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An inequality involving the generalized hypergeometric function and the arc length of an ellipse. (English) Zbl 0943.33002

A conjecture of M. Vuorinen that the Muir approximation is a lower approximation to the arc length of an ellipse is verified. Vuorinen conjectured that

f(x)= 2 F 1 (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, they show that

3 F 2 (-n,a,b;1+a+b,1-ε-n;1)>0

when 0<ab/(1+a+b)<ε<1 for all positive integers n.

33C20Generalized hypergeometric series, p F q