I came across the paper by Anderson et al. (2007) "Generalized Convexity and Inequalities" (here https://arxiv.org/abs/math/0701262). That paper, in particular, considers the following eight generalized convexity notions:
I have two questions.
It looks to me like there is a pattern with notations (A,G,H) which I don't quite see yet.
What motivated these notions (AA is an obvious one but what about others) and what are their applications? A quick google search has not really shed light on this. I am especially interested in where thhese notions may be used.
Section 2.2 (Examples) lists a few different mean functions: A is the arithmetic mean, G is the geometric mean and H is the harmonic mean. This is for A/H/G.
In Section 2.3, $MN$-convexity for a function $f$ if defined by requiring $f(M(x,y)) \leq N(f(x),f(y))$ for all $x,y$. For example, a function is $AG$ convex iff: $$f\left(\frac{x+y}{2}\right) \leq \sqrt{f(x)f(y)} \quad \forall x,y$$