A new look at classical inequalities involving Banach lattice norms

Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for ‘continuous’ many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results of this type and also by deriving some new results related to classical Popoviciu’s, Bellman’s and Beckenbach-Dresher’s inequalities.


Introduction
Let (Y , , ν) be a σ -finite measure space, and let L  (Y ) denote the space of ν-measurable functions defined and being finite a.e. on Y . A Banach subspace (E, · ) of L  (Y ) is a Banach lattice (Banach function space) on (Y , , ν) if, for every x ∈ E, y ∈ L  (Y ), |y| ≤ |x|, ν-a.e., it follows that y ∈ E and y ≤ x .
Moreover, the 'convexification' of E, denoted by E p , -∞ < p < ∞, p =  consists of all x ∈ L  (Y ) satisfying For the case p < , we assume that x = x(t) =  for all t ∈ Y .
Some classical inequalities are known to hold also in the frame of such Banach lattice norms. See, for example, [] and [].
It is also known that some classical inequalities for finite many functions (like those of Hölder and Minkowski) can be generalized to hold for continuous (infinitely) many functions. For such results in L p and l p -spaces, we refer the reader to the recent article [] and the references therein. We proved there the continuous versions of Popoviciu's and Bellman's inequalities.
However, there exists a generalization of Hölder's inequality in both of these directions simultaneously, see [] and also Lemma ..
The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. Some known results of this type which we need in this paper can be found in our Section . We now shortly discuss some elementary forms of the inequalities we consider to generalize as new contributions in this paper.
We assume that a i , b i , i = , , . . . , n, are positive numbers, c  , c  are positive numbers and f and g are positive functions on (Y , , ν).
A. Popoviciu's inequality: Let p, q ≥ ,  p +  q = . In the most elementary form it reads (see [ For more general forms, see, e.g., Theorem . which in particular shows that '>' in the assumptions of (.) and (.) can be replaced by '≥' . Some continuous forms of (.) and (.) were recently proved in [].
In Section  of this paper we present, prove and apply our main results concerning Popoviciu's inequality (see Theorems . and .). B There is also a more general integral form of this inequality, namely: which holds under proper conditions. For more general forms, see, e.g., Theorem .. Also here our result shows in particular that '>' in the assumptions of (.) and (.) can be replaced by '≥' . Some continuous forms of (.) and (.) were recently proved in []. Our main results related to this inequality are proved and discussed in Section . We remark that obviously Popoviciu's and Bellman's inequalities may be regarded as a type of reversed inequalities of Hölder's and Minkowski's inequalities, respectively.

C. Beckenbach-Dresher's inequality: In its most elementary form it reads: If
Especially for the case p = α/(αβ), α = β, we obtain the triangle inequality for the so-called Gini-means G defined by There are many generalizations of inequality (.). Of special importance as the background for this paper, we mention [], where also a version for the spaces E p is included.
In Section  of the present paper, we derive a new version of (.) which is both 'continuous' (containing infinitely many functions, e.g., sequences) and involving Banach lattice norms (see Theorem .). Moreover, we also derive a type of reversed inequality of the same general form (see Theorem .). Finally, Section  is reserved for some concluding remarks and results. Especially, we present new Popoviciu's inequality in the case of infinite interpolation families (see Theorem .), and the connection to Milne's inequality is pointed out (see Section .).

Preliminaries
It is known that if · E is a Banach function norm, then f (x, ·) E need not be a measurable function. But it is also known that if E has the Fatou property, then indeed f (x, ·) E is measurable (see []). Therefore, for simplicity, we assume that the considered Banach function spaces have the Fatou property. It is also known that in this situation E is a perfect space, i.e., E = E , where E denotes the second associate space of E.
We need the following simple generalization of Hölder's inequality.
Lemma . Let p, q = ,  p +  q = . If p > , then If p < , then (.) holds in the reverse direction. We have equality in (.) when g = cf p- .
A simple proof of this lemma in an even general symmetric form can be found in [], p. .
We also need the following more general form of Hölder's inequality (both continuous and involving Banach function norms).

measurable and define p by
A proof of this result can be found in []. We note that (.) is an inequality between generalized geometric means. We also need the following inequality.
We also need the following analogous version of Minkowski's inequality.
Since E has the Fatou property and p ≥ , we have that E p has the Fatou property, i.e., it is a perfect space, and the proof can be found in [], Chapter . Note that in the case E = L  (Y ) this is just the classical integral Minkowski inequality.
Remember that the Banach lattice E is p-convex or q-concave if there exists a positive constant M such that, for every finite set x  , x  , . . . , x n of elements in E, we have The smallest M satisfying the corresponding inequality is called constant of p-convexity, respectively, of q-concavity.
In [] the following fact appears: Let ρ and λ be function norms with the Fatou property and assume that there exists  ≤ p ≤ ∞ such that ρ is p-convex and λ is p-concave. Then there exists a constant C such that, for all measurable f (x, y), we have If we take ρ(h) to be X h(x) dx which is -convex with constant of convexity equal to  and λ = E , where E is -concave with constant of concavity equal to M, and follow the part of the proof of (.) in which the constant of convexity of the norm ρ(·) is equal to , we find that

