Dunkl-Williams inequality
Let $V$ be an inner product space^{} and $a,b\in V$. If $a\ne 0$ and $b\ne 0$, then
$$\parallel a-b\parallel \ge \frac{1}{2}(\parallel a\parallel +\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel}-\frac{b}{\parallel b\parallel}\parallel .$$ | (1) |
Equality holds if and only if $a=0$, $b=0$, $\parallel a\parallel =\parallel b\parallel $ or $a\parallel b\parallel =b\parallel a\parallel $. In fact, if (1) holds and $V$ is a normed linear space, then $V$ is an inner product space.
If $X$ is a normed linear space and $a\ne 0$ and $b\ne 0$ then
$$\parallel a-b\parallel \ge \frac{1}{4}(\parallel a\parallel +\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel}-\frac{b}{\parallel b\parallel}\parallel .$$ | (2) |
Equality holds if and only if $a=0$, $b=0$ or $a=b$. The constant $\frac{1}{4}$ is best possible. For example, let $X$ be the set of ordered pairs of real numbers, with norm of $({x}_{1},{x}_{2})$ equal to $|{x}_{1}|+|{x}_{2}|.$ Let $a=(1,\u03f5)$ and $b=(1,0)$ where $\u03f5$ is a small positive number. After a bit of routine calculation, it is easily seen that the best possible constant is $\frac{1}{4}$.
The inequality^{} (2) has been generalized in the case where $X$ is a normed linear space over the reals. In that case one can show:
$$\parallel a-b\parallel \ge {c}_{p}{({\parallel a\parallel}^{p}+{\parallel b\parallel}^{p})}^{1/p}\parallel \frac{a}{\parallel a\parallel}-\frac{b}{\parallel b\parallel}\parallel $$ | (3) |
where ${c}_{p}={2}^{-1-1/p}$ if $$ and ${c}_{p}=1/4$ if $p\ge 1$. The case $p=1$ is the Dunkl and Williams inequality.
If $X$ is a normed linear space and $$ then (3) holds with ${c}_{p}={2}^{-1/p}$ if and only if $X$ is an inner product space.
The inequality (2) can be improved slightly to get:
$$\parallel a-b\parallel \ge \frac{1}{2}\mathrm{max}(\parallel a\parallel ,\parallel b\parallel )\parallel \frac{a}{\parallel a\parallel}-\frac{b}{\parallel b\parallel}\parallel .$$ | (4) |
Equality holds in (4) if and only if $a$ and $b$ span an $\mathrm{\ell}_{2}{}^{1}$ in the underlying real vector space with $\pm {\parallel b-a\parallel}^{-1}(b-a)$ and $\pm {\parallel a\parallel}^{-1}a$ (or $\pm {\parallel b\parallel}^{-1}b$) as the vertices of the unit parallelogram.
References
- 1 C.F. Dunkl, K.S. Williams, A simple norm inequality. Amer. Math. Monthly, 71 (1) (1964) 53-54.
- 2 W.A. Kirk, M.F. Smiley, Another characterization of inner product spaces, Amer. Math. Monthly, 71, (1964), 890-891.
- 3 A.M. Alrashed, Norm inequalities and Characterizations of Inner Product Spaces, J. Math. Anal. Appl. 176, (2) (1993), 587-593.
- 4 J.L. Massera, J.J. Schäffer, Linear differential equations and functional analysis^{}, Annals of Math. 67 (2)(1958), 517-573. (on page 538)
- 5 L.M. Kelly, The Massera-Schäeffer equality, Amer. Math. Monthly, 73, (1966) 1102-1103.
Title | Dunkl-Williams inequality |
---|---|
Canonical name | DunklWilliamsInequality |
Date of creation | 2013-03-22 16:56:38 |
Last modified on | 2013-03-22 16:56:38 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 12 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 47A12 |