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On some Hölder-type inequalities with applications
- Mustapha Raïssouli^{1, 2} and
- Rabie Zine^{3}Email author
https://doi.org/10.1186/s13660-015-0616-5
© Raïssouli and Zine; licensee Springer. 2015
- Received: 25 August 2014
- Accepted: 27 February 2015
- Published: 7 March 2015
Abstract
In this paper, some mathematical inequalities of Hölder type are established. Applications for some operator inequalities as well as for functional inequalities in convex analysis are provided as well.
Keywords
- Hölder-type inequalities
- operator means
- operator inequalities
- functional inequalities
- convex analysis
1 Introduction
- (i)
C is such that \(u\in C\) and \(t\geq0 \Rightarrow tu\in C\); C is then called a cone of E.
- (ii)
C is such that \(u\in C\) and \(\lambda\in{\mathbb{K}} \Rightarrow \lambda u\in C\); C is sometimes called a generalized cone of E. Clearly, every generalized cone of E is a cone.
Let \(f:C\rightarrow{\mathbb{K}}\) be a map. If C is a generalized cone, we say that f is homogeneous of degree p if \(f(\lambda u)=|\lambda|^{p}f(u)\) for all \(u\in C\) and \(\lambda\in\mathbb{K}\). If C is a cone, f is called positively homogeneous of degree p if \(f(tu)=t^{p}f(u)\) for all \(u\in C\) and \(t\geq0\). Clearly, every homogeneous map of degree p (on a generalized cone) is positively homogeneous of the same degree p. The reverse is not always true.
Now, let C be a convex cone of E. A map \(\Phi:C\rightarrow {\mathbb{R}}\) is called sub-additive if \(\Phi(u+v)\leq\Phi(u)+\Phi(v)\) holds for all \(u,v\in C\). If C is equipped with an order ≺, the map Φ is said to be monotone if for all \(u,v\in C\) such that \(u\prec v\) we have \(\Phi(u)\leq\Phi(v)\).
The remainder of this paper is organized as follows: Section 2 is devoted to the presentation of our main results together with some related consequences. Section 3 displays a lot of examples illustrating the above theoretical results. In Section 4, we investigate some operator inequalities as applications of our main results. Section 5 is focused on another application for inequalities in convex analysis.
2 The main results
We use the same notations as previously. We start this section by stating the following lemma, which will be needed in the sequel.
Lemma 2.1
Proof
Now, our first main result may be presented.
Theorem 2.2
Proof
Remark 2.1
(i) With the assumptions of Theorem 2.2 the inequalities (1) and (2) are in fact equivalent. The implication ‘(2) ⇒ (1)’ follows by a simple application of the Young inequality. Similar statements can be made for the analog situation in the following results.
(iii) It is worth noticing that the functions f and g in the previous theorem, as well as in the following results, are not necessarily continuous.
Theorem 2.2 has many consequences whose certain of them are recited in what follows.
Corollary 2.3
Proof
Remark 2.2
In Corollary 2.3, if E is a locally convex space, we can take \(F=E^{*}\) algebraic (or topological) dual of E and h the duality map between E and \(E^{*}\). As an example explaining this situation, see Theorem 5.2 in Section 5.
Another corollary of Theorem 2.2 may be stated as well.
Corollary 2.4
Proof
We now state the following result.
Theorem 2.5
Proof
We end this section by stating the two following results, which extend Theorem 2.2 and Theorem 2.5, respectively.
Theorem 2.6
Proof
The statement of Corollary 2.4 can be included in the situation of the previous theorem. We omit all details of this point, leaving them for the reader.
Theorem 2.7
Proof
It is similar to that of Theorem 2.6. We omit all details leaving them to the reader. □
For an application of the previous theorem, see Section 4 below.
3 Some examples
A norm \(|\!|\!|\cdot|\!|\!|\) on \({\mathcal{B}}(H)\) is said to be unitarily invariant if it satisfies the invariance property \(|\!|\!|UTV|\!|\!|=|\!|\!|T|\!|\!|\) for all \(T\in{\mathcal{B}}(H)\) and for all unitary operators U and V.
Now we are in a position to state the following list of examples.
Example 3.1
Example 3.2
Now, let us observe the two following examples illustrating, particularly, the situation of Corollary 2.4.
Example 3.3
Example 3.4
Example 3.5
Example 3.6
4 Application to operator inequalities
We preserve the same notation as in the previous section. The following result, which is an operator version of Theorem 2.5, may be stated.
Theorem 4.1
Proof
From the above theorem, we immediately deduce the following corollary.
Corollary 4.2
Now, we will illustrate the above theorem with some applications.
The first result of application here may be stated as well.
Theorem 4.3
Proof
Remark 4.1
Theorem 4.4
Proof
Setting \(h(T,S)=(1+\lambda)T\), \(f(T,S)=\lambda TS^{-1}T\), and \(g(T,S)=T\, \sharp_{\lambda}\, S\), it is easy to see that h, f, and g are homogeneous, with respect to the first variable T, of degrees 1, 2, and \(1-\lambda\). The remainder of the proof is similar to that of Theorem 4.3. The details are simple and are omitted here. □
5 An application in convex analysis
Later, we shall need the following lemma.
Lemma 5.1
- (i)
If \(f(0)\leq0\) then \(f^{*}\) is a positive functional.
- (ii)
If f is homogeneous of degree \(p>1\) then \(f^{*}\) is homogeneous of degree \(p^{*}>1\), with \(p^{*}=p/(p-1)\).
Proof
(i) From (20) we immediately deduce that \(f^{*}(u^{*})\geq \operatorname{Re}\langle u^{*},u\rangle-f(u)\) for all \(u\in E\) and \(u^{*}\in E^{*}\). Taking \(u=0\) in this inequality we obtain \(f^{*}(u^{*})\geq0\) for all \(u^{*}\in E^{*}\).
Our main result of application in this section is the following.
Theorem 5.2
Proof
Now, we will illustrate the above result with the following example.
Example 5.1
The following example is also of interest.
Example 5.2
Declarations
Acknowledgements
This project was supported by the deanship of scientific research at Salman Bin Abdulaziz University under the research project ♯2014/01/1444. Many thanks to the anonymous referees for valuable comments and suggestions, which have been included in the final version of this manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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