# On some Hölder-type inequalities with applications

## 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:

1. (i)

C is such that $$u\in C$$ and $$t\geq0 \Rightarrow tu\in C$$; C is then called a cone of E.

2. (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 $$\Phi(u)\leq\Phi(v)$$.

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 semi-inner product if and only if

$$h(u,v)=\overline{h(v,u)}, \qquad h(\lambda u,v)=\lambda h(u,v)\quad \mbox{and} \quad h(u,\lambda v)=\overline{\lambda}h(u,v)$$

hold for all $$\lambda\in{\mathbb{C}}$$, $$u\in C_{1}$$ and $$v\in C_{2}$$. Clearly, every semi-inner 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

$$\inf_{t>0} \bigl(at^{p}+bt^{-q} \bigr)=(p+q) \biggl(\frac{b}{p} \biggr)^{\frac{p}{p+q}} \biggl( \frac{a}{q} \biggr)^{\frac{q}{p+q}}.$$

### 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

$$\phi'(t)=pat^{p-1}-qbt^{-q-1}$$

for all $$t>0$$, with $$\phi'(t)=0$$ if and only if

$$t=t_{0}= (qb/pa )^{1/(p+q)}.$$

$$\phi(t_{0})=(p+q) \biggl(\frac{b}{p} \biggr)^{\frac{p}{p+q}} \biggl(\frac{a}{q} \biggr)^{\frac{q}{p+q}}.$$

This, with the fact that

$$\lim_{t\rightarrow 0}\phi(t)=\lim_{t\rightarrow\infty}\phi(t)=\infty,$$

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

$$\forall(u,v)\in C_{1}\times C_{2},\quad h(u,v) \leq f(u)+g(v).$$
(1)

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

$$h(u,v)\leq \biggl(\frac{p}{r}f(u) \biggr)^{r/p} \biggl(\frac {p}{p-r}g(v) \biggr) ^{(p-r)/p}$$
(2)

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

$$t^{r}h(u,v)\leq t^{p}f(u)+g(v),$$

or equivalently

$$h(u,v)\leq t^{p-r}f(u)+t^{-r}g(v).$$

This means that the map $$t\mapsto t^{p-r}f(u)+t^{-r}g(v)$$, for $$t>0$$, is bounded below and so we can write

$$h(u,v)\leq\inf_{t>0} \bigl(t^{p-r}f(u)+t^{-r}g(v) \bigr).$$

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:

$$\biggl(\frac{1}{p}h(u,v) \biggr)^{p}\leq \biggl( \frac{1}{r}f(u) \biggr)^{r} \biggl(\frac{1}{p-r}g(v) \biggr) ^{p-r} \quad \mbox{if } p>0$$

and

$$\biggl(\frac{1}{-p}h(u,v) \biggr)^{p}\geq \biggl( \frac{1}{-r}f(u) \biggr)^{r} \biggl(\frac{1}{r-p}g(v) \biggr) ^{p-r} \quad \mbox{if } p< 0.$$

(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

$$\forall(u,v)\in C_{1}\times C_{2},\quad \operatorname{Re} \bigl(h(u,v) \bigr)\leq f(u)+g(v).$$
(3)

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 semi-inner product. Then the inequality

$$\bigl\vert h(u,v)\bigr\vert \leq \bigl(pf(u) \bigr)^{1/p} \bigl(p^{*}g(v) \bigr) ^{1/p^{*}}$$
(4)

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),

$$\operatorname{Re} \bigl(h(u,v) \bigr)\leq \bigl(pf(u) \bigr)^{1/p} \bigl(p^{*}g(v) \bigr) ^{1/p^{*}}$$

for all $$(u,v)\in C_{1}\times C_{2}$$, with $$p^{*}=p/(p-1)$$. 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 semi-inner product, we obtain after elementary manipulation

$$\bigl\vert h(u,v)\bigr\vert ^{2}\leq\bigl\vert h(u,v)\bigr\vert \bigl(pf(u) \bigr)^{1/p} \bigl(p^{*}g(v) \bigr)^{1/p^{*}}.$$

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

$$\sum_{i=1}^{n} h(u_{i},v_{i})\leq \Biggl(\frac{p}{r}\sum _{i=1}^{n}f(u_{i}) \Biggr)^{r/p} \Biggl(\frac{p}{p-r}\sum_{i=1}^{n}g(v_{i}) \Biggr) ^{(p-r)/p}$$
(5)

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

$$\tilde{h}(u,v):=\sum_{i=1}^{n}h(u_{i},v_{i}) \leq\sum_{i=1}^{n}f(u_{i})+\sum _{i=1}^{n}g(v_{i}):=\tilde{f}(u)+ \tilde{g}(v)$$

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

$$\forall(u,v)\in C_{1}\times C_{2} ,\quad h(u,v)\leq f(u,v)+g(u,v).$$

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

$$\biggl(\frac{1}{q-p}h(u,v) \biggr)^{q-p}\leq \biggl( \frac {1}{q-r}f(u,v) \biggr)^{q-r} \biggl(\frac{1}{r-p}g(u,v) \biggr) ^{r-p}$$
(6)

holds true for all $$(u,v)\in C_{1}\times C_{2}$$.

### Proof

Analogously to the proof of Theorem 2.2, we show that

$$h(u,v)\leq\inf_{t>0} \bigl(t^{p-r}f(u,v)+t^{q-r}g(u,v) \bigr).$$

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

$$\forall(u,v)\in C_{1}\times C_{2} ,\quad h(u,v)\prec f(u)+g(v).$$
(7)

Assume that f and h are as in Theorem  2.2. If $$\Phi :C\rightarrow[0,\infty)$$ is monotone sub-additive and homogeneous of degree $$s>0$$, then the inequality

$$\Phi \bigl(h(u,v) \bigr)\leq \biggl(\frac{p}{r}\Phi \bigl(f(u) \bigr) \biggr)^{r/p} \biggl(\frac{p}{p-r}\Phi \bigl(g(v) \bigr) \biggr) ^{(p-r)/p}$$
(8)

holds true for all $$(u,v)\in C_{1}\times C_{2}$$.

### Proof

With the fact that Φ is monotone and sub-additive, (7) implies that

$$\Phi \bigl(h(u,v) \bigr)\leq\Phi \bigl(f(u) \bigr)+\Phi \bigl(g(v) \bigr),$$

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

$$\forall(u,v)\in C_{1}\times C_{2} ,\quad h(u,v)\prec f(u,v)+g(u,v).$$
(9)

Assume that f and h are as in Theorem  2.5. If $$\Phi :C\rightarrow[0,\infty)$$ is monotone sub-additive and homogeneous of degree $$s>0$$, then the inequality

$$\biggl(\frac{1}{q-p}\Phi \bigl(h(u,v) \bigr) \biggr)^{q-p}\leq \biggl(\frac {1}{q-r}\Phi\bigl(f(u,v) \bigr) \biggr)^{q-r} \biggl(\frac{1}{r-p}\Phi \bigl(g(u,v) \bigr) \biggr) ^{r-p}$$
(10)

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 self-adjoint 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 self-adjoint 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 self-adjoint and $$S-T\in{\mathcal{B}}^{+}(H)$$. The space $${\mathcal{B}}(H)$$ is endowed with the classical operator norm, namely

$$\|T\|=\sup_{\|u\|=1}\|Tu\|.$$

It is well known that if T is positive then

$$\|T\|=\sup_{\|u\|=1}\langle Tu,u\rangle.$$

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

$$0\leq\|Tu-Sv\|^{2}=\bigl\langle \bigl(T^{*}T\bigr)u,u\bigr\rangle -2 \operatorname{Re}\langle Tu,Sv\rangle+\bigl\langle \bigl(S^{*}S\bigr)v,v\bigr\rangle ,$$

which, by Corollary 2.3 with $$p=p^{*}=2$$, yields

$$\bigl\vert \langle Tu,Sv\rangle\bigr\vert ^{2}\leq\bigl\langle |T|^{2}u,u\bigr\rangle \bigl\langle |S|^{2}v,v\bigr\rangle ,$$

where as usual $$|T|= (T^{*}T )^{1/2}$$. If $$S=T^{*}$$ then

$$\bigl\vert \bigl\langle T^{2}u,v\bigr\rangle \bigr\vert ^{2}\leq\bigl\langle |T|^{2}u,u\bigr\rangle \bigl\langle \bigl\vert T^{*}\bigr\vert ^{2}v,v\bigr\rangle ,$$

see [1]. If T is positive (self-adjoint) then

$$\bigl\vert \langle Tu,v\rangle\bigr\vert ^{2}\leq\langle Tu,u \rangle \langle Tv,v\rangle,$$

which is a well-known extension of the Cauchy-Schwarz inequality.

