A variant of Jensen–Steffensen's inequality and quasi-arithmetic means

A variant of Jensen–Steffensen's inequality is proved. Necessary and sufficient conditions for the equality in Jensen–Steffensen's inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights.