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... (2) This concept was introduced by Orlicz, see [17]. It is known as Orlicz-convexity but it is often named s-convexity of first kind [10] or s 1 -convexity [19]. Obviously, the domain D is (α, β)-convex in this case. ...

... The results in this section show that the inequalities derived in this paper establish the best boundaries in the set of the s-convex functions in the first sense. They are a subclass of many classes of generalized convex functions as shown in [19]. Hermite-Hadamard type inequalities occur in almost all cases. ...

Some Hermite-Hadamard type inequalities are derived for products of functions having Orlicz-convexity properties. We also obtain these inequalities via Riemann-Liouville fractional integrals for Orlicz-convex functions. These inequalities are as best as possible from the sharpness point of view, meaning that a sharpnesss class of functions is defined for each inequality, within the functions that are s-affine of first kind. Some special cases are discussed.

... (2) This concept was introduced by Orlicz, see [17]. It is known as Orlicz-convexity but it is often named s-convexity of first kind [10] or s 1 -convexity [19]. Obviously, the domain D is (α, β)-convex in this case. ...

... The results in this section show that the inequalities derived in this paper establish the best boundaries in the set of the s-convex functions in the first sense. They are a subclass of many classes of generalized convex functions as shown in [19]. Hermite-Hadamard type inequalities occur in almost all cases. ...

Some Hermite-Hadamard type inequalities are derived for products of functions having Orlicz-convexity properties. We also obtain these inequalities via Riemann-Liouville fractional integrals for Orlicz-convex functions. These inequalities are as best as possible from the sharpness point of view, meaning that a sharpness class of functions is identified, for each inequality, within the functions that are s-affine of first kind. Some special cases are discussed.

... With this the first sense of s-convexity becomes a close to the meaning of convexity and so the geometric explanation of s-convex function is easy to be compared with the geometry of convex function if some further restrictions are imposed to it. The proposed geometric description for s-convex curve in the first sense stated by Pinheiro [25][26][27][28][29][30] as follows: ...

In this work, we discuss the continuity of h -convex functions by introducing the concepts of h -convex curves ( h -cord). Geometric interpretation of h -convexity is given. The fact that for a h -continuous function f , is being h -convex if and only if is h -midconvex is proved. Generally, we prove that if f is h -convex then f is h -continuous. A discussion regarding derivative characterization of h -convexity is also proposed.

... With this the first sense of s-convexity becomes a close to the meaning of convexity and so the geometric explanation of s-convex function is easy to be compared with the geometry of convex function if some further restrictions are imposed to it. The proposed geometric description for s-convex curve in the first sense stated by Pinheiro [25]- [30] as follows: ...

In this work, we discuss the continuity of h-convex functions by introducing the concepts of h-convex curves (h-cord). Geometric interpretation of h-convexity is given. The fact that for a h-continuous function f, is being h-convex if and only if is h-midconvex is proved. Generally, we prove that if f is h-convex then f is h-continuous. A discussion regarding derivative characterization of h-convexity is also treated.

In this paper, we review the results of the paper H-H inequalityforS-convex functions, published in the prestigious academic vehicle IJPAM.Substantial part of those results will have to be nullified. Most of the time,the mistakes have been inherited from other authors’ work, so that we are alsoproviding argumentation for the nullification of those authors’ results in thispaper in an indirect way
On s-convexity and our paper with IJPAM. Available from: https://www.researchgate.net/publication/266172514_On_s-convexity_and_our_paper_with_IJPAM [accessed Dec 22, 2015].

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