Refinements for meaninequalities via the stabilizability concept
 Mustapha Raïssouli^{1}Email author
https://doi.org/10.1186/1029242X201255
© Raïssouli; licensee Springer. 2012
Received: 27 July 2011
Accepted: 7 March 2012
Published: 7 March 2012
Abstract
Exploring the stabilizability concept, recently introduced by Raïssouli, we give an approach for obtaining refinements of meaninequalities in a general point of view. Our theoretical study will be illustrated by a lot of examples showing the generality of our approach and the interest of the stabilizability concept.
AMS Subject Classification: 26E60.
Keywords
means refinements of meaninequalities stable and stabilizable means1 Introduction
 (i)
m(a, a) = a, for all a > 0;
 (ii)
m(a, b) = m(b, a), for all a, b > 0;
 (iii)
m(ta, tb) = tm(a, b), for all a, b, t > 0;
 (iv)
m(a, b) is an increasing function in a (and in b);
 (v)
m(a, b) is a continuous function of a and b.
The set of all means can be equipped with a partial ordering, called pointwise order, defined by, m_{1} ≤ m_{2} if and only if m_{1}(a, b) ≤ m_{2}(a, b) for every a, b > 0. We write m_{1} < m_{2} if and only if m_{1}(a, b) < m_{2}(a, b) for all a, b > 0 with a ≠ b. Clearly, m_{1} < m_{2} implies m_{1} ≤ m_{2}.
where min and max are the trivial means (a, b) ↦ min(a, b) and (a, b) ↦ max(a, b).
holds true for every real number t ∈ [0, 1] and all means m_{1} and m_{2}. Further, the inequality (1.5) is strict (in the above sense) if and only if t ∈(0, 1) and m_{1} ≠ m_{2}.
A mean m is called strict if m(a, b) is strictly monotonic increasing in a (and in b). Every strict mean m satisfies that, m(a, b) = a ⇒ a = b. It is easy to see that if m is a strict mean then so is m*. The means min and max are not strict while H, G, A, L, L*, I, I* are strict means.
In the literature, there are some families of means, called power means, which include the above familiar means. Precisely, let p be a real number, we recall the following:

The power binomial mean:$\left\{\begin{array}{c}{B}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right):={B}_{p}={\left(\frac{{a}^{p}+{b}^{p}}{2}\right)}^{1/p},\\ {B}_{1}=H,\phantom{\rule{2.77695pt}{0ex}}{B}_{1}=A,\phantom{\rule{2.77695pt}{0ex}}{B}_{0}:=\underset{p\to 0}{\text{lim}}{B}_{p}=G.\end{array}\right.$(1.9)

The power logarithmic mean:$\left\{\begin{array}{c}{L}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)={L}_{p}={\left(\frac{{a}^{p+1}{b}^{p+1}}{\left(p+1\right)\left(ab\right)}\right)}^{1/p},{L}_{p}\left(a,a\right)=a,\\ {L}_{2}=G,{L}_{1}=L,{L}_{0}=I,{L}_{1}=A.\end{array}\right.$(1.10)

The power difference mean:$\left\{\begin{array}{c}{D}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right):={D}_{p}=\frac{p}{p+1}\frac{{a}^{p+1}{b}^{p+1}}{{a}^{p}{b}^{p}},{D}_{p}\left(a,a\right)=a,\\ {D}_{2}=H,{D}_{1}={L}^{*},{D}_{1/2}=G,{D}_{0}=L,{D}_{1}=A.\end{array}\right.$(1.11)

The power exponential mean:$\left\{\begin{array}{c}{I}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right):={I}_{p}=\text{exp}\left(\frac{1}{p}+\frac{{a}^{p}\text{ln}a{b}^{p}\text{ln}b}{{a}^{p}{b}^{p}}\right),{I}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}a\right)=a,\\ {I}_{1}={I}^{*},{I}_{0}=G,{I}_{1}=I.\end{array}\right.$(1.12)

The second power logarithmic mean:$\left\{\begin{array}{c}{l}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right):={l}_{p}={\left(\frac{1}{p}\frac{{b}^{p}{a}^{p}}{\text{ln}b\text{ln}a}\right)}^{1/p},{l}_{p}\left(a,\phantom{\rule{2.77695pt}{0ex}}a\right)=a,\\ {l}_{1}={L}^{*},{l}_{0}=G,{l}_{\mathsf{\text{l}}}=L.\end{array}\right.$(1.13)
If m_{ p } stands for one of the above power means, it is well known that m_{ −∞ } = min and m_{+∞}= max. Further, all the above power means (also called means of order p) are strictly monotonic increasing in p, for fixed a, b > 0. Otherwise, it is easy to see that ${B}_{p}^{*}={B}_{p}$ for all real number p. We notice that these power means are included in a generalized family of means (not needed here), namely the Stolarsky mean of order 2, see [2] for instance.
