application of Cauchy–Schwarz inequality

In determining the perimetre of ellipse one encounters the elliptic integralMathworldPlanetmath


where the parametre ε is the eccentricity of the ellipse (0ε<1).  A good upper bound for the integralDlmfPlanetmath is obtained by utilising the–Schwarz inequality


choosing in it  f(t):=1  and  g(t):=1-ε2sin2t.  Then we get

0<0π21-ε2sin2t𝑑t 0π212𝑑t0π2(1-ε2sin2t)𝑑t

Thus we have the estimation


It is the better approximation for the perimetre of ellipse the smaller is its eccentricity, i.e. the closer the ellipse is to circle.  The accuracy is O(ε4)

Title application of Cauchy–Schwarz inequality
Canonical name ApplicationOfCauchySchwarzInequality
Date of creation 2013-03-22 18:59:42
Last modified on 2013-03-22 18:59:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Application
Classification msc 26A42
Classification msc 26A06
Synonym application of Cauchy-Schwarz inequality