Inequalities for averages of quasiconvex and superquadratic functions

Abstract

Abstract. For n ∈ Z + we consider the difference

Dynamic Inequalities On Time Scales

“…In this case, we say f Δ2 (t, s) is the (partial delta) derivative of f (t, s) at s. The function f is called rd-continuous in t if for every α 2 ∈ T 2 the function f (t, α 2 ) is rd-continuous on T 1 . The function f is called rd-continuous in s if for every α 1 ∈ T 1 the function f (α 1 , s) is rd-continuous on T 2 .…”

“…The concept of superquadratic functions in one variable, as a generalization of the class of convex functions was introduced by S. Abramovich, G. Jameson, and G. Sinnamon in [1,2].…”

“…First note that e Δ λ (t, t 0 ) = λ(t)e λ (t, t 0 ). For a given K > 1, the function Ke λ (t, t 0 ) is a solution of (5.3.7) if and only if λ is a root of the characteristic polynomial P (t, λ) defined by P (t, λ) := (λ + p(t)) e λ (t, δ − (h r , t))e 1 hence, by choosing a K 0 > 1 with…”

“…Shift Operators) Let T * be a nonempty subset of the time scale T including a fixed number t 0 ∈ T * such that there exist operators δ ± : [t 0 , ∞) T × T * → T * satisfying the following properties: P 1. The functions δ ± are strictly increasing with respect to their second arguments, i.e., if(T 0 , t), (T 0 , u) ∈ D ± := {(s, t) ∈ [t 0 , ∞) T × T * : δ ± (s, t) ∈ T * } , then T 0 ≤ t < u implies δ ± (T 0 , t) < δ ± (T 0 , u), P.2 If (T 1 , u), (T 2 , u) ∈ D − with T 1 < T 2 , then δ − (T 1 , u) > δ − (T 2 , u),…”

Klein's Trace Inequality and Superquadratic Trace Functions

“…itself. If f is differentiable and satisfies f (0) = f ′ (0) = 0, then one can easily see that the constant C s in the definition is necessarily f ′ (s), see [2].…”