Remarks on the Grunsky norm and \(p\)th root transformation (Q1301971)

Let \(S\) denote the class of functions \(f\) holomorphic and univalent in the unit disc \(E\), and normalized by the conditions \(f(0)= 0\), \(f'(0)=1\). Each function \(f\in S\) generates its Grunsky operator \(G_f= \{\sqrt{nm} \alpha_{n,m} \}_{n,m=1}^\infty: l^2\to l^2\), where \(\alpha_{n,m}\) are determined by the expansion \(\log [f(z)- f(\zeta)]/ (z-\zeta)= \sum_{n,m=0}^\infty \alpha_{n,m} z^n \zeta^m\), \(z,\zeta\in E\). Here \(l^2\) is the Hilbert space of all square-summable complex sequences \(X= (x_1, x_2,\dots)\), with the norm \(\|X\|= (\sum_{n=1}^\infty|x_n|^2)^{1/2}\). The author shows that the norm of the Grunsky operator generated by a univalent function does not decrease with the \(p\)th root transformation, i.e. there holds the theorem: Let \(f\in S\), \(p=2,3,\dots\), and let \(g\) be defined by the formula \(g(z)= \root p\of {f(z^p)}\), \(z\in E\). Then \(\|G_f\|\leq \|G_g\|\). This inequality is sharp for each \(p\) and over each subclass \(\{f\in S: \|G_f\|= k\}\), \(0\leq k\leq 1\).