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On analytic and meromorphic functions and spaces of $$Q_K$$-type. (English) Zbl 1048.30017
Let $$K$$ be a right-continuous nonnegative nondecreasing function on $$[0,\infty).$$ The authors define $$Q_K$$ to be the set of functions $$f$$ analytic in the unit disk $$\Delta$$ for which $| | f| | _K^2 :=\sup \int_{\Delta} K(g(z,a)) | f'(z)| ^2\,dxdy < \infty,$ where $$g$$ is the Green function of $$\Delta$$ and the sup is over $$a\in \Delta.$$ The set $$Q_K^{\#}$$ is defined to be the set of meromorphic functions in $$\Delta$$ for which the supremum above is finite when $$f'$$ is replaced by the spherical derivative $$f^{\#}$$ of $$f.$$ The sets $$Q_K$$ and $$Q_K^{\#}$$ are invariant under Möbius transformations mapping $$\Delta$$ onto itself, and the $$Q_K$$ are Banach spaces. For $$K(t) = t^p, 0<p<\infty,$$ these sets, denoted by $$Q_p$$ and $$Q_p^{\#}$$, have been the subject of a good deal of recent work. It is known, for example, that $$Q_1$$ coincides with the space $$BMOA$$ of analytic functions of bounded mean oscillation, and that for $$p>1$$ $$Q_p$$ coincides with the Bloch space. Moreover, $$Q_p^{\#}$$ coincides with the set of normal functions when $$1<p<\infty.$$ In the paper under review the authors develop some theory for general $$K.$$ For example: If $$K_1, K_2$$ satisfy the conditions for $$K,$$ if $$K_1(r) = o(K_2(r))$$ as $$r\rightarrow 0$$ and if a certain integral involving $$K_2$$ diverges, then $$Q_{K_2}$$ is properly contained in $$Q_{K_1}.$$

##### MSC:
 30D45 Normal functions of one complex variable, normal families 30D50 Blaschke products, etc. (MSC2000)
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