Fix . Let , then

**Proof:** Unfold and rewrite what we want to show as:

By Cauchy-Schwarz the left hand-side is less than

which can be written as

But, since ,

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Fix . Let , then

**Proof:** Unfold and rewrite what we want to show as:

By Cauchy-Schwarz the left hand-side is less than

which can be written as

But, since ,

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A more general version: if is a convex function on and is its gradient, then

,

i.e., is monotone. With you get the above. works too, if the gradient at 0 is defined as some vector in the subdifferential of .

Indeed, the restriction of to the line through and is a convex function of one real variable. Therefore, its derivative is nondecreasing, and this is exactly what the inequality says.

[PPC: I’ve edited the LaTeX, thanks. You forgot to write ‘latex’ after the dollar sign]I was vaguely hoping for the TeX markup to magically transform into formulas.