Abstract
Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given.
Highlights
Let f : I ⊆ R → R be a convex mapping defined on the interval I of real numbers and a, b ∈ I, with a < b, f a b
In the following we develop a Hadamard-type inequality for coordinated log-convex functions
1 b−a d−c bd exp yq dx dy ≤ L eap , ebp L ecp , edp
Summary
Let f : I ⊆ R → R be a convex mapping defined on the interval I of real numbers and a, b ∈ I, with a < b, f a b. 1.1 holds, this inequality is known as the Hermite-Hadamard inequality. Counterparts, generalizations and new Hadamard-type inequalities, see 1–8. A positive function f is called log-convex on a real interval I a, b , if for all x, y ∈ a, b and λ ∈ 0, 1 , f λx 1 − λ y ≤ f λ x f 1−λ y. If f is a positive log-concave function, the inequality is reversed. A function f is log-convex on I if f is positive and log f is convex on I. If f > 0 and f exists on I, f is log-convex if and only if f · f − f 2 ≥ 0
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