Generalizations of Hardy Type Inequalities by Abel–Gontscharoff’s Interpolating Polynomial

: In this paper, we extend Hardy’s type inequalities to convex functions of higher order. Upper bounds for the generalized Hardy’s inequality are given with some applications.

Theorem 1.Let u be a weight function, k(x, y) ≥ 0. Assume that k(x,y) K(x) u(x) is locally integrable on Ω 1 for each fixed y ∈ Ω 2 .Define v by v(y) := If φ is a convex function on the interval I ⊆ R, then the inequality φ( f (y))v(y)dµ 2 (y) (4) holds for all measurable functions f : Ω 2 → R, such that Im f ⊆ I, where A k 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 f ∈ L p (R + ) and R + = (0, ∞).The constant p p−1 p is sharp.More details about Hardy's inequality can be found in [4,5].
Inequality (5) can be interpreted as the Hardy operator H : H f (x) := 1 x x 0 f (t) dt, maps L p into L p with the operator norm p = p p−1 .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 n−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 The ordinary convex function is 1-convex.For more details see [6].In the second definition f : 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 φ (n) ≥ 0. (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]).
where Q n−1 is the Abel-Gontscharoff interpolating polynomial for two-points of degree n − 1, i.e., and the remainder is given by where G mn (u, t) is Green's function defined by Remark 1.For α ≤ t, u ≤ β the following inequalities hold

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 (Σ 1 , Ω 1 , µ 1 ) and (Σ 2 , Ω 2 , µ 2 ) be measure spaces with positive σ-finite measures.Let u : Ω 1 → R, be a weight function and v is defined by (3).Let A k f (x), K(x) be defined by ( 1) and ( 2) respectively, for a measurable function f : ) and G mn be defined by (6).Then Proof.Using Theorem 2 we can represent every function φ ∈ C n ([α, β]) in the form By an easy calculation, applying (8) in we get Since the summand for s = 0 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 [α, β] and Then If the reverse inequality in (9) holds, then the reverse inequality in (10) holds.
Remark 2. Notice that for n = 2 and 0 Therefore the assumption ( 9) is satisfied and then the inequality (10) holds.For an arbitrary n ≥ 3 and 0 ≤ m ≤ 1, we use Remark 1, i.e., we consider the following inequality: Ww conclude that the convexity of G mn (•, t) depends of a parity of n.If n is even, then ∂ 2 G mn (u,t) ≥ 0, i.e., G mn (•, t) 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 n, m ∈ N, (10) holds.(ii) If n − m is even, then the reverse inequality in (10) holds.

Proof.
(i) By Remark 1, the following inequality holds Then by Theorem 1 we have i.e., the assumption (9) is satisfied.By applying Theorem 4 we get (10).
, 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 n, (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.

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 g, h : [a, b] → R we consider the Chebyshev functional.In [12] authors proved the following theorems.

T(g
The constant The constant 1 2 in (13) is the best possible.
Under assumptions of Theorem 3 we define the function The Chebyshev functional is defined by Theorem 9. Suppose that all the assumptions of Theorem 3 hold.Also, let ( and L be defined as in (14).Then the following identity holds: where the remainder R(α, β; φ) satisfies the estimation .
Proof.Applying Theorem 7 for g → L and h → φ (n) we get .
The following Grüss type inequality also holds.
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 (p, q) be a pair of conjugate exponents, that is Then .
The constant on the right-hand side of ( 18) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.
Proof.We apply the Hölder inequality to the identity (7) and get where F (t) 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 G mn (u, t) defined by (6).For n = 2, m = 1, we have If we choose n = 2 and m = 1 in Theorem 11, we get the following corollary.
The constant on the right hand side of ( 21) is sharp for 1 < p ≤ ∞ and the best possible for p = 1.
Remark 4. If we additionally suppose that φ is convex, then the difference .
In sequel we consider some particular cases of this result.

Example 1.
Let and dµ 2 (y) by the Lebesque measures dx and dy, respectively, and let k(x, y) = 0 for x < y ≤ b.Then A k coincides with the Hardy operator H k defined by where If also u(x) is replaced by u(x)/x and v(x) by v(x)/x, then .
where the dual Hardy operator H k f is defined by where We continue with the following Example.

Example 3.
Let Ω 1 = Ω 2 = (0, ∞) and k(x, y) = 1, 0 ≤ y ≤ x, k(x, y) = 0, y > x, dµ 1 (x) = dx, dµ 2 (y) = dy and u(x) = 1 x (so that v(y) = 1 y ) we obtain the following result where A k is defined by We continue with the result that involves Hardy-Hilbert's inequality.If p > 1 and f is a non-negative function such that f ∈ Inequality ( 26) is sometimes called Hilbert's inequality even if Hilbert himself only considered the case p = 2.We also mention Pólya-Knopp's inequality, for positive functions f ∈ L 1 (R + ).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 f where k(x, y), K(x), u(x) and v(y) are defined as in Theorem 1 and k(x, y) ln f (y)dµ 2 (y).
At the end, we give interesting application.Using (10), under the assumptions of Theorem 4, we define the linear functional A : C n ([α, β]) → R by we denote the usual Lebesgue norms on space L p [a, b].