Jensen type inequalities for $Q$-class functions

Some inequalities of Jensen type for $Q$-class functions are proved. More precisely a refinement of the inequality $f\left(\frac{1}{P}\sum_{i=1}^{n}p_ix_i\right)\leq P\sum_{i=1}^{n}\frac{f(x_i)}{p_i}$ is given in which $p_1, \ldots, p_n$ are positive numbers, $P=\sum_{i=1}^{n}p_i$ and $f$ is a $Q$-class function. The notion of jointly $Q$-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite--Hadamard type inequalities related to $Q$-class functions are presented as well. DOI: 10.1017/S000497271100286