In the past years, enormous efforts by some authors has been devoted to refine various inequalities between means (called meaninequalities), see [2–10] for instance and the related references cited therein. Our fundamental goal in this article is to explore the stabilizability concept for obtaining a game of meaninequalities whose certain of them have been differently discussed in the literature. Our approach stems its importance in the following items:
First, by a united procedure we find some known meaninequalities and further other ones in a short and nice manner.
Second, by the same procedure, starting from an arbitrary lower and/or upper bounds of a stabilizable mean we show how to obtain in a recursive manner an infinity of lower and/or upper bounds of this mean. We also give, throughout a lot of examples, sufficient conditions for ensuring that the new bounds are refinements of the initial ones.
2 Background material about stabilizable means
For the sake of simplicity for the reader, we will recall in this section some basic notions and results stated by Raïssouli in an earlier article [1].
called the resultant meanmap of m_{1}, m_{2} and m_{3}.
A study investigating the elementary properties of the resultant meanmap has been stated in [1]. Here, we just recall the following result needed later.
Proposition 2.1. ([1], Proposition 1) The map (a, b) ↦ R(m_{1}, m_{2}, m_{3})(a, b) defines a mean, with the following properties:
The following result, which the proof is straightforward, is also of interest in what follows.
holds for all real number t ∈ [0, 1] and all means m, m'.
The following lemma will be needed in the sequel.
holds for all a, b > 0.
As proved in [1], and will be again shown throughout this article, the resultant meanmap stems its importance in the fact that it is a tool for introducing the stability and stabilizability notions as recalled in the following.
 (a)
Stable if $\mathcal{R}\left(m,m,m\right)=m$.
 (b)
Stabilizable if there exist two nontrivial stable means m _{1} and m _{2} satisfying the relation $\mathcal{R}\left({m}_{1},m,{m}_{2}\right)=m$. We then say that m is (m _{1}, m _{2})stabilizable.
In [1], Raïssouli stated a developed study about the stability and stabilizability of the standard and power means. In particular, he proved that if m is stable then so is m*, and if m is (m_{1}, m_{2})stabilizable then m* is $\left({m}_{1}^{*},{m}_{2}^{*}\right)$stabilizable. About the power standard means, the summarized results stated in [1] are recited in the following theorem.
Theorem 2.4. ([1], Theorems 1,3,4,5) For all real number p, the following statements are met:
(1) The power binomial mean B_{ p } is stable.
(2) The power logarithmic mean L_{ p } is (B_{ p } , A)stabilizable, while the power difference mean D_{ p } is (A, B_{ p } )stabilizable.
(3) The power exponential mean I_{ p } is (G, B_{ p } )stabilizable, while the second power logarithmic mean l_{ p } is (B_{ p } , G)stabilizable.
The following result, needed in the sequel, is immediate from the above.
Corollary 2.5. With the above, the following assertions are met:
(1) The arithmetic, geometric, and harmonic means A, G and H are stable.
(2) The logarithmic mean L is (H, A)stabilizable and (A, G)stabilizable while the identric mean I is (G, A)stabilizable.
(3) The mean L* is (A, H)stabilizable and (H, G)stabilizable while I* is (G, H)stabilizable.
