In this paper, we prove some basic properties of the discrete Muckenhoupt class $\mathcal{A}^{p}$ A p and the discrete Gehring class $\mathcal{G}^{q}$ G q . These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes.

In this paper, we study the structure of the discrete Muckenhoupt class $\mathcal{A}^{p}(\mathcal{C})$ A p ( C ) and the discrete Gehring class $\mathcal{G}^{q}(\mathcal{K})$ G q ( K ) . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if $u\in \mathcal{A}^{p}(\mathcal{C})$ u ∈ A p ( C ) then there exists $q< p$ q < p such that $u\in \mathcal{A}^{q}(\mathcal{C}_{1})$ u ∈ A q ( C 1 ) . Next, we prove that the power rule also holds, i.e., we prove that if $u\in \mathcal{A}^{p}$ u ∈ A p then $u^{q}\in \mathcal{A}^{p}$ u q ∈ A p for some $q>1$ q > 1 . The relation between the Muckenhoupt class $\mathcal{A}^{1}(\mathcal{C})$ A 1 ( C ) and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose.

In this paper, we will prove some fundamental properties of the discrete power mean operator M p u n = 1 / n ∑ k = 1 n   u p k 1 / p , for   n ∈ I ⊆ ℤ + , of order p , where u is a nonnegative discrete weight defined on I ⊆ ℤ + the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class B p q B of weights that satisfy the reverse Hölder inequality   M q u ≤ B M p u , for positive real numbers p , q , and B such that 0 < p < q and B > 1 . For applications, we will prove some self-improving properties of weights from B p q B and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.

Abstract. In this paper, we prove some new multiplicative dynamic inequalities of Opial type on a time scale T . The main results will be proved by using Hölder's inequality, the chain rule and some basic dynamic inequalities designed and proved for this purpose. As special cases, we will derive some continuous and discrete inequalities from the main results.Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40.

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class.