An Improvement of the Power-Mean Integral Inequality in the Frame of Fractal Space and Certain Related Midpoint-Type Integral Inequalities

First, we construct a reformative version of the power-mean integral inequality in the sense of fractal space. Second, we define what we named as the generalized [Formula: see text]-convex mappings, and investigate some related properties. Moreover, in accordance with the derived midpoint-type integral identities on fractal space, we establish certain improvements of the midpoint-type integral inequalities for mappings whose first-order derivatives in absolute value belong to the generalized [Formula: see text]-convex mappings. As applications in association with local fractional calculus, we acquire three inequalities considering [Formula: see text]-type arithmetic mean and [Formula: see text]-logarithmic mean, numerical integration, as well as probability density mappings, respectively.