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In this paper, we will prove some fundamental properties of the power mean operator M p g (t) = (1 ϒ (t) ∫ 0 t λ (s) g p (s) d s) 1 / p , for t ∈ I ⊆ R + , of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on I and ϒ (t) = ∫ 0 t λ (s) d s . Next, we will study the structure of the generalized class U p q (B) of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , for some p < q , p. q ≠ 0 , and B > 1 is a constant. For applications, we will prove some self-improving properties of weights in the class U p q (B) and derive the self improving properties of the weighted Muckenhoupt and Gehring classes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when T = R and T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new characterizations of weighted functions of dynamic inequalities containing a Hardy operator on time scales. These inequalities contain the characterization of Ariňo and Muckenhoupt when T = R , whereas they contain the characterizations of Bennett–Erdmann and Gao when T = N . [ABSTRACT FROM AUTHOR]

In this paper, we prove some properties of weighted Cesàro and Copson sequences spaces by establishing some factorization theorems. The results lead to two-sided norm discrete inequalities with best possible constants and also give conditions for the boundedness of the generalized discrete weighted Hardy and Copson operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the discrete Muckenhoupt class A p (C) and the discrete Gehring class G q (K) . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if u ∈ A p (C) then there exists q < p such that u ∈ A q (C 1) . Next, we prove that the power rule also holds, i.e., we prove that if u ∈ A p then u q ∈ A p for some q > 1 . The relation between the Muckenhoupt class 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. [ABSTRACT FROM AUTHOR]