We study convexity in the class of weighted quasi-arithmetic means. It turns out that their convexity depends only on the generator, neither on weights, nor on the number of variables. Connections between the convexity of a mean and the convexity of its increasing generators are considered. We prove that convex means are generated by convex strictly increasing functions. A simple example shows that the converse is not true, so the problem arises when this is the case. Some answers are given under regularity assumptions imposed on the generator.
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