Rättyä, J. \(n\)-th derivative characterisations, mean growth of derivatives and \(F(p,q,s)\). (English) Zbl 1064.30027 Bull. Aust. Math. Soc. 68, No. 3, 405-421 (2003). For \(0< p<\infty\), \(-2< q<\infty\) and \(0\leq s<\infty\). \(F(p,q,s)\) is defined as the set of all analytic functions, on \(D\) for which \[ \sup_{a\in D}\,\iint_D|f'(z)|^p(1-|z|^2)^qg^s(z, a)\,d\sigma_z<\infty, \] where \(g(z,a)= \log|1- \overline az|/|z- a|\). \(F(2,1,0)\) is the Hardy space \(H^2\), \(F(p,p,0)\) is the Bergman space \(L^p_a\), \(F(2,0,s)= Q_s\) and \(F(2,0,1)= \text{BMOA}\). The author generalizes known results of \(F(p,q,0)\) to \(F(p,q,s)\) for \(0\leq s<\infty\). In particular, he generalizes results of R. Yoneda [Characterizations of Bloch and Besov spaces by oscillations, Hokkaido Math. J. 29, No. 2, 409–451 (2000; Zbl 0984.46019)]. Moreover, he answers partially the question in the paper of R. Yoneda. Reviewer: Takahiko Nakazi (Sapporo) Cited in 23 Documents MSC: 30D45 Normal functions of one complex variable, normal families 30D50 Blaschke products, etc. (MSC2000) Keywords:analytic function; \(Q_p\) space; Bloch space; BMOA; derivative Citations:Zbl 0984.46019 PDFBibTeX XMLCite \textit{J. Rättyä}, Bull. Aust. Math. Soc. 68, No. 3, 405--421 (2003; Zbl 1064.30027) Full Text: DOI References: [1] Duren, Theory of H (1970) [2] DOI: 10.2307/2043708 · Zbl 0524.30023 [3] Aulaskari, Bull. Austral. Math. Soc. 58 pp 43– (1998) [4] DOI: 10.1006/jmaa.2000.7259 · Zbl 0980.30024 [5] Zygmund, Trigonometric series (1959) · JFM 58.0296.09 [6] Zhu, Operator theory in function spaces 139 (1990) · Zbl 0706.47019 [7] Yoneda, Hokkaido Math. J. 29 pp 409– (2000) · Zbl 0984.46019 [8] DOI: 10.1111/j.1749-6632.1996.tb33169.x [9] Stroethoff, Bull. Austral. Math. Soc. 54 pp 211– (1996) [10] Stroethoff, Bull. Austral. Math. Soc. 39 pp 405– (1989) [11] Rättyä, Ann. Acad. Sci. Fenn. Math. Dissertationes 124 pp 1– (2001) [12] Heittokangas, Ann. Acad. Sci. Fenn. Math. Dissertationes 122 pp 1– (2000) [13] Garnett, Bounded analytic functions 96 (1981) [14] DOI: 10.1016/0022-247X(72)90081-9 · Zbl 0246.30031 [15] Fàbrega, Ann. Inst. Fourier. (Grenoble) 46 pp 111– (1996) · Zbl 0840.32001 [16] Danikas, Function spaces and complex analysis (Joensuu, 1997) pp 9– (1999) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.