A coefficient estimate for nonvanishing \(H^ p\) functions (Q1117059)

Let \(B_ p\) denote the set of all functions \(f(z)=a_ 0+a_ 1z+..\). which are regular and non-vanishing in the unit disc, belong to \(H^ p\) and satisfy \(\| f\|_ p\leq 1\). Let any such f have the representation \(e^{i\gamma}\Omega (z)I(z)\) where \(\Omega\) (z) and I(z) are the outer and inner functions of f respectively. Let \(B_ p(n)\) denote the set of all \(f\in B_ p\) such that \(\Omega '(0)=\Omega ''(0)=...=\Omega^{(n-1)}(0)=0\). Hummel, Zalcman, and Scheinberg conjectured that if \(f\in B_ p\), then \(| a_ n| \leq (2/e)^{1/q}\) for all n where \(1/p+1/q=1\). By making clever use of the Pontryagin maximum principle, the authors prove that if \(f\in B_ p(2)\) then \(| a_ 2| \leq (2/e)^{1/q}\) (there is a misprint in the statement of the theorem in the paper) and if \(f\in B_ p(3)\) then \(| a_ 3| \leq (2/e)^{1/q}\).