In this article, we establish several inequalities for $(h,m)$-convex maps, related to weighted integrals, used in previous works. Throughout the work, we show that our results generalize several of the integral inequalities known from the literature.

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In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well.

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