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On some Höldertype inequalities with applications
Journal of Inequalities and Applications volume 2015, Article number: 93 (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.
Introduction
We begin by stating some notions needed. Let E be a linear vector space over \({\mathbb{K}}={\mathbb{R}},{\mathbb{C}}\) and let C be a nonempty subset of E. Consider the two following statements:

(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 subadditive 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 \(\Phi(u)\leq\Phi(v)\).
Let E and F be two linear vector spaces over \({\mathbb{K}}\), \(C_{1}\) and \(C_{2}\) be two nonempty subsets of E and F, respectively, and \(h:C_{1}\times C_{2}\rightarrow{\mathbb{K}}\) be a given map. If \(C_{1}\) is a cone, we say that h is positively homogeneous of degree r, with respect to the first variable, if \(h(tu,v)=t^{r}h(u,v)\) for all \(u\in C_{1}\), \(v\in C_{2}\) and \(t\geq0\). If \(C_{1}\) and \(C_{2}\) are generalized cones, we say that h is a semiinner product if and only if
hold for all \(\lambda\in{\mathbb{C}}\), \(u\in C_{1}\) and \(v\in C_{2}\). Clearly, every semiinner product map is positively homogeneous of degree 1 with respect to its two variables. The reverse is, in general, false.
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.
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
Let \(a,b\geq0\) and \(p,q>0\) be real numbers. Then we have
Proof
If \(a=0\) or \(b=0\), it is easy to see that \(\inf_{t>0}(at^{p}+bt^{q})=0\) and the desired equality holds. Assume that \(a,b>0\) and set \(\phi (t)=at^{p}+bt^{q}\) for \(t>0\). It is easy to see that
for all \(t>0\), with \(\phi'(t)=0\) if and only if
Further, simple computation leads to
This, with the fact that
yields the desired result. □
Now, our first main result may be presented.
Theorem 2.2
Let E and F be two linear vector spaces over \({\mathbb{K}}\), \(C_{1}\) is a cone of E and \(C_{2}\) is a nonempty subset of F. Let \(f:C_{1}\rightarrow[0,\infty)\), \(g:C_{2}\rightarrow[0,\infty)\), and \(h:C_{1}\times C_{2}\rightarrow{\mathbb{R}}\) be three maps such that
Assume that f is positively homogeneous of degrees p and h is positively homogeneous, with respect to the first variable, of degree r, with \(\min(p,0)< r<\max(p,0)\). Then the inequality
holds true for all \((u,v)\in C_{1}\times C_{2}\).
Proof
We present the proof for \(p>0\) (\(0< r<p\)), and that of the case \(p<0\) (\(p< r<0\)) can be stated in a similar manner. Replacing \(u\in C_{1}\) by \(tu\in C_{1}\), with \(t>0\), in (1) and using the positive homogeneity assumed in our statement, we obtain
or equivalently
This means that the map \(t\mapsto t^{pr}f(u)+t^{r}g(v)\), for \(t>0\), is bounded below and so we can write
Following Lemma 2.1, with \(a=f(u)\geq0\) and \(b=g(v)\geq0\), we immediately deduce, after a simple manipulation, the desired inequality. We then finished the 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.
(ii) If the map h comes with positive values then (2) can be written in the following equivalent form:
and
(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
Let E, F be two linear vector spaces and \(C_{1}\), \(C_{2}\) be two generalized cones of E and F, respectively. Let \(f:C_{1}\rightarrow[0,\infty)\), \(g:C_{2}\rightarrow[0,\infty)\) and \(h:C_{1}\times C_{2}\rightarrow{\mathbb{C}}\) be three maps such that
Assume that f is homogeneous of degree \(p>1\), g is homogeneous of degree \(p^{*}>1\), with \(1/p+1/p^{*}=1\), and h is a semiinner product. Then the inequality
holds for all \((u,v)\in C_{1}\times C_{2}\).
