An Inequality Involving the Generalized Hypergeometric Function and the Arc Length of an Ellipse

Abstract

In this paper we verify a conjecture of M. Vuorinen that the Muir approximation is a lower approximation to the arc length of an ellipse. Vuorinen conjectured that $f(x)={}_{2}F_{1}({\frac{1}{2}},-{\frac{1}{2}};1;x)-[(1+(1-x)^{3/4})/2]^{2/3}$ is positive for $x\in (0,1)$. The authors prove a much stronger result which says that the Maclaurin coefficients of f are nonnegative. As a key lemma, we show that ${}_{3}F_{2}(-n,a,b;1+a+b,1+\epsilon -n;1) > 0$ when $0 < ab/(1+a+b) < \epsilon < 1$ for all positive integers n.