Abstract
Some new inequalities for functions whose second derivatives in absolute value at certain powers are s-convex in the second sense are established. Two mistakes in a recently published paper are pointed out and corrected.
Highlights
In a recent paper [ ], Özdemir et al proved the following inequalities for functions whose second derivatives in absolute value at certain powers are s-convex in the second sense
1 Introduction It is well known that a function f : I → R, ∅ = I ⊂ R is called convex if f λx + ( – λ)y ≤ λf (x) + ( – λ)f (y) holds for all x, y ∈ I and λ ∈ [, ]
In [ ], the class of s-convex function in the second sense is defined in the following way: a function f : [, ∞) → R is said to be s-convex in the second sense if f λx + ( – λ)y ≤ λsf (x) + ( – λ)sf (y) holds for all x, y ∈ [, ∞), λ ∈ [, ] and for some fixed s ∈
Summary
In a recent paper [ ], Özdemir et al proved the following inequalities for functions whose second derivatives in absolute value at certain powers are s-convex in the second sense. It is a pity that Theorem in [ ] is not valid since nonnegative |f |q could not be an s-concave function for any fixed s ∈ ( , ) which has been mentioned in [ ], and it is the Hölder inequality but not the power mean inequality that has been used in proving Theorem of [ ].
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