Proof
Under our assumptions, we can apply the above theorem (with \(r=1\)) for obtaining, from (3),
for all \((u,v)\in C_{1}\times C_{2}\), with \(p^{*}=p/(p1)\). If in this inequality we replace u by \((h(v,u) )^{1/2}u\in C_{1}\) and v by \((h(u,v) )^{1/2}v\in C_{2}\) and we use the fact that f and g are homogeneous of degree p and \(p^{*}\), respectively, h being a semiinner product, we obtain after elementary manipulation
We can assume that \(h(u,v)\neq0\), since for \(h(u,v)=0\) the inequality (4) is obviously satisfied. We then deduce the desired result and this completes the 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
Let f, g, and h be as in Theorem 2.2. Then
holds true for all \(u_{1},u_{2},\ldots,u_{n}\in C_{1}\) and \(v_{1},v_{2},\ldots,v_{n}\in C_{2}\).
Proof
Condition (1) implies that
for all \(u=(u_{1},u_{2},\ldots,u_{n})\in C_{1}^{n}\) and \(v=(v_{1},v_{2},\ldots,v_{n})\in C_{2}^{n}\). It is easy to see that \(\tilde{f}:C_{1}^{n}\rightarrow [0,\infty)\) is positively homogeneous of degree p and \(\tilde {h}:C_{1}^{n}\times C_{2}^{n}\rightarrow{\mathbb{R}}\) is positively homogeneous, with respect to the first variable u, of degree r. Theorem 2.2 yields the desired inequality (5), and this completes the proof. □
We now state the following result.
Theorem 2.5
Let E and F be as above, \(C_{1}\) be a cone of E and \(C_{2}\) be a nonempty subset of F. Let \(f,g:C_{1}\times C_{2}\rightarrow[0,\infty )\) and \(h:C_{1}\times C_{2}\rightarrow[0,\infty)\) be three maps such that
Assume that further f, g, and h are positively homogeneous, with respect to the first variable, of degrees p, q, and r, respectively, with \(p< r<q\). Then the inequality
holds true for all \((u,v)\in C_{1}\times C_{2}\).
Proof
Analogously to the proof of Theorem 2.2, we show that
The desired inequality (6) follows by application of Lemma 2.1 in a similar manner as previous. The details are simple and are omitted here. □
We end this section by stating the two following results, which extend Theorem 2.2 and Theorem 2.5, respectively.
Theorem 2.6
Let \(C_{1}\), \(C_{2}\) be as in Theorem 2.2 and \((C,\prec)\) be an ordered cone of a certain linear space. Let \(f:C_{1}\rightarrow C\), \(g:C_{2}\rightarrow C\), and \(h:C_{1}\times C_{2}\rightarrow C\) be such that
Assume that f and h are as in Theorem 2.2. If \(\Phi :C\rightarrow[0,\infty)\) is monotone subadditive and homogeneous of degree \(s>0\), then the inequality
holds true for all \((u,v)\in C_{1}\times C_{2}\).
Proof
With the fact that Φ is monotone and subadditive, (7) implies that
with \(\Phi\circ f:C_{1}\rightarrow[0,\infty)\) homogeneous (with respect to the first variable) of degree ps and \(\Phi\circ h:C_{1}\times C_{2}\rightarrow[0,\infty)\) homogeneous of degree rs, with \(\min(\mathit{ps},0)<\mathit{rs}<\max(\mathit{ps},0)\) since \(s>0\). We can then use Theorem 2.2 and the desired inequality follows after a simple manipulation. □
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
Let \(C_{1}\), \(C_{2}\), and C be as in Theorem 2.6. Let \(f:C_{1}\times C_{2}\rightarrow C\), \(g:C_{1}\times C_{2}\rightarrow C\), and \(h:C_{1}\times C_{2}\rightarrow C\) be such that
Assume that f and h are as in Theorem 2.5. If \(\Phi :C\rightarrow[0,\infty)\) is monotone subadditive and homogeneous of degree \(s>0\), then the inequality
holds true for all \((u,v)\in C_{1}\times C_{2}\).
