1. Introduction and Preliminaries
Let
and
be measure spaces with positive
-finite measures. For a measurable function
, let
denote the linear operator
where
is measurable and non-negative kernel with
The following result was given in [
1] (see also [
2]), where
u is a positive function on
.
Theorem 1. Let u be a weight function, . Assume that is locally integrable on for each fixed . Define v by If ϕ is a convex function on the interval , then the inequalityholds for all measurable functions , such that , where is defined by (
1) and (
2).
Inequality (
4) is generalization of Hardy’s inequality. G. H. Hardy [
3] stated and proved that the inequality
holds for all
f non-negative functions such that
and
. The constant
is sharp. More details about Hardy’s inequality can be found in [
4,
5].
Inequality (
5) can be interpreted as the Hardy operator
maps
into
with the operator norm
.
In this paper, we consider the difference of both sides of the generalized Hardy’s inequality
and obtain new inequalities that hold for
n-convex functions.
Now, we recall
-convex functions. There are two parallel notations. First, is given by E. Hopf in 1926 and second by T. Popoviciu in 1934. E. Hopf defined that the function
f is
-convex if difference
is nonnegative. The ordinary convex function is 1-convex. For more details see [
6]. In the second definition
is
n-convex
, if its
n-th order divided differences
are nonnegative for all choices of
distinct points
. By second definition 0-convex function is nonnegative, 1-convex function is non-decreasing and 2-convex function is convex in the usual sense. If an
n-convex function is
n times differentiable, then
. (see [
7]).
An important role in the paper will be played by Abel–Gontscharoff interpolation, which was first studied by Whittaker [
8], and later by Gontscharoff [
9] and Davis [
10]. The Abel–Gontscharoff interpolation for two points and the remainder in the integral form is given in the following theorem (for more details see [
11]).
Theorem 2. Let , , and Thenwhere is the Abel–Gontscharoff interpolating polynomial for two-points of degree , i.e.,and the remainder is given bywhere is Green’s function defined by Remark 1. For the following inequalities hold 2. Generalizations of Hardy’s Inequality
Our first result is an identity related to generalized Hardy’s inequality. We apply interpolation by the Abel–Gontscharoff polynomial and get the following result.
Theorem 3. Let and be measure spaces with positive σ-finite measures. Let be a weight function and v is defined by (3). Let be defined by (1) and (2) respectively, for a measurable function and let and be defined by (6). Then Proof. Using Theorem 2 we can represent every function
in the form
By an easy calculation, applying (
8) in
, we get
Since
the summand for
in the first sum on the right hand side is equal to zero, so (
7) follows. □
We continue with the following result.
Theorem 4. Let all the assumptions of Theorem 3 hold, let ϕ be n-convex on and If the reverse inequality in (9) holds, then the reverse inequality in (10) holds.
Proof. We assumed that
is
n-convex, so
on
. We apply Theorem 3 and (
10). □
Remark 2. Notice that for and the function is convex on . Therefore the assumption (9) is satisfied and then the inequality (10) holds. For an arbitrary and , we use Remark 1, i.e., we consider the following inequality:Ww conclude that the convexity of depends of a parity of If n is even, then i.e., is convex and assumption (9) is satisfied. Also, the inequality (10) holds. For odd n we get the reverse inequality. For all other choices, the following generalization holds. Theorem 5. Suppose that all assumptions of Theorem 1 hold. Additionally, let , and be n-convex.
- (i)
If is odd, then the inequality (10) holds. - (ii)
If is even, then the reverse inequality in (10) holds.
Proof . - (i)
By Remark 1, the following inequality holds
In case
is odd (
is even), we have
i.e.,
is convex on
Then by Theorem 1 we have
i.e., the assumption (
9) is satisfied. By applying Theorem 4 we get (
10).
- (ii)
Similarly, if
is even, then
is concave on
, so the reversed inequality in (
9) holds and, hence, in (
10) as well.
□
Theorem 6. Suppose that all assumptions of Theorem 1 hold and let , , be n-convex and be defined by - (i)
If (10) holds and F is convex, then the inequality (4) holds. - (ii)
If the reverse of (10) holds and F is concave, then the reverse inequality (4) holds.
