A Monotonicity Property Involving 3 F 2 and Comparisons of the Classical Approximations of Elliptical Arc Length

Conditions are determined under which \,₃F₂(-n,a,b;a+b+2,ε -n+1;1) is a monotone function of n satisfying a b·\,₃F₂(-n,a,b;a+b+2,ε -n+1;1) ≥ a b·\,₂F₁(a,b;a+b+2;1). Motivated by a conjecture of Vuorinen [Proceedings of Special Functions and Differential Equations, K. S. Rao, R. Jagannathan, G. Vanden Berghe, J. Van der Jeugt, eds., Allied Publishers, New Delhi, 1998], the corollary that \,₃F₂(-n,-1/2,-1/2;1,ε -n+1;1) ≥ 4/π, for 1>ε ≥1/4 and n≥ 2, is used to determine surprising hierarchical relationships among the 13 known historical approximations of the arc length of an ellipse. This complete list of inequalities compares the Maclaurin series coefficients of 2 F 1 , with the coefficients of each of the known approximations, for which maximum errors can then be established. These approximations range over four centuries from Kepler's in 1609 to Almkvist's in 1985 and include two from Ramanujan.