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Brannan's conjecture for initial coefficients

Udaya C. Jayatilake
Journal / Anthology

Complex Variables and Elliptic Equations
Year: 2013
Volume: 58
Issue: 5
Page range: 685-694

In 1973, D. A. Brannan conjectured that odd Maclaurin coefficients An( y, , ) of the function (1žyz)/(1z) satisfy the inequality jAn(y, , )jjAn(1, , )j for all y,  and  such that jyj¼ 1, 40, 40. He verified that this is true when n¼3 and showed that this inequality is not true for even coefficients in general. This article deals with the special case of Brannan’s conjecture when ¼1. The case n¼5 with ¼1 was settled by J.G. Milcetich in 1989. For n¼7 and ¼1, Brannan’s conjecture was proved to be true by R.G. Barnard, K. Pearce and W. Wheeler in 1997. In this work we introduce a squaring procedure which allows us to reduce the proof of Brannan’s conjecture to the verification of positivity of polynomials of much smaller degree than An(y, , 1) which, in addition, have integer coefficients. In this case the positivity of polynomials can be verified using the Sturm sequence method. We wrote a short program in Mathematica code and used it to prove Brannan’s conjecture for ¼1, 0551 and all odd n51.

*Mathematics > Probability and Statistics

Brannan’s conjecture, Maclaurin coefficients