Abstract
In the article, we establish several Petrović-type inequalities for the harmonic h-convex (concave) function if h is a submultiplicative (super-multiplicative) function, provide some new majorizaton type inequalities for harmonic convex function, and prove the superadditivity, subadditivity, linearity, and monotonicity properties for the functionals derived from the Petrović type inequalities.
Highlights
Let ⊆ R be a nonempty interval. en a real-valued function : → R is said to be convex on if the inequality+ (1 − ) ≤ (≥) ( ) + (1 − ), (1)holds for all, ∈ and ∈ [0, 1]
It is well known that the convex functions have wide applications in pure and applied mathematics [1–21], many remarkable properties and inequalities can be found in the literature [22–48] via the theory of convexity
A great deal of generalizations, extensions, and variants have been made for convexity, for example, GA-convexity [49], GG-convexity [50], -convexity [51, 52], preinvex convexity [53], strong convexity [54–57], Schur convexity [58–60], and others [61–67]
Summary
Let ⊆ R be a nonempty interval. en a real-valued function : → R is said to be convex (concave) on if the inequality. En a real-valued function : → R is said to be convex (concave) on if the inequality. =1 holds for all nonnegative -tuples 1, 2, ⋅ ⋅ ⋅ , and 1, 2, ⋅ ⋅ ⋅ , such that ≤ ∑ =1 ≤ ( = 1, 2, ⋅ ⋅ ⋅ , ). We recall the de nitions of various convexities and the majorization relation between two -tuples. En a realvalued function : → R is said to be submultiplicative (supermultiplicative) on if the inequality. En a real-valued function : → R is said to be harmonic convex (concave) on if the inequality Let ⊆ R be an interval. en a real-valued function : → R is said to be harmonic convex (concave) on if the inequality
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