A proof of Campbell’s subordination conjecture

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− √ α2 + 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 affirmatively verify Campbell’s conjecture in Theorem 1.