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
\[
f(x)= {_2F_1}(1/2,-1/2; 1;x)- [(1+(1- x)^{3/4})/2]^{2/3}
\]
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, \({}_pF_q\) |