Abstract. 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) = 2 F 1 (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 3 F 2 (−n, a, b; 1 + a + b, 1 + − n; 1) > 0 when 0 < ab/(1 + a + b) < < 1 for all positive integers n.

In 1972, D.A. Brannan conjectured that all of the odd coefficients, a 2n+1 , of the power series (1 + xz) α /(1 − z) were dominated by those of the series (1 + z) α /(1 − z) for the parameter range 0 < α < 1, after having shown that this was not true for the even coefficients. He verified the case when 2n + 1 = 3. The case when 2n + 1 = 5 was verified in the mid-eighties by J.G. Milcetich. In this paper, we verify the case when 2n + 1 = 7 using classical Sturm sequence arguments and some computer algebra. 1980

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The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen’s inequality. The conjecture was also made independently by B. Conroy in connection with his work in number theory. A crucial and interesting lemma is proved which describes general coefficient-root relations for polynomials with nonnegative coefficients and for polynomials for which the case of equality holds in Descarte’s Rule of Signs.

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.