Skip to main content
MathOverflow

Return to Question

Added reference to another question
Source Link
mather
mather
  • 21
  • 4

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.

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 $f(x,y)f(y,x) \leq f(x,x)f(y,y)$. 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.

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.

Source Link
mather
mather
  • 21
  • 4

General bivariate functions that satisfy Cauchy-Schwarz

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 $f(x,y)f(y,x) \leq f(x,x)f(y,y)$. 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.