N.B. Throughout the article, we investigate some results of meaninequalities, under convenient assumptions, for the strict symbol < (in the above sense). By similar manner, all stated results remain still true when we replace < by ≤ in the hypotheses as in the related conclusions. Of course, this is not immediate since m_{1} < m_{2} is, as hypothesis and as conclusion, stronger than m_{1} ≤ m_{2}.
3 Refinements for meaninequalities: general approach
As already pointed before, this section displays some important applications of the above concepts for refining meaninequalities in a general point of view. Particular examples illustrating the generality of our approach and the interest of this study will be discussed. We first state the following result which is an improvement of that of Proposition 2.1.
Assume that one of the following three statements holds:
(i)${m}_{1}<{m}_{1}^{\prime},{m}_{2}^{\prime}$and${m}_{3}^{\prime}$are strict means,
(ii)${m}_{2}<{m}_{2}^{\prime},{m}_{1}$and${m}_{3}^{\prime}$are strict means,
(iii)${m}_{3}<{m}_{3}^{\prime}$, m_{1}and m_{2}are strict means.
holds for all a, b > 0 with a ≠ b.
 (i)Without loss the generality, let a, b > 0 with a < b. Then we have$\begin{array}{ll}\hfill \mathcal{R}\left({m}_{1},\phantom{\rule{2.77695pt}{0ex}}{m}_{2},\phantom{\rule{2.77695pt}{0ex}}{m}_{3}\right)\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}{m}_{1}\left({m}_{2}\left(a,\phantom{\rule{2.77695pt}{0ex}}{m}_{3}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)\right),\phantom{\rule{2.77695pt}{0ex}}{m}_{2}\left({m}_{3}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right),\phantom{\rule{2.77695pt}{0ex}}b\right)\right)\phantom{\rule{2em}{0ex}}\\ \le {m}_{1}\left({{m}^{\prime}}_{2}\left(a,{{m}^{\prime}}_{3}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)\right),{{m}^{\prime}}_{2}\left({{m}^{\prime}}_{3}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right),\phantom{\rule{2.77695pt}{0ex}}b\right)\right).\phantom{\rule{2em}{0ex}}\end{array}$(3.4)
This, with ${m}_{1}<{m}_{1}^{\prime}$, yields the desired result.
(ii), (iii) Similar to (i). We left the detail to the reader as simple exercise. □
Now, we are in position to state the following result which gives a refinement of a meaninequality m_{1} < m < m_{2} when m is (m_{1}, m_{2})stabilizable or (m_{2}, m_{1})stabilizable.
If m_{2} < m < m_{1}then the role of m_{1}and m_{2}in the above inequalities is reversed.
This, with the fact that m_{1} and m_{2} are stable and m is (m_{1}, m_{2})stabilizable, yields the desired result. □
Now, let us observe the following particular examples illustrating the situation of the above theorem.
Example 3.3. Now, consider the known inequalities G < I < A with I is (G, A)stabilizable.
Refinements of meaninequalities, even stronger than that of the above examples, are largely studied in the literature, see [2] and the related reference cited therein. As already pointed before, our approach gives a united procedure having a general point of view when we have to refine a mean double inequality m_{1} ≤ m ≤ m_{2} where the intermediary mean m is (m_{1}, m_{2})stabilizable or (m_{2}, m_{1})stabilizable. Further, the next theorem shows that our approach can be successively repeated in the aim to obtain more lower and/or upper bounds of a given stabilizable mean.
Theorem 3.3. Let m be a (m_{1}, m_{2})stabilizable mean with m_{1}and m_{2}are strict means. Let
This, with the fact that m is (m_{1}, m_{2})stabilizable, gives the desired result. □
This makes appear in (3.17) weak conditions of stabilizability, which we call substabilizability and superstabilizability of m_{3} and m_{4}, see [11]. For the moment, we will not give any answer about general sufficient conditions for ensuring the above refinement, but we just discuss (in the sections below) the response for some particular cases.
