Some new inequalities involving the Hardy operator

In this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on (0,b],0


INTRODUCTION
In 1928 G. H. Hardy (see [6]) himself proved the following generalization of his famous inequality from 1925 (see [5]) namely whenever ≥ 1 and < − 1. The constant is sharp. Here, and in the sequel in this paper, all functions are supposed to be nonnegative and measurable (and if some negative power of the function appears, then we assume that the function is strictly positive a.e.).
The first Hardy inequality was for the case = 0. The continued history of Hardy-type inequalities up to 2007 can be found in [8] and some complements in [9]. It was recently pointed out in [16] (see also [9]) that (1.1) is not a genuine generalization of this first inequality, because both are equivalent to the "fundamental" Hardy inequality , respectively.
The classical Hardy operator is defined by We are dealing with nonnegative functions. Hence, in order to prove (1.1) we need only to prove (1.2) and this, in its turn, is just to use Jensen's inequality to see that ( ( ( )) ≤ ( ( ( )) ) (1. 3) and reverse the order of integration. The same technique works also for finite intervals and we have that for 0 < ≤ ∞ the inequality ≥ 1 or < 0, holds and with the same substitutions, is equivalent to the inequality where 0 < 0 ≤ ∞ (formally 0 = −1− ) and ≥ 1, < − 1 or < 0, > − 1. The constants are sharp since the constant 1 in (1.4) is sharp. See [9], p. 340. In connection to (1.4) it is also natural to study the so-called Jensen functional as a measure of the so-called "Jensen gap". In Section 2 we shall discuss further the Jensen functional in more general situations and prove some new estimates of the Hardy type operator and a corresponding new Hardy-type inequality with this basic idea (see Theorem 2.8).
In Section 3 we prove some new reverse Hardy-type inequalities on the cone of non-increasing functions in the setting of a general Banach lattice = (0, ), 0 < ≤ ∞. As applications we get some well-known such reverse Hardy-type inequalities but now including also the case with finite intervals.
Finally, in Section 4 we study the problem, to find conditions which guarantee that the left-hand side of (1.1) is finite even in cases when do not satisfy the restriction ≥ 1, < − 1. Here we use some fairly new continuous generalizations of Bellman's inequality (see [14] and [15]) in a crucial way.

T WO-SIDED ESTIMATES OF THE JENSEN FUNCTIONAL AND HARDY OPERATORS
A simple form of the Jensen's inequality reads: Theorem 2.1. Let be an interval on , let ( ) be a weight function on such that ∫ ( ) dx = 1. If is a measurable function on and is a convex function on an open interval, which contains the image of , then ( It is also well known that in special cases of convex functions we can also find good upper estimates of the so-called "Jensen gap" see e.g. [1], the new book [13] and the references there. Note also that, by the Jensen inequality mentioned above, ( ) ≥ 0 and, since this inequality holds in the reversed direction for concave functions, that ( ) ≤ 0 in this case. For later purposes in this paper we need the following result of this type.
If instead is concave, then these inequalities hold in the reversed directions.
A discrete version of this lemma is proved in [3] and a related result can be found in [4] for isotonic normalized functionals. We present the following proof.
Proof. For a differentiable convex function , we have that and (2.1) hold in reversed direction when is concave. This follows e.g. from a result of O. Stolz, see e.g. Theorem 1.4.2 in [13]. Now, for convex, putting = ∫ ( ) ( ) dt and = ( ) we get ) .
After integrating over we get the second inequality, the first one is just Jensen's inequality.
The proof in the case of concave is analogous. □ Next we formulate the following consequence of the Jensen and reverse Jensen inequalities. (a) If ≥ 1 or < 0, then By using this lemma we can conclude that when ≥ 1 or < 0 the Jensen gap is nonnegative. In the case 0 < < 1 the Jensen gap is non-positive. Next we state the following two-sided estimate: Theorem 2.4. Let = ( ( )), 0 < < ∞, be the Hardy operator. Then we have the following two-sided estimates: Proof. (a) The first inequality is just Jensen's inequality (see Lemma 2.3). We use Lemma 2.2 with = [0, ] and ( ) = 1 , 0 ≤ ≤ , and obtain Apply these inequalities with the convex function ( ) = , ≥ 1 or < 0 to find that The proof is completely similar since both used inequalities hold in reversed direction in this case. □ Remark 2.5. Of course the right-hand side in (2.3) must be nonnegative when ≥ 1 or < 0. In fact, this is a special case of the Chebyshev inequality yielding for similarly ordered functions and . It holds in reversed direction if and are oppositely ordered. By applying this inequality with ( ) = 1 , 0 ≤ ≤ , for ( ) = −1 ( ) (hence and are similarly ordered when > 1 and oppositely ordered when < 1). We get in the case ≥ 1 or < 0 and the reverse inequality holds in the case 0 < ≤ 1.
Next we apply the inequality (2.2) for − 1 ≥ 1 and get multiply both sides of this inequality with ( ( )) and subtract ( ( ( ))) from both sides. So we get the first inequality. □ By using Corollary 2.6 we get the following refinement of the fundamental Hardy inequality (1.4) for ≥ 2: Lemma 2.7. Let 0 < ≤ ∞ and let be a measurable function on (0, ]. If ≥ 2, then Proof. Multiply (2.4) by dx , integrate over the interval [0, ] and reverse the order of integration in the last integral By using this lemma and the same substitution as before we can derive the following refinement of the Hardy inequality (1.5) when ≥ 2.

