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, withrespect to the absolute value, are (s, m)-convex and (alpha, m)-convex functions. These results are relatedto well-known Hermite-Hadamard type integral inequality, Simpson type integral inequality, and Jensentype inequality. In other words, new upper bounds for these inequalities using the indicated classes ofconvex functions have been obtained. These estimates are obtained using a direct definition for a convexfunction, classical integral inequalities of Holder and power mean types. Along with the new outcomes, thepaper presents results confirming the existing in literature upper bound estimates for integral inequalities(in particular well known in literature results obtained by U. Kirmaci in [7] and M.Z. Sarikaya and N. Aktanin [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)