Brannan%26apos;s conjecture for initial coefficients

作者：Jayatilake Udaya C^*

来源：Complex Variables and Elliptic Equations, 2013, 58(5): 685-694.

DOI：10.1080/17476933.2011.605445

• Brannan%26apos
• s conjecture
• Maclaurin coefficients
• 30B
• 30C50

In 1973, D. A. Brannan conjectured that odd Maclaurin coefficients A(n)(y,,) of the 1+yz)/(1z) satisfy the inequality |A(n)(y,,)||A(n)(1,,)| for all y, and such that |y|=1, %26gt;0, %26gt;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%26apos;s conjecture when =1. The case n=5 with =1 was settled by J.G. Milcetich in 1989. For n=7 and =1, Brannan%26apos;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%26apos;s conjecture to the verification of positivity of polynomials of much smaller degree than A(n)(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%26apos;s conjecture for =1, 0%26lt;%26lt;1 and all odd n51.

• 出版日期2013-5-1