### Example 3.2

Let $$T,S,X\in{\mathcal{B}}(H)$$ be three operators. Then we have

\begin{aligned}& M:=\left (\begin{array}{@{}c@{\quad}c@{}} T & X^{*} \\ X & S \end{array} \right ) \mbox{ is positive in } {\mathcal{B}}(H\oplus H) \\& \quad \Leftrightarrow\quad \forall u,v\in H,\quad \bigl\vert \langle Xu,v\rangle\bigr\vert ^{2}\leq \langle Tu,u\rangle \langle Sv,v\rangle. \end{aligned}

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

$$\langle Tu,u\rangle+2\operatorname{Re}\langle Xu,v\rangle+\langle Sv,v\rangle \geq0$$

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

$$|a||b|\leq\frac{1}{p}|a|^{p}+\frac{1}{p^{*}}|b|^{p^{*}}$$

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 $${\mathbb{C}}^{n}$$):

$$\sum_{i=1}^{n}|a_{i}||b_{i}| \leq \Biggl(\sum_{i=1}^{n}|a_{i}|^{p} \Biggr)^{1/p} \Biggl(\sum_{i=1}^{n}|b_{i}|^{p^{*}} \Biggr)^{1/p^{*}},$$

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

$$\forall s\in\Omega,\quad \bigl\vert f(s)\bigr\vert \bigl\vert g(s)\bigr\vert \leq\frac{1}{p}\bigl\vert f(s)\bigr\vert ^{p}+ \frac {1}{p^{*}}\bigl\vert g(s)\bigr\vert ^{p^{*}}.$$

The map Φ defined by $$\Phi(\psi)=\int_{\Omega}\psi(s)\, ds$$, for ψ Lebesgue-integrable on Ω, is linear and monotone. It follows that if $$f\in L^{p}(\Omega)$$ and $$g\in L^{p^{*}}(\Omega)$$ then we have

$$\int_{\Omega}\bigl\vert (fg) (s)\bigr\vert \, ds\leq\int _{\Omega}\bigl\vert f(s)\bigr\vert ^{p}\, ds+\int _{\Omega}\bigl\vert g(s)\bigr\vert ^{p^{*}}\, ds.$$

By Theorem 2.6, we then deduce the Hölder inequality in integration:

$$\int_{\Omega}\bigl\vert (fg) (s)\bigr\vert \, ds\leq \biggl( \int_{\Omega}\bigl\vert f(s)\bigr\vert ^{p}\, ds \biggr)^{1/p} \biggl(\int_{\Omega}\bigl\vert g(s)\bigr\vert ^{p^{*}}\, ds \biggr)^{1/p^{*}}.$$

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 p-convergent series.

### Example 3.5

For all (Hermitian) positive definite matrices A and B and every $$p\in(1,\infty)$$, with $$p^{*}=p/(p-1)$$, we have [3]

$$\operatorname{tr}(AB)\leq\frac{1}{p}trA^{p}+\frac{1}{p^{*}} \operatorname{tr}B^{p^{*}}.$$

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

$$\operatorname{tr}\sum_{i=1}^{m}A_{i}B_{i} \leq \Biggl(\sum_{i=1}^{m} \operatorname{tr}A_{i}^{p} \Biggr)^{1/p} \Biggl(\sum _{i=1}^{m}\operatorname{tr}B_{i}^{p^{*}} \Biggr)^{1/p^{*}}.$$
(11)

See [4], Theorem 4.1, pp.3-4, 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 [57]

$$|\!|\!|TXS|\!|\!|\leq\frac{1}{p}\bigl|\!\bigl|\!\bigl|T^{p}X\bigr|\!\bigr|\!\bigr|+\frac{1}{p^{*}}\bigl|\!\bigl|\!\bigl|XS^{p^{*}} \bigr|\!\bigr|\!\bigr|$$
(12)

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

$$|\!|\!|TXS|\!|\!|\leq\bigl|\!\bigl|\!\bigl|T^{p}X\bigr|\!\bigr|\!\bigr|^{1/p} \bigl|\!\bigl|\!\bigl|XS^{p^{*}}\bigr|\!\bigr|\!\bigr|^{1/p^{*}},$$

which is stronger than (12). According to Corollary 2.4, (12) implies that

$$\sum_{i=1}^{n}|\!|\!|T_{i}XS_{i} |\!|\!|\leq \Biggl(\sum_{i=1}^{n}\bigl|\!\bigl|\!\bigl|T_{i}^{p}X\bigr|\!\bigr|\!\bigr|\Biggr)^{1/p} \Biggl(\sum_{i=1}^{n}\bigl|\!\bigl|\!\bigl|XS_{i}^{p^{*}}\bigr|\!\bigr|\!\bigr|\Biggr)^{1/p^{*}}$$

