A proof of Campbell's subordination conjecture

Roger W. Barnard, Kent Pearce
2009 Complex Variables and Elliptic Equations  
In the early 70's, D.M. Campbell published three papers on majorizationsubordination results for locally univalent functions. In particular, he showed that if F is linearly invariant of order α and if f is subordinate to F on {z : |z| < 1}, then f is majorized by F on {z : |z| < m(α)} where m(α) = α + 1 − √ α 2 + 2α, provided α ≥ 1.65. He conjectured, in fact, that this result also held for 1.65 > α ≥ 1. We review Campbell's proof and why the restriction α ≥ 1.65 arose in the proof. We then
more » ... rmatively verify Campbell's conjecture in Theorem 1.
doi:10.1080/17476930802669686 fatcat:lhedhdyt4nbxtnzr2tot5uo5aq