Hardy-type inequalities in fractional h-discrete calculus

The first power weighted version of Hardy’s inequality can be rewritten as ∫0∞(xα−1∫0x1tαf(t)dt)pdx≤[pp−α−1]p∫0∞fp(x)dx,f≥0,p≥1,α<p−1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{0}^{\infty } \biggl( x^{\alpha -1} \int _{0}^{x} \frac{1}{t ^{\alpha }}f(t)\,dt \biggr) ^{p}\,dx\leq \biggl[ \frac{p}{p-\alpha -1} \biggr] ^{p} \int _{0}^{\infty }f^{p}(x)\,dx,\quad f\geq 0,p\geq 1, \alpha < p-1, $$\end{document} where the constant C=[pp−α−1]p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C= [ \frac{p}{p-\alpha -1} ] ^{p}$\end{document} is sharp. This inequality holds in the reversed direction when 0≤p<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\leq p<1$\end{document}. In this paper we prove and discuss some discrete analogues of Hardy-type inequalities in fractional h-discrete calculus. Moreover, we prove that the corresponding constants are sharp.

Integral inequalities have always been of great importance for the development of many branches of mathematics and its applications. One typical such example is Hardy-type inequalities, which from the first discoveries of Hardy in the twentieth century now have been developed and applied in an almost unbelievable way, see, e.g., monographs [23] and [24] and the references therein. Let us just mention that in 1928 Hardy [25] proved the following inequality: for 1 ≤ p < ∞ and α < p -1 and where the constant [ p p-α-1 ] p is best possible. Inequality (1.1) is just a reformulation of the first power weighted generalization of Hardy's original inequality, which is just (1.1) with α = 0 (so that p > 1) (see [26] and [27]). Up to now there is no sharp discrete analogue of inequality (1.1). For example, the following two inequalities were claimed to hold by Bennett([28,; see also [29, p. 407 n , a n ≥ 0, n , a n ≥ 0, n , a n ≥ 0, for 0 < q < 1, p ≥ 1 and α < 1 -1/p, where λ := 1 -1/pα. The main aim of this paper is to establish the h-analogue of the classical Hardy-type inequality (1.1) in fractional h-discrete calculus with sharp constants which is another discrete analogue of inequality (1.1).
The paper is organized as follows: In order not to disturb our discussions later on some preliminaries are presented in Sect. 2. The main results (see Theorem 3.1 and Theorem 3.2) with the detailed proofs can be found in Sect. 3.

Preliminaries
We state the some preliminary results of the h-discrete fractional calculus which will be used throughout this paper.
Let fg : T a → R. Then the product rule for h-differentiation reads (see [34]) The chain rule formula that we will use in this paper is which is a simple consequence of Keller's chain rule [35,Theorem 1.90]. The integration by parts formula is given by (see [34]) the following.

Definition 2.3
We say that a function g : T a − → R, is nonincreasing (respectively, non- Definition 2.4 (see [34]) Let t, α ∈ R. Then the h-fractional function t (α) h is defined by where is Euler gamma function, t h / ∈ {-1, -2, -3, . . .} and we use the convention that division at a pole yields zero. Note that Hence, by (2.1) we find that holds for all x, y ∈ (0, ∞) and 0 < u < 1.
Next, we will derive some properties of the h-fractional function, which we need for the proofs of the main results, but which are also of independent interest.
Proof By using Definition 2.4 we get It is well known that the gamma function is log-convex (see, e.g., [37], p. 21). Hence, so we have proved that (2.7) holds wherever 1 ≤ p < ∞. Moreover, for 0 < p < 1, so we conclude that (2.8) holds for 0 < p < 1. The proof is complete.

Main results
Our h-integral analogue of inequality (1.1) reads as follows.
Our second main result is the following h-integral analogue of the reversed form of (1.1) for 0 < p < 1.
To prove Theorem 3.1 we need the following lemma, which is of independent interest.
Proof Let α < p-1 p and 1 ≤ p < ∞. Since (x) > 0 for x > 0, and using Definition 2.4, we have h . Then by using (2.5) we find that and and By using the fact that (x + hαh) (
Finally, we will prove that the constant [ p p-αp-1 ] p is the best possible in inequality (3.1). Let x, a ∈ T 0 be such that a < x, and consider the test function f β (t) = t (β) h χ [a,∞) (t), t > 0, for β = -1 pε. Then from (2.4), (2.5) and (2.7) it follows that which means that (3.1) holds even with equality in this case. The proof is complete.