# Schmidt's inequality

## by Shayne Waldron

## Abstract:

The main result is the computation of the best constant in the
Wirtinger-Sobolev inequality

$$\norm{f}_p\le C_{p,q,\gth}\,(b-a)^{1+{1\over p}-{1\over q}}\norm{Df}_q,$$

where

$$f(\gth)=0,$$

and $\gth$ is some point in $[a,b]$, or, equivalently,
the determination of the norm of the (bounded) linear map

$$A:L_q[a,b]\to L_p[a,b]$$

given by

$$Af(x):=\int_\gth^xf(t)\,dt.$$

This and other results are seen to be closely related to an inequality of
Schmidt 1940.
The method of proof is elementary, and should be the main point of interest
for most readers since it clearly illustrates a technique that can be applied
to other situations. These include the generalisations of Hardy's inequality
where $\gth=a$ and $\norm{\cdot}_p$, $\norm{\cdot}_q$ are replaced by weighted
$p$, $q$ norms, and higher order Wirtinger-Sobolev inequalities involving
boundary conditions at a single point.

**Keywords:**
Schmidt's inequality,
Hardy-type inequalities,
Wirtinger-Sobolev inequalities,
Poincar\'e inequalities,
H\"older's inequality,
$n$-widths,
isoperimetric calculus of variations problems

**Math Review Classification:**
41A44, 41A80, 47A30 (primary), 34B10, 34L30 (secondary)

**Length:**

**Comment:**
See Project Hermite
for related work

**Last updated:**
20 May 1997

**Status:**
Appeared in East Journal on Approximations, Volume 3, Number 2 (1997), 11-29

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