Hardy-Leindler-Type Inequalities via Conformable Delta Fractional Calculus

In this article, some fractional Hardy-Leindler-type inequalities will be illustrated by utilizing the chain law, Hölder ’ s inequality, and integration by parts on fractional time scales. As a result of this, some classical integral inequalities will be obtained. Also, we would have a variety of well-known dynamic inequalities as special cases from our outcomes when α = 1 .


Introduction
The Hardy discrete inequality is known as (see [1]) where lðrÞ > 0 for all r ≥ 1.

Basic Concepts
In this part, we introduce the essentials of conformable fractional integral and derivative of order α ∈ ½0, 1 on time scales that will be used in this article (see [12][13][14][15]). For a time scale T , we define the operator σ : T ⟶ T , as Also, we define the function μ : T ⟶ 0, ∞Þ by Finally, for any τ ∈ T , we refer to the notation ξ σ ðτÞ by ξðσðτÞÞ, i.e., ξ σ = ξ ∘ σ. In the following, we define conformable α-fractional derivative and α-fractional integral on T .

Main Results
Here, we will exemplify our major results in this article. In the pursuing theorem, we will exemplify Leindler's inequality (7) for fractional time scales as follows.
In the pursuing theorem, we will exemplify Leindler's inequality (8) on fractional time scales as follows.
In the pursuing theorem, we will exemplify Leindler's inequality (9) for fractional time scales as follows.