An inequality involving the generalized hypergeometric function and the arc length of an ellipse

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) = F-2(1) (1/2, -1/2; 1; x) [(1 + (1-x)(3/4))/2](2/3) is positive for x is an element of (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 F-3(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.