The \(H^ p\) spaces of some classes of univalent functions. (English) Zbl 0614.30016

Fekete, Otto

Prepr., ”Babes-Bolyai” Univ., Fac. Math., Res. Semin. 5, 45-50 (1986).

The author determines the Hardy classes for subclasses \(S_ n(\alpha)\) and \(M_{n,\alpha}(\beta)\) of univalent functions defined by the differential operator \(D^ n\) [G. S. Sălăgean, Lect. Notes Math. 1013, 362-372 (1983; Zbl 0531.30009)] and their derivatives. For instance, he proves:
Theorem 1. Let \(f\in S_ n(\alpha)\), \(n\in N\cup \{0\}\), \(\alpha\in [0,1)\), then \[ f^{(n+1)}\in H^ p,\quad p<1/(3-2\alpha);\quad f^{(n)}\in H^ p,\quad p<1/(2-2\alpha); \]
\[ f^{(n-1)}\in H^ p,\quad p<1/(1-2\alpha)\text{ if } \alpha \leq 1/2 \] and \[ f^{(n- 1)}\in H^{\infty}\text{ if } \alpha >1/2,\quad f^{(n-2)}\in H^ p,\quad p<\infty \text{ if } \alpha =0 \] and \[ f^{(n-2)}\in H^{\infty}\text{ if } \alpha \neq 0,...\quad f\in H^{\infty}. \]

Reviewer: Ren Fuyao

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30D55	\(H^p\)-classes (MSC2000)

Keywords:

starlike function; convex function

Citations:

Zbl 0531.30009