{p}\) and \(\beta \geq 2/3\).
Similarly, it is proved in [5] that the inequality
holds true for every distinct positive number a and b if and only if \(p\geq \ln ( \frac{3}{2} ) /\ln ( \frac{e}{2} ) \approx 1.3214\), and that the reverse inequality holds true for every distinct positive number a and b if and only if \(p\leq 6/5= 1.2\), this generalizes the results of [8] and [12].
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
Motivated by the works [4] and [13], we consider the two parameter family of means \(Q_{t,s}(a,b)\) defined for \(s\geq 1\), \(t\in [0,1/2]\), and any positive real numbers a and b by
Indeed, the authors in [4, 13] compare the identric mean to
see [4, Theorem 1.1], and to
see [13, Theorem 1.1].
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].
Indeed, in Corollary 3.1 we prove that, for distinct positive real numbers a and b and given \(s\ge 1\), the function \(t\mapsto Q _{t,s}(a,b)\) is continuous and increasing. Further, for \(s\geq 1\) and every distinct positive number a and b, we have
Thus, for given \(s\geq 1\), it is natural to consider the sets
Because \(t\mapsto Q_{t,s}(a,b)\) is increasing, it is clear that \(\mathcal{L}_{s}\) and \(\mathcal{U}_{s}\) are both intervals. In Theorem 4.1 we prove that, for \(s\geq 1\), \(\mathcal{L}_{s}=[0,p _{s}]\), and \(\mathcal{U}_{s}=[q_{s},1/2]\), where
This extends the results of Chu et al. [4] and Wang et al. [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
For \(s\geq 1\) and \(u\in [0,1]\), we consider the real function \(f_{u,s}\) defined on \([0,1)\) by

(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
We consider only the case \(u\in (0,1]\), since \(f_{0,s}\) is independent of s and positive on \((0,1)\). It is straightforward to see that \(f_{u,s}^{\prime }(x)=h_{u,s}(x)/x^{2}\), where
and that
where \(T_{u,s}\) is the trinomial defined by
Noting that \(T_{u,s}(1)=(1u)^{2}\geq 0\) and \(T_{u,s}(0)=13su\), we see that we have two cases:

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. □
Next we introduce the set \({\mathcal{D}}\) defined as follows:
It is sufficient to consider couples \((a,b)\) from \({\mathcal{D}}\), since the considered means are symmetric functions of their arguments. Lemma 3.2 explains why the family of functions studied in Lemma 3.1 is important to our study.
Lemma 3.2
Consider \((a,b)\in {\mathcal{D}}\) and let \(v=\frac{ab}{a+b}\).

(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
Indeed, (a) follows from the simple fact that
To see (b), we note that
This concludes the proof of Lemma 3.2. □
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
Combining (a) and (b) from Lemma 3.2, we see immediately that if \(f_{u,s}\) is the function defined in Lemma 3.1 then, for every \((a,b)\in {\mathcal{D}}\), we have
This explains the importance of the family of functions in Lemma 3.1 to our study.
4 The main theorem
In this section we prove our main result which states that, for \(s\ge 1\) and \(u,v\in [0,1/2]\), the double inequality \(Q_{u,s}(a,b)< I_{(}a,b)< Q_{v,s}(a,b)\) holds for all distinct positive real numbers a and b if and only if
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
Let s be a real number such that \(s\geq 1\), and define the sets
Then
Proof
First note that
So, using Corollary 3.1, we see that \(t\in {\mathcal{L}}_{s}\) if and only if \(f_{(12t)^{2},s}(x)<0\) for every \(x\in (0,1)\). However, according to Lemma 3.1, this is equivalent to \((12t)^{2}+(2/e)^{2/s} \geq 1\) or \((1\sqrt{1(2/e)^{2/s}})/2\geq t\). This proves that
Similarly, using Corollary 3.1, we see that \(t\in {\mathcal{U}} _{s}\) if and only if \(f_{(12t)^{2},s}(x)>0\) for every \(x\in (0,1)\). Again, by Lemma 3.1, this is equivalent to \(3s(12t)^{2}\leq 1\) or \((11/\sqrt{3s})/2\leq t\). This proves
and achieves the proof of Theorem 4.1. □
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])
The necessary and sufficient condition on p, q from \([0,1/2]\) to have
for every distinct positive number a and b is that
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])
The necessary and sufficient condition on p, q from \([0,1/2]\) to have
for every distinct positive number a and b is that
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
For every positive number a and b, we have
Proof
Indeed, for \(s\geq 1\), let
Using Theorem 4.1, for every \((a,b)\in {\mathcal{D}}\), we have
This can be written as
Using Lemma 3.2, this is equivalent to
where \(v=(ab)/(a+b)\). Now, letting s tend to +∞, we obtain
which is the conclusion of Corollary 4.1. □
In fact, because of the “limit argument” in the proof of Corollary 4.1, we lost the strict inequalities for distinct positive real arguments. However, studying the family of functions \((g_{t})_{t\in (0,+\infty )}\) defined by
using similar arguments to those used in Lemma 3.1, we can prove the following exact version of Corollary 4.1, which extends the results of Seiffert [11] and those of Sándor and Trif [10].
Theorem 4.4
The necessary and sufficient condition on p, q from \((0,+\infty )\) to have
is that \(p\leq \frac{1}{6}\) and \(q\geq \ln (\frac{e}{2})\).
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.
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Kouba, O. Sharp twoparameter bounds for the identric mean. J Inequal Appl 2018, 322 (2018). https://doi.org/10.1186/s1366001819172
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DOI: https://doi.org/10.1186/s1366001819172