Proof . - (i)
Let (
10) holds. If
F is convex, then by Theorem 1 we have
which, changing the order of summation, can be written in form
We conclude that the right-hand side of (
10) is nonnegative and the inequality (
4) follows.
- (ii)
Similar to (i) case.
□
Remark 3. Note that the function is convex on for each i.e.,for each - (i)
If (10) holds, for and for and then the right hand side of (10) is non-negative, i.e., the inequality (4) holds. - (ii)
If the reverse of (10) holds, for and for and , then the right hand side of (10) is negative, i.e., the reverse inequality in (4) holds.
3. Upper Bound for Generalized Hardy’s Inequality
The following estimations for Hardy’s difference is given in the previous section, under special conditions in Theorem 6 and Remark 3.
In this section, we present upper bounds for obtained generalization. We recall recent results related to the Chebyshev functional. For two Lebesgue integrable functions
we consider the Chebyshev functional.
With we denote the usual Lebesgue norms on space .
In [
12] authors proved the following theorems.
Theorem 7. Let be a Lebesque integrable function and be an absolutely continuous function with . Then we have the inequality The constant in (12) is the best possible.
Theorem 8. Assume that is monotonic nondecreasing on and is absolutely continuous with . Then we have the inequality The constant in (13) is the best possible.
Under assumptions of Theorem 3 we define the function
by
The Chebyshev functional is defined by
Theorem 9. Suppose that all the assumptions of Theorem 3 hold. Also, let and be defined as in (14). Then the following identity holds:where the remainder satisfies the estimation Proof. Applying Theorem 7 for
and
we get
Therefore, we have
where the remainder
satisfies the estimation. Now from the identity (
7) we obtain (
15). □
The following Grüss type inequality also holds.
Theorem 10. Suppose that all the assumptions of Theorem 3 hold. Let on and be defined as in (14). Then the identity (15) holds and the remainder satisfies the bound Proof. By applying Theorem 8 for
and
we obtain
Since
using the identities (
7) and (
17) we deduce (
16). □
We continue with the following result that is an upper bound for generalized Hardy’s inequality.
Theorem 11. Suppose that all the assumptions of Theorem 3 hold. Let be a pair of conjugate exponents, that is , Then The constant on the right-hand side of (18) is sharp for and the best possible for . Proof. We apply the Hölder inequality to the identity (
7) and get
where
is defined as in (
14).
The proof of the sharpness is analog to one in proof of Theorem 11 in [
13]. □
We continue with a particular case of Green’s function
defined by (
6). For
we have
If we choose and in Theorem 11, we get the following corollary.
Corollary 1. Let and be a pair of conjugate exponents, that is , Then The constant on the right hand side of (21) is sharp for and the best possible for . Remark 4. If we additionally suppose that ϕ is convex, then the difference is non-negative and we have In sequel we consider some particular cases of this result.
Example 1. Let , replace and by the Lebesque measures and , respectively, and let for . Then coincides with the Hardy operator defined bywhere If also is replaced by and by , then Example 2. By arguing as in Example 1 but and with kernels such that for we obtain the following resultwhere the dual Hardy operator is defined bywhere We continue with the following Example.
Example 3. Let and and (so that ) we obtain the following resultwhere is defined by Example 4. By arguing as in Example 3 but only with , we obtain the following result We continue with the result that involves Hardy–Hilbert’s inequality.
If
and
f is a non-negative function such that
, then
Inequality (
26) is sometimes called Hilbert’s inequality even if Hilbert himself only considered the case
.
Example 5. Let , replace and by the Lebesque measures and , respectively. Let and . Then and . Let , replace with then the following result followswhere We also mention Pólya–Knopp’s inequality,
for positive functions
. Pólya–Knopp’s inequality may be considered as a limiting case of Hardy’s inequality since (
27) can be obtained from (
5) by rewriting it with the function
f replaced with
and then by letting
.
Example 6. By applying (22) with , and f replaced by we obtain thatwhere and are defined as in Theorem 1 and At the end, we give interesting application.
Using (
10), under the assumptions of Theorem 4, we define the linear functional
by
If
is
n-convex, then
by Theorem 4. Using the positivity and the linearity of functional
A we can get corresponding mean-value theorems. We may also obtain new classes of exponentially convex functions and get new means of the Cauchy type applying the same method as given in [
14,
15,
16,
17,
18,
19,
20,
21].