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.
Some examples
This section is devoted to the presentation of some examples illustrating the above theoretical results. We need more notations. In what follows, H denotes a complex Hilbert space with its inner product \(\langle\cdot,\cdot\rangle\) and its associate norm \(\\cdot\\). The notation \({\mathcal{B}}(H)\) refers to the algebra of linear bounded operators defined from H into itself. A selfadjoint operator \(T\in{\mathcal{B}}(H)\) is positive (in short, \(T\geq0\)) if \(\langle Tu,u\rangle\geq0\) for all \(u\in H\). We denote by \({\mathcal{B}}^{+}(H)\) (resp. \({\mathcal{B}}^{+*}(H)\)) the convex cone of all selfadjoint positive (resp. invertible) operators \(T\in{\mathcal{B}}(H)\). As usual, for \(T,S\in{\mathcal{B}}(H)\) we write \(T\leq S\) if and only if T, S are selfadjoint and \(ST\in{\mathcal{B}}^{+}(H)\). The space \({\mathcal{B}}(H)\) is endowed with the classical operator norm, namely
It is well known that if T is positive then
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
Let \(T,S\in{\mathcal{B}}(H)\). Then we have
which, by Corollary 2.3 with \(p=p^{*}=2\), yields
where as usual \(T= (T^{*}T )^{1/2}\). If \(S=T^{*}\) then
see [1]. If T is positive (selfadjoint) then
which is a wellknown extension of the CauchySchwarz inequality.
Example 3.2
Let \(T,S,X\in{\mathcal{B}}(H)\) be three operators. Then we have
See [2], p.284, for a direct method. Here, we simply proceed as follows. By definition, M is positive in \({\mathcal{B}}(H\oplus H)\) if and only if
for all \(u,v\in H\). According to Corollary 2.3, with Remark 2.1, we immediately deduce the desired aim.
Now, let us observe the two following examples illustrating, particularly, the situation of Corollary 2.4.
Example 3.3
Let a, b be complex numbers and \(p,p^{*}>1\) with \(1/p+1/p^{*}=1\). The inequality
is known as the Young inequality. We are in the situation of Corollary 2.4 with \(r=1\). We then immediately deduce the following Hölder inequality (in \({\mathbb{C}}^{n}\)):
valid for all complex numbers \(a_{i}\) and \(b_{i}\), \(1\leq i\leq n\).
Example 3.4
Let Ω be a nonempty open subset of \({\mathbb{R}}^{n}\) and \(f,g:\Omega\rightarrow{\mathbb{C}}\). The Young inequality asserts that
The map Φ defined by \(\Phi(\psi)=\int_{\Omega}\psi(s)\, ds\), for ψ Lebesgueintegrable on Ω, is linear and monotone. It follows that if \(f\in L^{p}(\Omega)\) and \(g\in L^{p^{*}}(\Omega)\) then we have
By Theorem 2.6, we then deduce the Hölder inequality in integration:
See also Example 5.1 (Section 5 below) for another point of view for proving this inequality. We leave to the reader the routine task of obtaining the Hölder inequality in \(l_{p}\), the space of pconvergent series.
Example 3.5
For all (Hermitian) positive definite matrices A and B and every \(p\in(1,\infty)\), with \(p^{*}=p/(p1)\), we have [3]
According to Corollary 2.4, we deduce that for all \(A_{i}\), \(B_{i}\), \(i=1,2,\ldots,m\) (Hermitian) positive definite matrices, we have
See [4], Theorem 4.1, pp.34, for a direct (but long) proof of (11) by using the spectral mapping theorem and some existing lemmas.
