Generalizations of Hardy’s Type Inequalities via Conformable Calculus

In this paper, we derive some new fractional extensions of Hardy’s type inequalities. The corresponding reverse relations are also obtained by using the conformable fractional calculus from which the classical integral inequalities are deduced as special cases at α=1.


Introduction
In 1925 Hardy [1] employed the calculus of variations and proved the inequality where ℎ ≥ 0 and integrable over any finite interval (0, ) and ℎ is integrable and convergent over (0, ∞) and > 1. The constant ( /( − 1)) is the best possible.
In recent years, several scholars have examined fractional inequalities by using the fractional derivative of Caputo and Riemann-Liouville; we refer to the papers [3][4][5][6][7] for these results.
The original motivation for this paper is obtaining the fractional forms of some extensions of Hardy's type inequalities and their reverses using conformable fractional calculus, and as a special case, we put α = 1 to get the generalized ones.
The paper is structured as follows: In Section 2, we will present some concepts for the conformable fractional calculus and also the Hӧlder's inequality for -fractional differentiable functions that will represent our key outcomes. In Section 3, we shall set out generalizations of Hardy's type inequalities and revers relations in each case for -fractional differentiable functions.

Key Concepts and Lemmas
In this section, we present some basic definitions concerning the conformable fractional calculus that will be used throughout the paper. For the latest findings on conformable derivatives and integrals, we refer to [8,9].
for all s and 0 < ≤ 1 Now, we state some lemmas which play important roles in our proofs of the main results. First, the integration by parts formula is given in the following lemma: Lemma 1. Suppose that , : [0, ] → ℝ be two functions such that is -differentiable and 0 < ≤ 1 then: Next, we state Hӧlder's and reversed Hӧlder's inequality for -conformable functions, which will mainly be used to prove the results of this paper. Lemma 2. Let ℎ, : [0, ] → ℝ be a continuous function and 0 < ≤ 1. Then: where > 1 1/ + 1/ = 1. This inequality is reversed if 0 < < 1 and if < 0 or < 0.

Hardy's Type Inequalities of A Fractional Order
In this section, we state and prove the main outcomes of this paper and we begin with the following theorem: non-decreasing and 0 < ≤ ∞, then: Proof. We start with the following identity: Since ℎ is non-decreasing, then we have: which is (8). For ( ) = , ≥ 1, we have (9). The proof is complete. □ The theorem below is the generalization of Hardy's inequality (1) on conformable calculus.
Proof. The proof follows from Theorem 6 for = 1.
Proof. The proof follows from Theorem 7 for = 1. □

Applications
Lyapunov's inequality is an important result in mathematics with many different applications see ( [21,22] and the reference therein). The result, as proved by Lyapunov in 1907 [23], asserts that if ( ) is real and continuous functions on [a, b], then a necessary condition for the boundary value problem: ( ) + ( ) ( ) = 0, < < , ( ) = ( ) = 0, to have nontrivial solutions is given by:

Data Availability:
No data were used to support this study.