A monotonicity property involving F-3(2) and comparisons of the classical approximations of elliptical arc length

Conditions are determined under which F-3(2) (-n, a, b; a + b + 2, epsilon- n + 1; 1) is a monotone function of n satisfying ab.F-3(2)(-n, a, b; a + b + 2, epsilon - n + 1; 1) greater than or equal to ab.F-2(1) (a, b; a + b 2; 1). Motivated by a conjecture of Vuorinen [ Proceedings of Special Funct ions and Differential Equations, K. S. Rao, R. Jagannathan, G. Vanden Bergh e, J. Van der Jeugt, eds., Allied Publishers, New Delhi, 1998], the corolla ry that F-3(2) (-n, -1/2, -1/2; 1, epsilon - n + 1; 1) greater than or equa l to 4/pi, for 1 > epsilon greater than or equal to 1/4 and n greater than or equal to 2, is used to determine surprising hierarchical relationships a mong the 13 known historical approximations of the arc length of an ellipse . This complete list of inequalities compares the Maclaurin series coeffici ents of F-2(1) with the coefficients of each of the known approximations, f or which maximum errors can then be established. These approximations range over four centuries from Kepler's in 1609 to Almkvist's in 1985 and includ e two from Ramanujan.