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

$$\forall T,S\in C ,\quad h(T,S)\leq f(T,S)+g(T,S).$$
(13)

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

$$\biggl(\frac{1}{q-p}\bigl\langle h(T,S)u,u\bigr\rangle \biggr)^{q-p}\leq \biggl(\frac {1}{q-r}\bigl\langle f(T,S)u,u\bigr\rangle \biggr)^{q-r} \biggl(\frac{1}{r-p}\bigl\langle g(T,S)u,u \bigr\rangle \biggr) ^{r-p}$$
(14)

holds true for all $$T,S\in C$$.

### Proof

By definition, the operator inequality (13) is equivalent to

$$\forall u\in H ,\quad \bigl\langle h(T,S)u,u\bigr\rangle \leq\bigl\langle f(T,S)u,u\bigr\rangle +\bigl\langle g(T,S)u,u\bigr\rangle .$$
(15)

The inequality (14) is true for $$u=0$$. Now, fixing $$0\neq u\in E$$, this inequality can be written as

$$h_{u}(T,S)\leq f_{u}(T,S)+g_{u}(T,S),$$

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

$$\biggl(\frac{1}{q-p}\bigl\Vert h(T,S)\bigr\Vert \biggr)^{q-p}\leq \biggl(\frac{1}{q-r}\bigl\Vert f(T,S)\bigr\Vert \biggr)^{q-r} \biggl(\frac{1}{r-p}\bigl\Vert g(T,S)\bigr\Vert \biggr)^{r-p}.$$
(16)

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

$$T\, \sharp_{\lambda}\, S=S^{1/2} \bigl(S^{-1/2}TS^{-1/2} \bigr)^{1-\lambda}S^{1/2} =T^{1/2} \bigl(T^{-1/2}ST^{-1/2} \bigr)^{\lambda}T^{1/2}=S\, \sharp _{1-\lambda}\, T,$$

while their weighted arithmetic mean is

$$T\oplus_{\lambda} S=(1-\lambda)T+\lambda S=S\oplus_{1-\lambda}T.$$

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

$$H_{\lambda}(T,S)=\frac{T\, \sharp_{\lambda}\, S+T\, \sharp_{1-\lambda}\, S}{2}.$$

This operator mean interpolates $$T\, \sharp\, S$$ and $$T\oplus S$$ [8], in the sense that

$$T\, \sharp\, S\leq H_{\lambda}(T,S)\leq T\oplus S$$
(17)

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

$$\bigl(\bigl\langle (T\, \sharp\, S )u,u\bigr\rangle \bigr)^{2}\leq\bigl\langle (T\, \sharp _{\lambda}\, S )u,u\bigr\rangle \bigl\langle (T\, \sharp_{1-\lambda}\, S )u,u\bigr\rangle \leq \bigl(\bigl\langle (T \oplus S )u,u\bigr\rangle \bigr)^{2}$$
(18)

hold true for all $$u\in H$$.

### Proof

Let us set

$$h(T,S)=2 T\, \sharp\, S, \qquad f(T,S)=T\, \sharp_{\lambda}\, S, \qquad g(T,S)=T\, \sharp _{1-\lambda}\, S.$$

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

\begin{aligned}& \biggl(\frac{1}{|2\lambda-1|}\bigl\langle (T\, \sharp\, S )u,u\bigr\rangle \biggr)^{|2\lambda-1|} \\& \quad \leq \biggl(\frac{1}{|2\lambda-1|}\bigl\langle (T\, \sharp_{\lambda}\, S )u,u \bigr\rangle \biggr)^{|\lambda-1/2|} \biggl(\frac{1}{|2\lambda-1|}\bigl\langle (T \, \sharp_{1-\lambda}\, S )u,u\bigr\rangle \biggr)^{|\lambda-1/2|}, \end{aligned}

which after simple reduction yields the left side of (5). The right side of (18) follows by a simple application of the arithmetic-geometric mean inequality with (17), and this completes the proof. □

From the above theorem, we immediately deduce the following inequality:

$$\|T\, \sharp\, S\|^{2}\leq\|T\, \sharp_{\lambda}\, S\|\|T \, \sharp_{1-\lambda}\, S\|\leq\| T\oplus S\|^{2}.$$

