On weighted means and their inequalities

In \cite{PSMA}, Pal et al. introduced some weighted means and gave some related inequalities by using an approach for operator monotone functions. This paper discusses the construction of these weighted means in a simple and nice setting that immediately leads to the inequalities established there. The related operator version is here immediately deduced as well. According to our constructions of the means, we study all cases of the weighted means from three weighted arithmetic/geometric/harmonic means, by the use of the concept such as stable and stabilizable means. Finally, the power symmetric means are studied and new weighted power means are given.


Introduction
The mean inequalities arise in various contexts and attract many mathematicians by their developments and applications. It has been proved throughout the literature that the mean-theory is useful in theoretical point of view as well as in practical purposes.
1.1. Standard weighted means. As usual, we understand by (binary) mean a map m between two positive numbers such that min(a, b) ≤ m(a, b) ≤ max(a, b) for any a, b > 0. Continuous (symmetric/homogeneous) means are defined in the habitual way. If m is a mean we define its dual by m * (a, b) = (m(a −1 , b −1 )) −1 . It is easy to see that if m is continuous,(resp. symmetric, homogeneous) then so is m * . Of course, m * * = m for any mean m. It is obvious that, (iii) implies (ii). The mean m := m 1/2 is called the associated symmetric mean of m v . It is not hard to check that, if m v is a weighted mean then so is m * v . The standard weighted means are recalled in the following. The weighted arithmetic mean a∇ v b = (1−v)a+vb, the weighted geometric mean a♯ v b = a 1−v b v and the weighted harmonic mean a! v b = (1 − v)a −1 + vb −1 −1 . For v = 1/2 they coincide with a∇b, a♯b and a!b, respectively. These weighted means satisfy for any a, b > 0 and v ∈ [0, 1]. These weighted means are all strict provided v ∈ (0, 1).

Two weighted means.
Recently, Pal et al. [4] introduced a class of operator monotone functions from which they deduced other weighted means, namely the weighted logarithmic mean defined by and the weighted identric mean given by (1.4) and Otherwise, Furuichi and Minculete [5] gave a systematic study from which they obtained a lot of mean-inequalities involving L v (a, b) and I v (a, b). Some of their inequalities are refinements and reverses of (1.5) and (1.6).
The outline of this paper will be organized as follows: In Section 2 we give simple forms for L v (a, b) and I v (a, b) and mean-inequalities are obtained in a fast and nice way. We also deduce two other weighted means from L v (a, b) and I v (a, b). Section 3 is devoted to investigate a general approach in service of weighted means. We then obtain more weighted means in another point of view. Section 4 displays the operator version of the previous weighted means as well as their related inequalities. In Section 5 we recall the standard power means known in the literature and we use, in Section 6, our approach for obtaining some new weighted means associated to the previous power means. 2. Another point of view for defining L v (a, b) and I v (a, b) We preserve the same notations as in the previous section. The expressions (1.3) and specially (1.4) seem to be hard in computational context. We will see that we can rewrite them in other forms having convex characters.
The key idea of this section turns out the following result. (2.1) Proof. Starting from the middle expression of (2.1) and using the definition of L(a, b) and a♯ v b we get the desired result after simple algebraic manipulations. By similar way we get (2.2). The details are straightforward and therefore omitted here.
In what follows we will see that the inequalities (1.5) and (1.6) can be immediately deduced from (2.1) and (2.2), respectively. In fact we will prove more.
Proof. The two right inequalities of (2.3) are those of (1.5). We will prove them again by using (2.1). Indeed, (2.1) with the help of (1.1) and then (1.2) yields We now prove the two left inequalities of (2.3). Again, (2.1) with (1.1) and then (1.2), implies that We left to the reader the routine task for proving (2.4) in a similar manner. (ii) From (2.1) and (2.2), it is immediate to see that L v (a, b) and I v (a, b) are binary means in the sense that they satisfy the conditions itemized in [4]. In order to emphasize even more the importance of (2.1) and (2.2) we will present below more results. These results investigate some inequalities refining the right inequalities in (2.3) and (2.4). We need the following lemma. (2.5) Proof. We prove the first inequality in (2.5). Since the map The second and third inequalities of (2.5) follow from (1.1).
To prove the second inequality of (2.6), we write by using the previous lemma This, with (2.2), yields the second inequality of (2.6). To prove the first inequality of (2.6) we write after a simple computation. The proof is finished.

