Multivalent functions and \(Q_K\) spaces (Q1777722)

Let \(K:[0,\infty)\to[0,\infty)\) be a right-continuous and nondecreasing function. The space \(Q_K\) (\(Q_{K,0}\)) consists of analytic functions \(f\) defined in the unit disk \(\Delta=\{z:| z| <1\}\) for which \(\sup_{a\in\Delta}\iint_\Delta| f'(z)| ^2K(g(z,a))\,dA(z)<\infty\), (\(\lim_{| a| \to1}\iint_\Delta| f'(z)| ^2K(g(z,a))\,dA(z)=0\)) where \(g(z,a)\) is the Green function of~\(\Delta\) with singularity at~\(a\). In order to avoid a trivial case it is supposed that \(\int_1^\infty K(t)e^{-2t}\,dt\) is convergent. For \(0<p<\infty\), \(K(t)=t^p\) gives the space \(Q_p\) (\(Q_{p,0}\)). If \(p=1\), \(Q_1=\) BMOA (\(Q_{1,0}=\) VMOA) and for all \(p>1\), \(Q_p={\mathcal B}\), the Bloch space (\(Q_{p,0}={\mathcal B}_0\), the little Bloch space). Let \(n(w,f)\) denote the number of roots of the equation \(f(z)=w\) in \(\Delta\) counted according to their multiplicity. For an analytic function~\(f\) in~\(\Delta\) let \(p(\rho)=\frac1{2\pi}\int_0^{2\pi}n(\rho e^{i\phi},f)\,d\phi\). If \(\int_0^R p(\rho)\,d(\rho^2)\leq qR^2\), \(R>0\), or \(p(R)\leq q\), \(R>0\), where \(q\) is a positive number, then \(f\) is areally mean \(q\)-valent or circumferentially mean \(q\)-valent, respectively. The author's main results are the following theorems: Theorem 1. Let \(f\) be an areally mean \(q\)-valent function in~\(\Delta\). If \[ \int_0^1\left(\log\frac1{1-r}\right)^2(1-r)^{-1}K \left(\log\frac1r\right)r\,dr<\infty,\tag{1} \] then (i) \(f\in\mathcal B\) if and only if \(f\in Q_K\) (ii) \(f\in\mathcal B_0\) if and only if \(f\in Q_{K,0}\). This generalizes \textit{C. Pommerenke's} Satz 1 in [Commun. Math. Helv. 52, 591--602 (1977; Zbl 0369.30012)] and reviewer's \textit{P. Lappan}, \textit{J. Xiao} and \textit{R. Zhao} Theorem 6.1 in [J. Math. Anal. Appl. 209, No. 1, 103--121 (1997; Zbl 0892.30030)]. For circumferentially mean \(q\)-valent functions the author proves Theorem 2. Let \(f\) be a circumferentially mean \(q\)-valent and nonvanishing function in~\(\Delta\). If (1) holds, then \(\log f\in Q_K\). This result should be compared with \textit{A. Baernstein's} Theorem 2 in [Mich. Math. J. 23, 217--223 (1976; Zbl 0331.30014)].