# application of Cauchy–Schwarz inequality

In determining the perimetre of ellipse one encounters the elliptic integral

 $\int_{0}^{\frac{\pi}{2}}\!\!\sqrt{1-\varepsilon^{2}\sin^{2}t}\;dt,$

where the parametre $\varepsilon$ is the eccentricity of the ellipse ($0\leqq\varepsilon<1$).  A good upper bound for the integral is obtained by utilising the http://planetmath.org/node/1628Cauchy–Schwarz inequality

 $\left|\int_{a}^{b}fg\right|\;\leqq\;\sqrt{\int_{a}^{b}f^{2}}\,\sqrt{\int_{a}^{% b}g^{2}}$

choosing in it  $f(t):=1$  and  $g(t):=\sqrt{1-\varepsilon^{2}\sin^{2}t}$.  Then we get

 $\displaystyle 0\;<\;\int_{0}^{\frac{\pi}{2}}\!\!\sqrt{1-\varepsilon^{2}\sin^{2% }t}\;dt$ $\displaystyle\;\leqq\;\sqrt{\int_{0}^{\frac{\pi}{2}}1^{2}\,dt}\sqrt{\int_{0}^{% \frac{\pi}{2}}\left(1-\varepsilon^{2}\sin^{2}t\right)\,dt}$ $\displaystyle\;=\;\sqrt{\frac{\pi}{2}}\sqrt{\int_{0}^{\frac{\pi}{2}}\left(1-% \varepsilon^{2}\cdot\frac{1-\cos 2t}{2}\right)\,dt}$ $\displaystyle\;=\;\frac{\pi}{2}\sqrt{1-\frac{\varepsilon^{2}}{2}}.$

Thus we have the estimation

 $\int_{0}^{\frac{\pi}{2}}\!\!\sqrt{1-\varepsilon^{2}\sin^{2}t}\;dt\;\leqq\;% \frac{\pi}{2}\sqrt{1-\frac{\varepsilon^{2}}{2}}.$

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 $O(\varepsilon^{4})$

Title application of Cauchy–Schwarz inequality ApplicationOfCauchySchwarzInequality 2013-03-22 18:59:42 2013-03-22 18:59:42 pahio (2872) pahio (2872) 5 pahio (2872) Application msc 26A42 msc 26A06 application of Cauchy-Schwarz inequality