Campbell's conjecture on a majorization-subordination result for convex functions
\textit{D. M. Campbell} [Trans. Am. Math. Soc. 198, 297-306 (1974; Zbl 0293.30015)] has shown that any f which is analytic in the unit disc, satisfies \(f'(0)\geq 0\) and is subordinate to an F belonging to the class of normalized analytic locally univalent functions of order \(\alpha\), \(1.65\leq\alpha,\) also satisfies \(| f'(z)|\leq | F'(z)|\) in \(| z| <\alpha +1+(\alpha^ 2+2\alpha)^{{1\over2}}.\) He conjectured that the theorem is true for \(1\leq\alpha <1.65,\) too. The authors confirm this for \(\alpha =1\), i.e. for the case of a convex F. Their method seems not to be applicable in the case \(1<\alpha <1.65.\)
scientific article; zbMATH DE number 3873649