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In the Wikipedia proof of the Minkowski inequality (http://en.wikipedia.org/wiki/Minkowski_inequality), the following inequality is used:
$$|f+g|^p\leq2^{p-1}(|f|^p+|g|^p).$$
I was just wondering if this inequality has a name or if this is too "first principles" to warrant a name.

Thanks!

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  • $\begingroup$ If I read the article correctly it's just using convexity. $\endgroup$
    – Michael Hoffman
    Commented Nov 2, 2009 at 7:53
  • $\begingroup$ Thanks Michael, I was under the impression that this was generally true! $\endgroup$
    – mornington
    Commented Nov 2, 2009 at 8:10
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    $\begingroup$ It is generally true. It is the composition of the triangle inequality |f+g| </= |f|+|g| composed with the power mean inequality (a+b)/2 </= ((a^p+b^p)/2)^(1/p) for a=|f| and b=|g| which is a consequence of convexity. $\endgroup$
    – Philipp Lampe
    Commented Nov 2, 2009 at 8:19

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It is not named after a person, but I would argue that the inequality does have a name, "the convexity inequality for xp". If you look at the definition of a convex function, the statement that f(x) = xp is convex is exactly the statement that ((x+y)/2)p ≤ (xp+yp)/2. (Plus either that f is continuous or that the weighted version also holds). It is also correct to say "is a consequence of convexity", but it seems better to call it the statement of convexity than a consequence of convexity.

The multivariate form of the convexity inequality is named after a person; it is Jensen's inequality.

All of this may seem like a pat answer, but it works. It is a theorem that you get a valid norm if xp is replaced by any non-negative convex function φ(x) with suitable behavior at 0 and ∞. The resulting norm is called an Orlicz norm and the resulting Banach space is called an Orlicz space.

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