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) = 2F1( 1 2 ,− 1 2 ; 1;x) − [(1 + (1 − x)3/4)/2]2/3 is positive for x ∈ (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 3F2(−n, a, b; 1 + a + b, 1 + − n; 1) > 0 when 0 < ab/(1 + a + b) < < 1 for all positive integers n.