Example 3.6
Let \(\!\!\cdot\!\!\) be a unitarily invariant norm. The inequality [5–7]
holds for all \(T,S\in{\mathcal{B}}^{+}(H)\), \(X\in{\mathcal{B}}(H)\), with \(1/p+1/p^{*}=1\). By Corollary 2.3, (12) is equivalent to
which is stronger than (12). According to Corollary 2.4, (12) implies that
for all \(T_{i},S_{i}\in{\mathcal{B}}^{+}(H)\), \(i=1,2,\ldots,n\), and \(X\in{\mathcal{B}}(H)\).
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
Let C be a cone of \({\mathcal{B}}^{+}(H)\) and \(f,g,h:C \times C\rightarrow C\) be three operator maps such that
Assume that further f, g, and h are positively homogeneous, with respect to the first variable, of degrees p, q, and r, respectively, with \(p< r<q\). Then the inequality
holds true for all \(T,S\in C\).
Proof
By definition, the operator inequality (13) is equivalent to
The inequality (14) is true for \(u=0\). Now, fixing \(0\neq u\in E\), this inequality can be written as
where \(h_{u},f_{u},g_{u}:{\mathcal{B}}^{+}(H)\times{\mathcal{B}}^{+}(H)\rightarrow(0,\infty)\) are the three quadratic forms of (15), respectively. Obviously, we can then apply Theorem 2.5 here for obtaining the desired result after a simple manipulation, and this completes the proof. □
From the above theorem, we immediately deduce the following corollary.
Corollary 4.2
With the same hypotheses as in Theorem 4.1 we have
Now, we will illustrate the above theorem with some applications.
• Let λ be a real number such that \(0\leq\lambda\leq 1\) and \(T,S\in{\mathcal{B}}^{+*}(H)\). The power geometric mean \(T\, \sharp_{\lambda}\, S\) of T and S is defined by
while their weighted arithmetic mean is
For \(\lambda=1/2\), we simply write \(T\, \sharp\, S\) and \(T\oplus S\), respectively.
The Heinz operator mean of T and S is defined by
This operator mean interpolates \(T\, \sharp\, S\) and \(T\oplus S\) [8], in the sense that
holds true for all \(\lambda\in[0,1]\) and \(T,S\in{\mathcal{B}}^{+*}(H)\).
The first result of application here may be stated as well.
Theorem 4.3
With the above, the inequalities
hold true for all \(u\in H\).
Proof
Let us set
It is easy to see that h, f, and g are positively homogeneous, with respect to the first variable T, of degrees \(r=1/2\), \(p=1\lambda\), and \(q=\lambda\), respectively. If \(\lambda=1/2\), the left side of (18) is an equality. Now, considering the two cases \(0\leq\lambda <1/2\) and \(1/2<\lambda\leq1\), Theorem 4.1 yields
which after simple reduction yields the left side of (5). The right side of (18) follows by a simple application of the arithmeticgeometric mean inequality with (17), and this completes the proof. □
From the above theorem, we immediately deduce the following inequality:
Remark 4.1
The above inequalities can be written, respectively, in the following forms:
where for two real numbers \(a,b>0\), \(a\, \sharp\, b=\sqrt{ab}\) is the geometric mean of a and b.
• A second application here is stated as follows. Let T, S, and \(T\, \sharp_{\lambda}\, S\) be as above, \(0\leq\lambda\leq1\). The inequality
is known as the operator entropy inequality; see [6] for instance. The following result may be stated.
Theorem 4.4
With the above, for all \(u\in H\) we have
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. □
An application in convex analysis
We need here more notions. Let E be a locally convex space and \(E^{*}\) its topological dual with the bracket duality \(\langle\cdot,\cdot\rangle\). Let \(f:E\rightarrow \widetilde{\mathbb{R}}:=\mathbb{R}\cup\{\infty\}\) be a functional not identically equal to ∞. The effective domain of f is \(\operatorname{dom} f=\{ u\in E, f(u)<\infty\}\), and the conjugate of f is the functional \(f^{*}:E^{*}\rightarrow \widetilde{\mathbb{R}}\) defined through [9]
Later, we shall need the following lemma.
