Let \(a\) and \(b\) be the semiaxes of an ellipse, \(a\not=b\). The perimeter of the ellipse is given by \[ L(a,b)=4\int_0^{\pi/2}\,\sqrt{a^2\cos^2t+b^2\sin^2t}\,dt. \] During the past few centuries, many easily computable approximations to \(L(a,b)\) have been suggested by a number of mathematicians.
The goal of the paper is to present several bounds for the perimeter of the ellipse in terms of harmonic, arithmetic, root-square means \[ H(a,b)=\frac{2ab}{a+b}\,, \quad A(a,b)=\frac{a+b}2\,, \quad S(a,b)=\sqrt{\frac{a^2+b^2}2}\,, \] which improve some well-known results. For instance, let \[ C(a,b)=\frac{5A(a,b)-H(a,b)}4\,, \qquad D(a,b)=\frac{A(a,b)+S(a,b)}2\,, \] then \[ \frac{\pi(5C-D)}2<L(a,b)<\frac{2(16-2(1+\sqrt{2})\pi)C-2(16-5\pi)D}{3-2\sqrt{2}}\,. \]