Have there been any study of general bivariate functions $f:X \times X \to \mathbb{R}$ that satisfy $f(x,y)^2 \leq f(x,x)f(y,y)$. This comes up as a function I'm working with satisfies the asymmetric version $f(x,y)f(y,x) \leq f(x,x)f(y,y)$ that shows up in this question, but it is assumed $f$ is an inner product already. Ideally, since having an inner product implies C-S, could there be some requirements on $f$ to make it an inner product (besides the obvious axioms of an inner product)? Any results on these type of functions would be amazing.
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