6)
where < − 1 and Proof. We use Lemma 2.7 with ( ) = We want to replace by In the integral we make the substitution = and get In particular, and then Next we make the substitution = and get Then the middle term in Theorem 2.8 will be equal to The proof is complete. □ Remark 2.9. By using other estimates of Jensen functionals than that in Lemma 2.2 we obtain other refinements of the second estimate in Corollary 2.6. For example, if we use the fairly new estimate (see [1]).

ON REVERSED HARDY INEQUALITIES
As M. Milman writes in [12] the subtle point is that Hardy's operator = ( ) is not invertible on (0, ∞)-spaces, and therefore it is not possible to find a reverse Hardy inequality of the form for any finite ( ) > 0 and holding for all positive functions.
On the other hand, if is positive and non-increasing, we trivially have that ( ) ≤ ( ( )), and that e.g. the inequality (1.1) holds for positive non-increasing functions with ( ) = 1. It turns out, however, that this inequality can be considerably sharpened and the obtained inequality is sharp. For example Gehring's lemma reads: Results about Gehring's lemma in Orlicz spaces and the connection to interpolation and approximation spaces can be found e.g. in [10,11], etc.
Here we first consider the possibility to prove a reverse type estimate for the Hardy operator for a general r.i. space. Remember that the Banach lattice is -concave if there exists a positive constant such that, for every finite set The smallest constant satisfying (3.2) is called the constant of -concavity. As it was mentioned in [15], from a result of A. Shep we have that if is 1-concave with constant of concavity equal to M, then where [ , ] denotes the characteristic function of the interval [ , ].
By integrating we get that Therefore, according to the Schep estimate (3.3) we have that The proof is complete. □

Corollary 3.3. If > 1 and ( ) is a positive and non-increasing function on
Proof. Apply Theorem 3.2 with = 1 (0, ) and we find that and the proof follows. □ Remark 3.4. For = ∞ we have just Gehring's inequality (3.1).
Next we recall the following generalization of Lemma 3.1: Lemma 3.5. Let > 1, let < − 1 and let ( ) be a positive and non-increasing function. Then, for , 0 < ≤ ∞, This is a special case of Theorem 3.II in [2] (the case = 0). The constant is sharp so also the constant in Gehring's lemma is sharp.
Moreover, we observe that by following the proof of Theorem 3.2 we find that the following generalization fitting to Lemma 3.5 holds: Remark 3.7. By making the similar calculations as in the proof of Corollary 3.3 we get the following weighted version of the inequality (3.4): for ≥ 1, < − 1, yielding for all positive and non-increasing functions. Especially, for = ∞ we get (3.5).

INEQUALITIES, BASED ON NEW FORMS OF BELLMAN'S INEQUALIT Y
A recently presented continuous form of Bellman's inequality reads (see [14]): Then, for ≥ 1, Using this theorem we can derive the first main result in this section: On the other hand the change of variables = implies that

dy.
Hence, according to Theorem 4.1 the proof is complete. □ Remark 4.3. We note that (1.1) holds only under the restriction < − 1 i.e. we can estimate the quantity dx only under this restriction. However, according to Theorem 4.2 this quantity can be estimated also without this restriction if we instead put the following restriction on : There exists a function 0 ( ) such that 0 ( ) > 1 +1 ∫ ∞ 0 ( ) dy.
The following more general continuous version of the Bellman inequality in Banach lattices was proved in [15].
Theorem 4.4. Let and be measure spaces, let ( , ) be a positive measurable function on × and assume that ≥ 1 and that 0 ( ) is a function on such that 0 ( ) > ‖ ( , ⋅)‖ , where is a Banach function space on for all ∈ . Assume that has Fatou property. Then the inequality holds provided that all integrals exist.
By using this theorem we can get a Banach lattice variant of Theorem 4.2.
With this in mind we can formulate the last main result in this section. On the other hand the change of variables = implies that Hence, by Theorem 4.4 the proof is complete. □