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

Barnard, Roger W.; Pearce, Kent; Richards, Kendall C.

SIAM J. Math. Anal. 31, No. 3, 693-699 (2000).

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 \[ f(x)= {_2F_1}(1/2,-1/2; 1;x)- [(1+(1- x)^{3/4})/2]^{2/3} \] 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, they show that \[ {_3F_2}(- n,a,b;1+ a+ b,1-\varepsilon- n;1)> 0 \] when \(0< ab/(1+ a+ b)<\varepsilon< 1\) for all positive integers \(n\).

Reviewer: Som Prakash Goyal (Jaipur)

MSC:

33C20

Generalized hypergeometric series, \({}_pF_q\)

Keywords:

generalized hypergeometric function; approximations; elliptic arc length