Two other weighted means.
A natural question arises from the previous subsection: do we have a weighted mean M v (a, b) such that We can also put the following question: do we have a weighted mean In what follows we will answer the two preceding questions. Recall that L * denotes the dual of the logarithmic mean L and L * v is the dual of the weighted logarithmic mean L v , as previously defined. Similar sentence for I * and I * v . We will establish the following result.
Proof. We can of course assume that v ∈ (0, 1). If in (2.1) we replace a and b by a −1 and b −1 , respectively, then we get Taking the inverses side by side and using the definition of the weighted harmonic mean we infer that Now, let us set If v ∈ (0, 1) is fixed, for any a > 0 and x > 0 it is easy to see that there exists a unique b > 0 such that a♯ v b = x. This means that M is well-defined by (2.12).
It follows that M is the dual of the logarithmic mean L. Following (2.11) and (2.7), the associated weighted mean M v of M is such that is the dual of the weighted logarithmic mean L v . We left to the reader the task for proving (2.10) in a similar manner.
Remark 2.7. After this, let us observe the following question: is L v the unique weighted mean satisfying (2.1)? In the next section, we will answer this question via a general point of view. Similar question can be put for (2.2), (2.9) and (2.10).

Weighted means in a general point of view
As already pointed before, we will investigate here a study that shows how to construct some weighted means in a general point of view.
3.1. Position of the problem. Let a, b > 0 and v ∈ [0, 1]. Let p v and q v be two weighted means. We write ap v b := p v (a, b) and aq v b := q v (a, b) for the sake of simplicity. As previous, p := p 1/2 and q := q 1/2 and we write apb := p(a, b) and aqb = q(a, b). To fix the idea and for the first time, we can choose p v and q v among the three standard weighted means i.e.
Our general problem reads as follows: do we have a weighted mean To answer this question, it is in fact enough to justify that there exists one and only one (symmetric) mean M such that Our aim here is to answer the previous question in its general form. We need to recall some notions and results as background material that we will summarize in the next subsection.
3.2. Stable and stabilizable means. We recall here in short the concept of stable and stabilizable means introduced in [6,7,9]. Let m 1 , m 2 and m 3 be three given symmetric means. For all a, b > 0, the resultant mean-map of m 1 , m 2 and m 3 is defined by, [6] A symmetric mean m is called stable if R(m, m, m) = m and stabilizable if there exist two nontrivial stable means m 1 and m 2 such that R(m 1 , m, m 2 ) = m. We then say that m is (m 1 , m 2 )-stabilizable. If m is stable then so is m * , and if m is (m 1 , m 2 )-stabilizable then m * is (m * 1 , m * 2 )-stabilizable. The tensor product of m 1 and m 2 is the map, denoted m 1 ⊗ m 2 , defined by A symmetric mean m is called cross mean if the map m ⊗2 := m ⊗ m is symmetric in its four variables. Every cross mean is stable, see [6], and the converse still an open problem. It is worth mentioning that, the operator version of the previous concepts as well as their related results has been investigated in a detailed manner in [10]. It has been proved there that every cross operator mean is stable but the converse does not in general hold provided that the Hilbert operator-space is of dimension greater than 2.
The following results will be needed later, see [6,7,9]. For more examples and properties about stable and stabilizable means we can consult [6,7,9,10]. See also Section 5 below.
Theorem 3.2. Let m 1 and m 2 be two symmetric means such that m 1 ≤ m 2 (resp. m 2 ≤ m 1 ). Assume that m 1 is a strict cross mean. Then there exists one and only one (m 1 , m 2 )-stabilizable mean m such that m 1 ≤ m ≤ m 2 (resp. m 2 ≤ m ≤ m 1 ).

3.3.
The main result. Now, we are in the position to answer our previous question as recited in the following result. Theorem 3.3. Let a, b > 0 and v ∈ [0, 1]. Let p v and q v be two weighted means such that p := p 1/2 and q := q 1/2 are stable. Assume that q is a strict cross mean. Then there exists one and only one weighted mean M v (a, b) such that (3.1) holds. Further, M := M 1/2 is the unique (q, p)-stabilizable mean.
Proof. As already pointed before, it is enough to consider (3.2). Following the previous subsection, (3.2) can be written as This means that M is (q, p)-stabilizable. According to Theorem 3.2, such M exists and is unique. Since p v and q v are given, we then deduce the existence and uniqueness of M v satisfying (3.1). The proof is finished.
Following Theorem 3.1, the symmetric means a∇b, a♯b, a!b are cross means and so stable. From the preceding theorem we immediately deduce the following corollary.
then we have the same conclusion as in the previous theorem.
The following examples discuss these cases in details.
We can show this separately for every case by checking (3.1) or use Theorem 3.3 when combined with Theorem 3.1. The details are immediate and therefore omitted here for the reader.
We have two cases left to see, namely , which we will discuss in the two following examples, respectively.  3) is a second weighted logarithmic mean which we will denote by L v . Its explicit form is given by . By similar way as previous, we show that the associated mean M is here given by M = L * the dual logarithmic mean. The associated weighted mean M v is defined by Also, from this latter relation we can verify that The details are immediate and therefore omitted here.
The previous examples are summarized in TABLE 1.

