## 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)

### MSC:

 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
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