Lemma 5.1
Let \(f:E\rightarrow\widetilde{\mathbb{R}}\) be a functional not identically equal to ∞.

(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 \(p^{*}=p/(p1)\).
Proof
(i) From (20) we immediately deduce that \(f^{*}(u^{*})\geq \operatorname{Re}\langle u^{*},u\ranglef(u)\) for all \(u\in E\) and \(u^{*}\in E^{*}\). Taking \(u=0\) in this inequality we obtain \(f^{*}(u^{*})\geq0\) for all \(u^{*}\in E^{*}\).
(ii) Let \(0\neq\lambda\in\mathbb{R},\mathbb{C}\) be fixed. By definition we have
It is easy to verify that \((p^{*}1)p=p^{*}\) and so
Summarizing, we have shown that
From this equality, with the fact that \(f^{*}\) is always lower semicontinuous, we deduce
We then have \(f^{*}(0)\leq0\). Since f is homogeneous, \(f(0)=0\) and, by (i) we have \(f^{*}(0)\geq0\). This, with (22), implies that \(f^{*}(0)=0\) and (21) is also satisfied for \(\lambda=0\), and this completes the proof. □
Our main result of application in this section is the following.
Theorem 5.2
Let \(f:E\rightarrow\widetilde{\mathbb{R}}\) be a positive functional such that \(f^{*}\) is positive, too. Assume that f is homogeneous of degree \(p>1\). Then, for all \(u\in E\) and \(u^{*}\in E^{*}\), we have
Proof
From (20) we immediately deduce that
for all \(u\in E\) and \(u^{*}\in E^{*}\). If \(f(u)<\infty\) then (24) is equivalent to
If \(f(u)=\infty\) then \(f(u)+f^{*}(u^{*})=\infty\) and so (25) is also satisfied. In all cases we have
We can apply Corollary 2.3 with \(h(u,u^{*})=\langle u,u^{*}\rangle\) and \(g=f^{*}\). The desired result follows. □
Now, we will illustrate the above result with the following example.
Example 5.1
Let \(p>1\) and \(E=L^{p} (\Omega )\) be equipped with the classical norm
The topological dual of E is \(E^{*}=L^{p^{*}} (\Omega )\), with \(1/p+1/p^{*}=1\). Take \(f(u)=\frac{1}{p}\u\_{p}^{p}\) for which we have \(f^{*}(u^{*})=\frac{1}{p^{*}}\u^{*}\_{p^{*}}^{p^{*}}\) [10]. According to (23), we immediately obtain the classical Hölder inequality in \(L^{p}(\Omega)\), namely: \(\langle u,u^{*}\rangle\leq\u\_{p}\u^{*}\ _{p^{*}}\) for all \(u\in L^{p} (\Omega )\) and \(u^{*}\in L^{p^{*}} (\Omega )\). Similarly we can obtain the Hölder inequality in \({\mathbb{C}}^{n}\) and in \(l_{p}\), the space of pconvergent series. For \(p=2\), the above is reduced to the CauchySchwarz inequality.
The following example is also of interest.
Example 5.2
Let E be a Hilbert space and T be a (selfadjoint) positive operator from E into itself. Take \(f=f_{T}\) defined by
We know that [10]
where \(T^{+}\) denotes the pseudoinverse of T. This, with (23), implies that
holds for all \(u\in E\) and \(u^{*}\in \operatorname{ran} T^{1/2}\). In particular, if, moreover, T is invertible then
holds for all \(u,u^{*}\in E\).
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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.
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Raïssouli, M., Zine, R. On some Höldertype inequalities with applications. J Inequal Appl 2015, 93 (2015). https://doi.org/10.1186/s1366001506165
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Keywords
 Höldertype inequalities
 operator means
 operator inequalities
 functional inequalities
 convex analysis