Operator Version
The operator version of the previous weighted means as well as their related operator inequalities have been also discussed in [4]. By using their approach for operator monotone functions and referring to the Kubo-Ando theory [3], they studied the analogs of L v (a, b) and I v (a, b) when the positive real numbers a and b are replaced by positive invertible operators.
Here, and with (2.1) and (2.2), we don't need any more tools for giving in an explicit setting the operator versions of L v (a, b) and I v (a, b). Before exploring this, let us recall a few basic notions about operator means.
Let H be a complex Hilbert space and let B(H) be the C * -algebra of bounded linear operators acting on H.  [1,2,11,12] and the related references cited therein. Some examples of operator monotone functions will be considered below.
Following the Kubo-Ando theory [3], there exists a unique one-to-one correspondence between operator means and operator monotone functions. More precisely, an operator mean m in the Kubo-Ando sense is such that for some positive monotone increasing function f m on (0, ∞). The function f m in (4.1) is called the representing function of the operator mean m. An operator mean in the Kubo-Ando sense is called operator monotone mean. Let A, B ∈ B + * (H) and v ∈ [0, 1]. As standard examples of operator monotone means, the following are known in the literature as the weighted arithmetic mean, the weighted geometric mean and the weighted harmonic mean of A and B, respectively. If v = 1/2 they are simply denoted by A∇B, A♯B and A!B, respectively. The previous operator means satisfy the following double inequality The weighted logarithmic mean and the weighted identric mean of A and B can be, respectively, defined through: 3) is also valid for any A, B ∈ B + * (H) and v ∈ [0, 1]. For the sake of information, the logarithmic mean L(A, B) previously defined can be also alternatively given by one of the following integral forms: It is worth mentioning that (4.6) Since all the involved operators in (4.6) and (4.7) are operator means in the sense of (4.1) then by Theorem 2.2 we immediately deduce the following result as well.  By the same arguments as previous, the operator version of Theorem 2.5 is immediately given in the following statement. (4.11)

Power symmetric means
This section deals with some weighted means for power symmetric means in one or two parameters. Let a, b > 0 and p, q be two real numbers. We recall the following: • The power binomial mean defined by: • The power logarithmic mean defined by: • The power difference mean given by: • The second power logarithmic mean defined through: (5.5) In particular, L −1 (a, b) = L * (a, b), L 0 (a, b) = a♯b and L 1 (a, b) = L(a, b).
• The previous power means are included in the so-called Stolarsky mean S p,q : , S p,q (a, a) = a, (5.6) in the sense that All the previous power means are symmetric in a and b. Also, remark that S p,q is symmetric in p and q. Otherwise, the power binomial mean B p is stable for any p ∈ R and the following result holds, see [8].
Theorem 5.1. For any p, q ∈ R, the Stolarsky mean S p,q is B q−p , B p -stabilizable.
The previous theorem when combined with (5.7) and a simple argument of continuity immediately implies the following, see also [6].
Corollary 5.2. For all real number p, the following assertions hold: Now, let us observe the following remark which is of interest.
Remark 5.3. Since S p,q = S q,p we can also say that S p,q is B p−q , B q -stabilizable. This, with (5.7), implies also that, L p is (B −p , B p+1 )-stabilizable, D p is (!, B p+1 )stabilizable, L p is (B −p , B p )-staabilizable and no news for I p . Obviously, (i) and (ii) of Corollary 5.2 are simpler than these latter statements.

Some new weighted power means
In this section we will investigate the weighted means of the previous power means. The weighted power binomial mean can be immediately given by This, with the results presented in the preceding section, will allow us to construct some new weighted power means. Recall that, m v is called weighted mean if it satisfies the conditions: m v is a mean for any v ∈  is a M-weighted mean.
Proof. It is straightforward. The details are simple and therefore omitted here for the reader.
Applying the previous simple result to the preceding power means, we will immediately obtain their associated weighted power means. We present these in the following examples. We begin by the S p,q -weighted mean and we then deduce the other weighted power means as particular cases.
Example 6.2. By Theorem 5.1, S p,q is (B q−p , B p )-stabilizable. By Theorem 6.1, an S p,q -weighted mean is given by S p,q;v (a, b) = B q−p;v S p,q B p;v (a, b), a , S p,q B p;v (a, b), b .
(6.2) Utilizing (5.2) with (5.1), the explicit form of S p,q;v (a, b) is given by (for a = b) provided that p = 0, q = 0, p = q, where we write B p;v := B p;v (a, b) for simplifying the writing. The three cases p = 0, q = 0 and p = q will be presented later. Example 6.3. Since S p,q is also (B p−q , B q )-stabilizable, see Remark 5.3, another S p,q -weighted mean is given by S p,q;v (a, b) = B p−q;v S p,q B q;v (a, b), a , S p,q B q;v (a, b), b , or, in explicit form, if p = 0, q = 0, p = q and a = b, S p,q;v (a, b) = q p Example 6.4. (i) By Corollary 5.2, L p is (B p , ∇)-stabilizable. By Theorem 6.1, the L p -weighted mean is given by By (5.2) and (5.1), or just using the relation L p = S 1,p+1 with (6.2), we obtain the explicit form of L p;v : (ii) Similarly, since D p is (∇, B p )-stabilizable, we then deduce that the D pweighted mean is given by