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We prove some relations between the discrete Gehring classes 풢q and discrete Muckenhoupt classes 풜p. Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between 풢1, 풜∞, and 풜1 for nonincreasing and nondecreasing weights. Finally, we establish some general results by introducing the notions of 풢φ classes defined for a nonnegative convex function φ. [ABSTRACT FROM AUTHOR]

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, by using the Riccati technique which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new sufficient conditions for oscillation of the second order nonlinear neutral dynamic equation (r(t)((x(t) + p(t)x(τ(t)))Δ)γ)Δ + q(t)xα(δ(t)) + v(t)xβ(η(t)) = 0; on time scales where γ, α, β, are quotient of odd positive integers. [ABSTRACT FROM AUTHOR]

In this paper, we will establish some new sufficient condition for oscillation of solutions of a certain class of first-order neutral delay difference equations of the form Δ (xn -- pnxn-1) + qnxn-τγ = 0, where γ is a quotient of odd positive integers. We will consider the sublinear and super linear cases. The results will be obtained by using the oscillation theorems of second order delay difference equations. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

In this paper, we prove some new characterizations of weights in some Hardy-type inequalities on time scales. The results as special cases contain the results due to Beesack and Heinig, Leindler and Bloom and Kerman. Some new integral and discrete inequalities related to Copson's, Flett's, Bliss's and Bennett's will be formulated. The main results will be proved by using new generalizations of Opial's type inequalities, Hölder's inequality, Minkowski's inequality and the chain rule on time scales. [ABSTRACT FROM AUTHOR]