Using the notion of s-ϕ-convex functions as generalization of convex functions, we estimate the difference between the middle and right terms in Hermite- Hadamard-Fejer inequality for differentiable mappings.

En este artículo establecemos algunos nuevos resultados relacionados a desigualdades del tipo Hermite-Hadamard para funciones cuya segunda derivada es s − ϕ−convexa.

There is a strong connection between convexity and inequalities. So, techniques from each concept applies to the other due to the strong correlation between them; specifically, in the past few years. In this attempt, we consider the Hermite–Hadamard inequality and related inequalities for the class of functions whose absolute value of the third derivative are quasi-convex functions. Finally, the applications of our findings for special functions and particular functions are pointed out.

In this paper we introduce the notion of s−j−convex functions as generalization of convex functions. Some basic results under various conditions for the function j are investigated. Moreover, we establish Ostrowski type inequalities for twice differentiable mappings which are s−j−convex.

La desigualdad integral de Jensen tiene mucha importancia en cuanto a sus aplicaciones en diferentes campos de las matemáticas. En este artículo encontramos una nueva desigualdad tipo Jensen para funciones cuya segunda derivada en valor absoluto es $varphi$-convexa.