Some new integral inequalities for (s, m)-convex and (α, m)-convex functions

Abstract

The paper considers several new integral inequalities for functions the second derivatives of which, with respect to the absolute value, are ( s,m )-convex and ( α,m )-convex functions. These results are related to well - known Hermite-Hadamard type integral inequality, Simpson type integral inequality, and Jensen type inequality. In other words, new upper bounds for these inequalities using the indicated classes of convex functions have been obtained. These estimates are obtained using a direct definition for a convex function, classical integral inequalities of Ho¨lder and power mean types. Along with the new outcomes, the paper presents results confirming the existing in literature upper bound estimates for integral inequalities (in particular well known in literature results obtained by U. Kırmacı in [7] and M.Z. Sarıkaya and N. Aktan in [35]). The last section presents some applications of the obtained estimates for special computing facilities (arithmetic, logarithmic, generalized logarithmic average and harmonic average for various quantities).