On univalent Bloch functions (Q5946689)

Let \(\Delta=\{z:| z| <1\}\) be the unit disc of the complex plane \(\mathbb{C}\) and let \(\varphi_a(z)=\frac{a-z}{1-\bar{a}z}\), \(a\in\Delta\), be the Möbius transformation of \(\Delta\) . Let \(g(z,a)=\log\left| \frac{1-\bar{a}z}{z-a}\right| \) be the Green's function of \(\Delta\) with logarithmic singularity at \(a\in\Delta\). The space \(Q_p\) consists of all functions \(f\) analytic in \(\Delta\) for which \[ \sup_{a\in\Delta}\iint_{\Delta}| f'(z)| ^2(g(z,a))^p\,dA(z)<\infty, \] where \(dA(z)\) is the Euclidean area element on \(\Delta\). It is known that \(Q_q\subsetneq Q_1=\text{ BMOA}\subsetneq Q_p=\mathcal{B}\) for \(0<q<1<p<\infty\), where BMOA denotes the space of all analytic functions of bounded mean oscillation and \(\mathcal{B}\) denotes the Bloch space \(\{f:f\text{ analytic on }\Delta \text{ and }\sup_{z\in\Delta}(1-| z| ^2)| f'(z)| <\infty\}\). A non-constant function \(T:[0,\infty)\to[0,\infty)\) is an essentially increasing function if there exists an \(r_0\in(0,\infty)\) such that \(T\) is bounded in \([0,r_0]\) and increasing in \((r_0,\infty)\) with \(T(r_0)>0\). The log-order of the function \(T(r)\) is defined as \[ \rho=\overline{\lim}_{r\to\infty}\frac{\log^+\log^+T(r)} {\log r}, \] where \(\log^+ x=\max\{\log x,0\}\). If \(0<\rho<\infty\), the log-type of the function \(T(r)\) is defined as \[ \sigma=\overline{\lim}_{r\to\infty}\frac{\log^+ T(r)} {r^{\rho}}. \] The main result in this paper is the following: Theorem. Let \(f\) be an analytic univalent function in \(\Delta\) and let \(T:[0,\infty)\to[0,\infty)\) be an essentially increasing function such that \(T(t)=O((t\log\frac{1}{t})^p)\) as \(t\to 0\) for some \(p>0\). Suppose that the log-order \(\rho\) and the log-type \(\sigma\) of \(T\) satisfy one of the following conditions: (i) \(0\leq\rho<1\), (ii) \(\rho=1\) and \(\sigma<2\). Then \(f\in\mathcal{B}\) if and only if \[ \sup_{a\in\Delta}\iint_{\Delta}| f'(z)| ^2 T(g(z,a))\,dA(z)<\infty. \] By choosing \(T(t)=t^p\) the author proves as a corollary: If \(f\) is an analytic function in \(\Delta\), then \(f\in\mathcal B\) if and only if \(f\in Q_p\) for all \(p\), \(0<p<\infty\).