Abstract
Some inequalities for weighted and integral means of convex functions on linear spaces
Highlights
Let X be a real linear space, x, y ∈ X, x = y and let [x, y] := {(1 − λ) x + λy, λ ∈ [0, 1]}be the segment generated by x and y
The following inequality is the well-known Hermite-Hadamard integral inequality for convex functions defined on a segment [x, y] ⊂ X : x+y f (x) + f (y)
Motivated by the above results, we establish in this paper some upper and lower bounds for the difference p (τ ) f ((1 − τ ) x + τ y) dτ − p (τ ) dτ f ((1 − τ ) x + τ y) dτ where f is a convex function on C and x, y ∈ C, with x = y while p : [0, 1] → R is a Lebesgue integrable function such that τ
Summary
The following inequality is the well-known Hermite-Hadamard integral inequality for convex functions defined on a segment [x, y] ⊂ X : x+y f (x) + f (y) f. In the recent paper [7] we established the following refinements and reverses of Fejer’s inequality for functions defined on linear spaces: Theorem 1.1. Motivated by the above results, we establish in this paper some upper and lower bounds for the difference p (τ ) f ((1 − τ ) x + τ y) dτ − p (τ ) dτ f ((1 − τ ) x + τ y) dτ where f is a convex function on C and x, y ∈ C, with x = y while p : [0, 1] → R is a Lebesgue integrable function such that τ. ≤ p (s) ds ≤ p (s) ds for all τ ∈ [0, 1]
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