An Inequality Involving the Generalized Hypergeometric Function and the Arc Length of an Ellipse

https://doi.org/10.1137/S0036141098341575

Richards, Kendall C.

2018-02-27

2018-02-27

2000

Barnard, R. W., Pearce, K., & Richards, K. C. (January 01, 2000). An Inequality Involving the Generalized Hypergeometric Function and the Arc Length of an Ellipse. Siam Journal on Mathematical Analysis, 31, 3, 693-699.

https://collections.southwestern.edu/s/suscholar/item/263

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( 12 , − 12 ; 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.

English

Siam Journal on Mathematical Analysis

Hypergeometric

Approximations

Elliptical arc length

Article

Faculty Scholarship