### Abstract:

In 1973, D. A. Brannan conjectured that odd Maclaurin coefficients A n (y, α, β) of the function (1 + yz)α/(1 − z)β satisfy the inequality |A n (y, α, β)| ≤ |A n (1, α, β)| for all y, α and β such that |y| = 1, α > 0, β > 0. He verified that this is true when n = 3 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 β = 1. The case n = 5 with β = 1 was settled by J.G. Milcetich in 1989. For n = 7 and β = 1, 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 A n (y, α, 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 β = 1, 0 < α < 1 and all odd n ≤ 51.