N.B. Let m_{ p } ∈ {l_{ p } , L_{ p } , I_{ p } , D_{ p } } be a power mean. Henceforth, when we say
"Let m_{1} and m_{2} be two means such that m_{1} < m_{ p } < m_{2} for some p", it should be understood in the following sense,
"Let p be a real number and assume that there exist two means m_{1} := m_{1}(p) and m_{2} := m_{2}(p) satisfying that m_{1} < m_{ p } < m_{2}".
4 Refinements for bounding the means l_{ p }and L
Since l_{ p } is (B_{ p } , G)stabilizable, we then will be interested by bounds of l_{ p } in terms of B_{ p } and G.
valid for all a, b > 0 and p ≠ 0.
We begin by regarding bounds of l_{ p } in a convexgeometric form ${B}_{p}^{\alpha}{G}^{1\alpha}$ as well:
from which the desired double inequality (4.3) follows. □
Proof. Taking p = 1 in the above theorem, with the fact that l_{1} = L and B_{1} = A, we immediately obtain the announced result. □
Let us now examine the following examples in the aim to illustrate the above theoretical results.
which refines the arithmeticlogarithmicgeometric mean inequality G < L < A.
If moreover α < (>)1/3 then (4.12) refines (4.11).
which holds when α < 1/3. For the reversed inequalities, the same arguments as previous study, so completes the proof. □
If we get p = 1 in the above theorem, we immediately obtain the following result.
If moreover α < 1/3 then (4.18) refines (4.17).
Theorem 4.3 tells us that every given bound of l_{ p } in a convexgeometric form yields another bound of l_{ p } in an analogs, but different, form. Illustrating this latter point, we will deduce a better bound of l_{ p } than the above ones. Precisely, we may state the next result.
The proof is similar for p < 0. Taking p = 1 in (4.19) we obtain (4.20), so completes the proof. □
To understand the interest of the above theorem, let us observe the following example.
which refines A^{1/3}G^{2/3} < L.
Remark 4.1. The inequality (4.20) was proved by Leach and Sholander [6], while (4.26) has been shown by Sāndor [10]. These two inequalities were proved by different methods therein while together obtained here via the same approach. In the same sense, other examples will be seen later (see Remarks 4.4, 4.5, and 5.1).
Remark 4.2. As well known, inequality (4.20) is the best possible in the sense that the constant α = 1/3 cannot be improved in A^{ α }G^{1−α}< L. This latter point rejoins the fact that if we apply Corollary 4.4 to (4.20) we obtain the same inequality.
Remark 4.3. By virtue of the relationships (4.1), it has been possible to begin by stating and proving the results of the above corollaries and then to deduce those of the corresponding theorems (with discussion on p). Details of this latter point are omitted for the reader.
Now, we will be interested by bounds of l_{ p } in a convexarithmetic expression as well:
By virtue of the identity (4.6), we obtain the desired result after simple manipulations. □
As in the above, taking p = 1 in the latter theorem we immediately obtain the following result.
Theorem 4.6 has many interesting consequences. For instance, we give the two following corollaries.
If α > 1=3 then (4.34) refines (4.33).
we obtain the announced result after substituting this latter inequality in (4.35). If α > 1/3, it is easy to see by similar manner as previous that (4.34) refines (4.33) and the proof is completed. □
converges to 1/3 and the desired result follows as previous. We omit the routine detail here. □
Remark 4.4. The inequality (4.38) was differently proved by Carlson [12] and here obtained by the same approach as (4.20) and (4.26).
Let us illustrate the above theoretical examples with the following examples.
Of course, we can combine some the above results to improve the lower and upper bounds of L. The following example explains this situation.
The reader can easily verify that this latter double inequality refines the initial one, so proving our desired aim.
and the desired inequality follows by combining (4.46) and (4.47) with a simple reduction. □
Taking p = −1 in the above theorem, with the fact that l_{−1} = L* = G^{2}/L and B_{−1} = H = G^{2}/A, we immediately obtain the next result.
If moreover α > 1/3 then (4.49) refines (4.48).
