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Zbl 0943.33002
Barnard, Roger W.; Pearce, Kent; Richards, Kendall C.
An inequality involving the generalized hypergeometric function and the arc length of an ellipse. (English)
[J] SIAM J. Math. Anal. 31, No.3, 693-699 (2000). ISSN 0036-1410; ISSN 1095-7154/e

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 $n$.
[Som Prakash Goyal (Jaipur)]

MSC 2000:: *33C20 Generalized hypergeometric series

Keywords: generalized hypergeometric function; approximations; elliptic arc length

Cited in: Zbl 1009.33005

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