On \((j,k)\)-symmetrical functions. (English) Zbl 0838.30004

Let \(X\), \(Y\) be two vector spaces over \(\mathbb{C}\) and let \(U\subseteq X\) be a \(k\)-fold symmetric set, i.e. \(\varepsilon U= U\) when \(\varepsilon= \exp({2\pi i\over k})\). A function \(f\in {\mathcal F}(U, Y):= Y^U\) is called \((j, k)\)-symmetric if \(f(\varepsilon z)= \varepsilon^j f(z)\) for every \(z\in U\). Using the operation \[ G^l_k= k^{- 1} \sum^{k- 1}_{j= 0} \varepsilon^{- lj} L^j_k, \] where \((L_k f)(z)= f(\varepsilon z)\), \(z\in U\), and \(L^j_k\) is the \(j\)th iterate of \(L_k\), the authors show that every function \(f\in {\mathcal F}(U, Y)\) has a unique representation as a sum \(\sum^{k- 1}_{j= 0} y_j\) of \((j, k)\)-symmetric functions \(y_j\) (take \(y_j= G^j_k(x)\)).
In the more concrete setting of holomorphic selfmaps of the open unit disk, the authors obtain best estimations like \(|G^j_k f(z)|\leq |z|^l\), \((l= 1, 2,\dots, k- 1)\), generalizing results of A. Pfluger [Elem. Math. 40, 46-47 (1985; Zbl 0566.30021)]. Several integral estimations, such as \(|\int_U f(z) d\lambda(z)|\leq {r^{k+ 1} k\over k+ 1}\), where \(U= \bigcup_{0\leq s\leq r} sE_k\), \(E_k= \{1, \varepsilon, \varepsilon^2,\dots, \varepsilon^{k- 1}\}\) and \(f(0)= 0\) are derived.
Reviewer: R.Mortini (Metz)


30A99 General properties of functions of one complex variable
30A10 Inequalities in the complex plane
32A99 Holomorphic functions of several complex variables


Zbl 0566.30021
Full Text: EuDML EMIS