Popoviciu type inequalities involving Banach function norms
Our first main result reads as follows.
provided that all integrals which occur in (.) exist.
Next we use Lemma . to conclude that We combine the above two inequalities and obtain (.). The proof is complete.
Next we state the following complementary result.
We can get the result like a corollary from inequality (.), by taking f  (x) = c p  for x ∈ X  and f  (x) = c q  and f (x, y) = f p (y) for x ∈ X  , f (x, y) = g q (y) for x ∈ X  .
If, for instance, c p f p E = , g q g q E ≥ , we have to show that c  c  ≥ fg . For this purpose we just use Lemma ., namely (b) The case  < p <  can be treated by using the same Lemma ., which says that which together with Hölder's inequality gives us the wanted inequality.
(c) The case p <  can be proved similarly (just interchange the roles of f and g and p and q, respectively).
Remark . Note that Theorem . in particular means that inequalities (.) and (.) hold also if '>' in these inequalities are replaced by '≥' .
We also state a generalization of Theorem .(a).

Corollary . Let E be a Banach function space
Proof In the case when c p f p E , g q g q E are strictly positive, we can get like corollary from Theorem .. First we take u( Look, for instance, to the third expression in inequality (.) which is the third expression in (.). Even simpler we see that the two first terms in (.) coincide with the corresponding terms in (.).
For the purpose, we just use Lemma ., namely The proof is complete.
Hence, for  < r, s < , the following chain of inequalities holds: If we compare inequalities (.) and (.), we can see that in the case  < r, s <  inequality (.) is better than inequality (.). Moreover, since in the case r, s >  inequality (.) holds in the reversed direction when r > , in the case r, s >  inequality (.) is stronger than inequality (.).
. Note that here we do not need the condition E = E because of Remark after Theorem . from [].

Bellman type inequalities involving Banach function norms
Our first main result in this case reads as follows.
Theorem . Let X and Y be measure spaces, let f (x, y) be a positive measurable function on X × Y and assume that p ≥  and f  (x) is a function on X such that f

where E is a Banach function space on Y for all x ∈ X. Assume that E has the Fatou property. Then
provided that all integrals exist. If E is -concave with constant of concavity ,  < p <  or p <  and f p (x, ·) E > , then inequality (.) holds in the reverse direction.
Proof Let p ≥ . We consider the following form of Minkowski's integral inequality: We choose and obtain that Next, by using (.) to the second term in the above inequality, we find that If  < p < , then first we use reverse to inequality (.) and then instead of (.) we use (.) for M = .
The proof in the case p <  is similar.
Remark . Inequality (.) can be written as follows: If E = L  , then we get the result of the first part of the continuous Bellman inequality for p ≥  proved in [], Theorem ..
Next, we state the following Bellman type inequalities.
We follow the idea of the proof of Theorem . from [] using the discrete Minkowski inequality We note that the right-hand side in (.) is equal to c  + c  , (a  + b  ) p coincides with the term on the left-hand side in (.) and, by (.), By using these facts and taking pth power of both sides in (.), we get (.).
(b) All inequalities above hold in the reverse direction in this case, and the proof follows by just doing obvious modifications of the proof of (a).
Remark . Note that similarly as in our previous section, Theorem . in particular means that inequalities (.) and (.) hold also if '>' in the statements of these inequalities are replaced by '≥' .

Direct and reverse Beckenbach-Dresher type inequalities involving Banach function norms
The following result concerning Beckenbach-Dresher's inequality was announced in []. For completeness, we give here also the proof.
Theorem . Let E, F be Banach function spaces with the Fatou property. If  < u < ,  < p, q ≤  and E is -concave with constant of concavity equal to M, F is -concave with constant of concavity equal to N , then the inequality holds with C = M -u N u- , providing all above integrals exist. If u > , q ≤  ≤ p, q =  and F is -concave with constant of concavity equal to N, then inequality (.) holds in the reverse direction with C = N u- .
If u < , p ≤  ≤ q, p =  and E is -concave with constant of concavity equal to M, then inequality (.) holds in the reverse direction with C = M -u .
Let  < u < , E be -concave with constant of concavity equal to M, and let F be concave with constant of concavity equal to N . Then Let u > , E be just a Banach function space, and let F be -concave with constant of concavity equal to N . Then we have and the statement follows by using reverse Hölder's inequality.
Let u < , E be -concave with constant of concavity equal to M, and let F be just a Banach function space. Then and as before we use reverse Hölder's inequality to complete the proof.
Remark . Inequality (.) can be rewritten as follows: Theorem . Let f  (x) > f (x, ·) E p for all x ∈ X, let g(x, z) be a positive measurable function on X × Z and assume that g  (x) is a function on X such that g  (x) > g(x, ·) F q for all x ∈ X, where E is a Banach function space on Y for all x ∈ X with the Fatou property and F is a Banach function space on Z for all x ∈ X with the Fatou property.
If  < u < , p ≥  or p <  and q ≥  or q < , then the following continuous reverse type version of Beckenbach-Dresher's inequality holds: If u ≥ ,  < p ≤  and q ≥  or q < , then reverse inequality (.) holds. If u < ,  < q ≤  and p ≥  or p < , then reverse inequality (.) holds. In the cases when p < , an additional condition on the function space E is that it should be concave with constant of concavity ; in the cases when q < , an additional condition on the function space F is that it should be concave with constant of concavity .
Proof In the proof we use Theorem . and then Hölder's inequality. Denote f (z  ) at z  of certain analytic vector-valued functions f (z) on D whose boundary values