Generalisations of integral inequalities of Hermite-Hadamard type through convexity (Q2865136)

Several Hermite-Hadamard-type inequalities are proven for non-negative functions on \([0, \infty)\) whose derivatives are \(s-(\alpha, m)\) convex, which is defined by the inequality NEWLINE\[NEWLINE f(\mu x + (1-\mu) y) \leq (\mu^{\alpha s }) f(x) + m (1-\mu^{\alpha s}) f \left( \frac{y}{m}\right) NEWLINE\]NEWLINE for \(\alpha, m \in [0,1]\) and \(s \in (0,1]\). The authors appear to intend this definition to be a generalization of \((\alpha, m)\)-convexity and \(s\)-convexity. However, since \(\alpha s\) can be treated as any number in \([0,1]\), this definition appears to be nearly identical to the definition of \((\alpha,m)\) convexity.NEWLINENEWLINE Applications to special means and the trapezoidal method of numerical integration are presented.