Dunkl-Williams inequality

Let V be an inner product spaceMathworldPlanetmath and a,bV. If a0 and b0, then

a-b12(a+b)aa-bb. (1)

Equality holds if and only if a=0, b=0, a=b or ab=ba. 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 a0 and b0 then

a-b14(a+b)aa-bb. (2)

Equality holds if and only if a=0, b=0 or a=b. The constant 14 is best possible. For example, let X be the set of ordered pairs of real numbers, with norm of (x1,x2) equal to |x1|+|x2|. Let a=(1,ϵ) and b=(1,0) where ϵ is a small positive number. After a bit of routine calculation, it is easily seen that the best possible constant is 14.

The inequalityMathworldPlanetmath (2) has been generalized in the case where X is a normed linear space over the reals. In that case one can show:

a-bcp(ap+bp)1/paa-bb (3)

where cp=2-1-1/p if 0<p1 and cp=1/4 if p1. The case p=1 is the Dunkl and Williams inequality.

If X is a normed linear space and 0<p1 then (3) holds with cp=2-1/p if and only if X is an inner product space.

The inequality (2) can be improved slightly to get:

a-b12max(a,b)aa-bb. (4)

Equality holds in (4) if and only if a and b span an 21 in the underlying real vector space with ±b-a-1(b-a) and ±a-1a (or ±b-1b) as the vertices of the unit parallelogram.


  • 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 analysisMathworldPlanetmathPlanetmath, 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