### Remark 4.1

The above inequalities can be written, respectively, in the following forms:

\begin{aligned}& \bigl\langle (T\, \sharp\, S )u,u\bigr\rangle \leq\bigl\langle (T\, \sharp_{\lambda }\, S )u,u\bigr\rangle \, \sharp\, \bigl\langle (T\, \sharp_{1-\lambda}\, S )u,u\bigr\rangle \leq \bigl\langle (T\oplus S )u,u\bigr\rangle , \\& \|T\, \sharp\, S\|\leq\|T\, \sharp_{\lambda}\, S\|\, \sharp\, \|T\, \sharp_{1-\lambda}\, S\| \leq\|T\oplus S\|, \end{aligned}

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

$$(1+\lambda)T\leq\lambda TS^{-1}T+T\, \sharp_{\lambda}\, S$$
(19)

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

\begin{aligned}& \bigl(\langle Tu,u\rangle \bigr)^{1+\lambda}\leq \bigl(\bigl\langle TS^{-1}Tu,u\bigr\rangle \bigr)^{\lambda}\bigl\langle (T\, \sharp_{\lambda}\, S )u,u\bigr\rangle , \\& \|T\|^{1+\lambda}\leq\bigl\| TS^{-1}T\bigr\| ^{\lambda}\|T \, \sharp_{\lambda}\, S\|. \end{aligned}

### 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]

$$\forall u^{*}\in E,\quad f^{*}\bigl(u^{*} \bigr)= {\sup_{u\in E}} \bigl(\operatorname{Re}\bigl\langle u,u^{*}\bigr\rangle -f(u) \bigr).$$
(20)

Later, we shall need the following lemma.

### Lemma 5.1

Let $$f:E\rightarrow\widetilde{\mathbb{R}}$$ be a functional not identically equal to ∞.

1. (i)

If $$f(0)\leq0$$ then $$f^{*}$$ is a positive functional.

2. (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 $$u^{*}\in E^{*}$$.

(ii) Let $$0\neq\lambda\in\mathbb{R},\mathbb{C}$$ be fixed. By definition we have

\begin{aligned} f^{*}\bigl(\lambda u^{*}\bigr) =&\sup_{u\in E} \bigl( \operatorname{Re}\bigl\langle u,\lambda u^{*} \bigr\rangle -f(u) \bigr) \\ =&\sup _{u\in E} \bigl(\operatorname{Re}\bigl\langle \lambda| \lambda|^{p^{*}-2}u,\lambda u^{*} \bigr\rangle -f \bigl(\lambda| \lambda|^{p^{*}-2}u \bigr) \bigr) \\ =&\sup_{u\in E} \bigl(|\lambda|^{p^{*}}\operatorname{Re} \bigl\langle u,u^{*} \bigr\rangle -|\lambda|^{(p^{*}-1)p}f(u) \bigr). \end{aligned}

It is easy to verify that $$(p^{*}-1)p=p^{*}$$ and so

$$f^{*}\bigl(\lambda u^{*}\bigr)=|\lambda|^{p^{*}}\sup_{u\in E} \bigl(\operatorname{Re}\bigl\langle u,u^{*} \bigr\rangle -f(u) \bigr)=| \lambda|^{p^{*}}f^{*}\bigl(u^{*}\bigr).$$

Summarizing, we have shown that

$$\forall u^{*}\in \operatorname{dom} f^{*} , \forall\lambda\neq0,\quad f^{*}\bigl(\lambda u^{*}\bigr)=|\lambda|^{p^{*}}f^{*}\bigl(u^{*}\bigr).$$
(21)

From this equality, with the fact that $$f^{*}$$ is always lower semi-continuous, we deduce

$$\forall u^{*}\in \operatorname{dom} f^{*} ,\quad f^{*}(0)\leq\liminf _{\lambda\rightarrow 0}f^{*}\bigl(\lambda u^{*}\bigr)=\liminf_{\lambda\rightarrow 0} \bigl(|\lambda|^{p^{*}}f^{*}\bigl(u^{*}\bigr) \bigr)=0.$$
(22)

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

$$\bigl\vert \bigl\langle u,u^{*}\bigr\rangle \bigr\vert \leq \bigl(pf(u) \bigr)^{1/p} \bigl(p^{*}f^{*}\bigl(u^{*}\bigr) \bigr)^{1/p^{*}}.$$
(23)

### Proof

From (20) we immediately deduce that

$$\operatorname{Re}\bigl\langle u,u^{*}\bigr\rangle -f(u)\leq f^{*} \bigl(u^{*}\bigr)$$
(24)

for all $$u\in E$$ and $$u^{*}\in E^{*}$$. If $$f(u)<\infty$$ then (24) is equivalent to

$$\operatorname{Re}\bigl\langle u,u^{*}\bigr\rangle \leq f(u)+f^{*} \bigl(u^{*}\bigr).$$
(25)

