An Exploration of Brannan’s Coefficient Conjecture

Complex analysis of a single variable is a type of mathematical analysis that observe properties for holomorphic functions. Over decades of development, complex analysis nowadays has played a more and more important role in algebraic geometry, fluid dynamics, quantum mechanics, and so on. This paper is to elaborate the Brannan’s coefficient conjecture, which was first proposed by D.A Brannan, J.G. Clunie, and W.E Kirwam in their paper on the coefficient problem for functions of bounded boundary rotation in 1973. The purpose of this paper is to give a detailed review and exploration of complicated but elegant proofs by D. Aharonov and S. Friedland. This paper mainly focuses on proving the inequality (1+αx)n(1−x)p≤(1+x)n(1−x)p for p > 0 |α| = 1 within condition n ≥ 1, which was derived from Brannan’s coefficient conjecture. The proof of the inequality is done by dividing the techniques of complex numbers, trigonometric functions, binomial expansions, and other mathematical formulas into three different cases.