Proof. We left it to the reader as an interesting exercise. □
We end this section by stating another result showing how to obtain a lower bound of the logarithmic mean L when we start from an upper bound of its dual L*. In fact, since L* is (A, H)stabilizable then we search bounds of L* in terms of A and H. Precisely, we have the following.
If moreover α > 1/3 then (4.53) refines (4.52).
which after reduction yields the desired result.
The general relation m* = G^{2}/m valid for all mean m, gives in particular, L* = G^{2}/L, H = G^{2}/A and A = G^{2}/H. Substituting this in the latter inequality, we obtain the desired result. □
Remark 4.5. The inequality (4.57) was differently proved by Chen [5] and shown here by the same approach as (4.20), (4.26), and (4.38), so proving the interest of this study. Further, we notice that it is easy to verify that (4.57) is stronger than (4.51).
5 Refinements for bounding the means I_{ p }and I
In this section, we will state some refinements for the power exponential mean I_{ p } in a parallel manner to those for l_{ p } already presented in the above section. We immediately deduce some refinements for the identric mean I. The proofs of the results announced here are often similar to that of the above and we omit the routine details to not lengthen this article.
We begin by stating the following lemma which will be needed later.
where we set B_{ p } := B_{ p } (a, b) for the sake of simplicity.
By computations as previous we easily deduce the desired result.
Starting from a double inequality m_{1} < I_{ p } < m_{2}, we may choose convenient means m_{1} and m_{2} giving easy computations with the fact that I_{ p } is (G, B_{ p } )stabilizable. It is easy to see that B_{ p } < I_{ p } < G for p < 0, with reversed inequalities if p > 0. Then, as for l_{ p } , bounds of I_{ p } in the form ${B}_{p}^{\alpha}{G}^{1\alpha}$ (resp., αB_{ p } + (1 − α)G) exist for some α ∈ [0, 1]. The following result is an analog of Theorem 4.1 from l_{ p }to I_{ p }.
so completes the proof. □
Taking p = 1 in the above theorem, with the fact that B_{1} = A and I_{1} = I, we deduce the
following result for bounding the identric mean I.
which refines the arithmeticidentricgeometric mean inequality G < I < A.
If moreover α < (>)2/3 then (5.14) refines (5.13).
Proof. Similar to that of the above. We left the detail for the reader as an interesting exercise. □
As previously, taking p = 1 in the above theorem we immediately obtain the following result.
If moreover α < 2/3 then (5.16) refines (5.15).
which converges to 2/3. We conclude by analogs arguments as previous. □
Remark 5.1. The inequality (5.18) has been proved by different methods, see [5] for comparison. We left the reader to state analogs ways about inequality (5.17) as in Remark 4.2.
As the reader can verify it, analog of Theorem 4.6 for I_{ p } has length expression and makes appear hard computations.
We left to the reader the routine task for considering other meaninequalities, involving the standard means, in the aim to obtain more lower and/or upper bounds for a stabilizable mean, eventually with some related refinements. As example, we can state the following.
We omit the proofs of the above results here. Of course, for the proof of Theorem 5.7 we use the fact that L_{ p } is (B_{ p }, A)stabilizable while that of Theorem 5.8 uses D_{ p } is (A, B_{ p } )stabilizable. Some consequences can also be derived from the above theorems in a similar manner as previous. In particular for p = 0, Theorem 5.7 coincides with Corollary 5.3 while Theorem 5.8 is reduced to Corollary 4.7. We left all these details to the interested reader.
In summary, the stability and stabilizability concepts are good tool for obtaining a lot of meaninequalities in a short and nice manner. In particular, some meaninequalities, already differently proved by many authors in the literature, have been here obtained as consequences via a procedure having a general point of view. This shows the interest of this study derived from the stabilizability concept.
has not been considered here. In fact, Raïssouli [1] conjectured that the mean P defined by (5.24) is not stabilizable and this problem remains open. In this direction, we indicate a recent article [11] for further comments about this latter point.
Declarations
Acknowledgements
The author would like to thank the anonymous referees for their useful comments and suggestions.
Authors’ Affiliations
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