If $$f(u)=\infty$$ then $$f(u)+f^{*}(u^{*})=\infty$$ and so (25) is also satisfied. In all cases we have

$$\forall u\in E , \forall u^{*}\in E^{*}, \quad \operatorname{Re}\bigl\langle u,u^{*} \bigr\rangle \leq f(u)+f^{*}\bigl(u^{*}\bigr).$$

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

$$\forall u\in L^{p} (\Omega ) ,\quad \|u\|_{p}= \biggl(\int _{\Omega }\bigl\vert u(t)\bigr\vert ^{p}\, dt \biggr)^{1/p}.$$

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 p-convergent series. For $$p=2$$, the above is reduced to the Cauchy-Schwarz inequality.

The following example is also of interest.

### Example 5.2

Let E be a Hilbert space and T be a (self-adjoint) positive operator from E into itself. Take $$f=f_{T}$$ defined by

$$\forall u\in E ,\quad f(u)=f_{T}(u):=(1/2)\langle Tu,u\rangle =(1/2) \bigl\Vert T^{1/2}u\bigr\Vert ^{2}.$$

We know that [10]

$$(f_{T})^{*}\bigl(u^{*}\bigr)=(1/2)\bigl\Vert \bigl(T^{1/2}\bigr)^{+}u^{*}\bigr\Vert ^{2} \quad \mbox{if } u^{*} \in \operatorname{ran} T^{1/2}, \qquad (f_{T})^{*}\bigl(u^{*} \bigr)=\infty \quad \mbox{otherwise},$$
(26)

where $$T^{+}$$ denotes the pseudo-inverse of T. This, with (23), implies that

$$\bigl\vert \bigl\langle u,u^{*}\bigr\rangle \bigr\vert \leq\bigl\Vert T^{1/2}u\bigr\Vert \bigl\Vert \bigl(T^{1/2} \bigr)^{+}u^{*} \bigr\Vert$$

holds for all $$u\in E$$ and $$u^{*}\in \operatorname{ran} T^{1/2}$$. In particular, if, moreover, T is invertible then

$$\bigl\vert \bigl\langle u,u^{*}\bigr\rangle \bigr\vert ^{2}\leq \langle Tu,u\rangle \bigl\langle T^{-1}u^{*},u^{*}\bigr\rangle$$

holds for all $$u,u^{*}\in E$$.

## References

1. Dragomir, SS: Some inequalities generalizing Kato’s and Furuta’s results. Filomat 28(1), 179-195 (2014)

2. Kittaneh, F: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 24, 283-293 (1988)

3. Magnus, JR: A representation theorem for $$(\operatorname{tr}A^{p} )^{1/p}$$. Linear Algebra Appl. 95, 127-134 (1987)

4. Zhou, H: On some trace inequalities for positive definite Hermitian matrices. J. Inequal. Appl. 2014, 64 (2014)

5. Ando, T: Matrix Young inequalities. Oper. Theory, Adv. Appl. 75, 33-38 (1995)

6. Furuta, T: Comprehensive survey on an order preserving operator inequality. Banach J. Math. Anal. 7(1), 14-40 (2013)

7. Mond, B, Pearce, CEM, Pečarić, J: The logarithmic mean is a mean. Math. Commun. 2, 35-39 (1997)

8. Kittaneh, F, Krnić, M, Lovričević, N, Pečarić, J: Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators. Publ. Math. (Debr.) 80(3-4), 465-478 (2012)

9. Raïssouli, M, Chergui, M: Arithmetico-geometric and geometrico-harmonic means of two convex functionals. Sci. Math. Jpn. 55(3), 485-492 (2002)

10. Aubin, JP: L’analyse Non Lineaire et ses Motivations Economiques. Masson, Paris (1985)

## 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.

## Author information

Authors

### Corresponding author

Correspondence to Rabie Zine.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors worked in coordination. Both authors carried out the proof, read, and approved the final version of the manuscript.

## Rights and permissions

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.

Reprints and Permissions

Raïssouli, M., Zine, R. On some Hölder-type inequalities with applications. J Inequal Appl 2015, 93 (2015). https://doi.org/10.1186/s13660-015-0616-5

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/s13660-015-0616-5

### Keywords

• Hölder-type inequalities
• operator means
• operator inequalities
• functional inequalities
• convex analysis