A Monotonicity Property Involving ₃F₂ and Comparisons of the Classical Approximations of Elliptical Arc Length

Abstract

Abstract. Conditions are determined under which 3 F 2 (−n, a, b; a + b + 2, ε − n + 1; 1) is a monotone function of n satisfying ab· 3 F 2 (−n, a, b; a + b + 2, ε − n + 1; 1) ≥ ab· 2 F 1 (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 3 F 2 (−n, − 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.

A Note on the Hypergeometric Mean Value

“…Applying Lemma 1, which is a general positivity result involving 3 F 2 and is of independent interest (see [6,7,8]), we have obtained the following Theorem 1. Suppose a ∈ (0, 1) and b, c > 0.…”

“…For an extensive bibliography and history see [1,3,4]. Hypergeometric functions, which have many of the classical special functions as special cases, have been found useful in resolving several current problems as noted in [5,7,8,10,11]. Given real numbers α, β, and γ with γ = 0, −1, −2, .…”

“…In [7], [8], R. Barnard, K. Pearce, and K. Richards proved very recently that the following inequalities are true for all r ∈ [0, 1]:…”

Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind

“…The first inequality of (1.2) is due to Qiu and Shen [13] and the second inequality of (1.2) is due to Barnard et al [2]. In [4,5], the authors proved that the inequalities…”

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

“…Remarks The ideas and techniques used to prove Lemma 2.1 and Theorem 1.1 will be used in [5] to determine surprising hierarchial relationships among the 13 historical approximations to L(a, b) discussed in [2]. These approximations range over four centuries from Kepler's in 1642 to Almkvist's in 1985 and include two from Ramanujan.…”