Radial growth of \(C^2\) functions satisfying Bloch type condition (Q1890216)

Let \(\mathcal{B}\) be the Bloch space of all holomorphic functions \(f\) on the unit disc \(\mathbb{D}\) which satisfy \[ \| f\| _{\mathcal{B}} = | f(0)| + \sup_{z \in \mathbb{D}}(1- | z| ^2)| f'(z)| < \infty. \] The celebrated law of the iterated logarithm of \textit{N. G. Makarov} [Proc. Lond. Math. Soc. (3) 51, 369--384 (1985; Zbl 0573.30029)] states that if \(f \in \mathcal{B}\), then \[ \limsup_{r \to 1} \frac{f(r\zeta)}{\sqrt{\log \frac{1}{1 -r}\log\log\log \frac{1}{1 - r}}} \leq C \| f\| _{\mathcal{B}}, \] for almost every \(\zeta \in \mathbb{D}\), where \(C\) is a universal constant. In fact, \textit{C. Pommerenke} [Boundary behaviour of conformal maps. Springer Verlag, Berlin (1992; Zbl 0762.30001)] proved that this inequality is true for \(C=1\) but it is false for \(C \leq 0.685\). \textit{M. J. González} and \textit{P. Koskela} [Complex Var. Theory Appl. 46, 59--72 (2001; Zbl 1031.31003)] studied the radial growth of \(C^2\) functions on the unit ball \(\mathbb{B}^n\) of \(\mathbb{R}^n\) which satisfy \[ | \nabla u(x)| ^2 + | u(x) \triangle u(x)| \leq \frac{c}{(1 - | x| )^2\left( \log \frac{2}{1 - | x| }\right)^{\gamma}} \tag{1} \] for all \(x \in \mathbb{B}^n\), where \(c > 0 \) and \(\gamma \leq 1\). Specifically, they proved the following result: Theorem A. Let u be a \(C^2\) function on \(\mathbb{B}^n\) satisfying (1). Then, for almost all \(\zeta\), \(| \zeta| = 1\), \[ \limsup_{r \to 1}\frac{| u(r\zeta)| }{\sqrt{\left(\log \frac{1}{1 - r} \right)^{1 - \gamma}\log \log \frac{1}{1 - r}}} \leq c_1 \] if \(\gamma < 1\); and \[ \limsup_{r \to 1}\frac{| u(r\zeta)| }{\log \log \frac{1}{1 - r}} \leq c_2 \] if \(\gamma = 1\). The constants \(c_1\) and \(c_2\) depend only on \(n\), \(c\), \(\gamma\). In the paper under review, the authors extend theorem A by González and Koskela. For such purpose, let \(\varphi\) be a positive, continuous and nondecreasing function on the interval \([0,1)\) satisfying \(\varphi(1 - r/2) \leq A \varphi (1 - r)\) for every \(r \in (0,1)\) with a constant \(A \geq 1\) and \[ \int_0^1 (1 - t) \varphi(t) \,dt = \infty . \] Set \(\Phi(r) = \int_0^r (1 - t) \varphi(t) \,dt .\) Theorem 1. Let u be a \(C^2\) function on \(\mathbb{B}^n\) with \(u(0) = 0\) such that \[ {\mathcal{A}}_u (x) = | \nabla u(x)| ^2 + | u(x) \triangle u(x)| \leq \varphi (| x| ), \;x \in \mathbb{B}^n.\tag{2} \] Then \[ \limsup_{r \to 1}\frac{| u(r\zeta)| }{\sqrt{\Phi(r) \log \log \frac{1}{1 - r}}} \leq \sqrt{A}, \] where \(A\) is as above, and \[ \liminf_{r \to 1}\frac{| u(r\zeta)| }{\sqrt{\Phi(r) \log \log \frac{1}{1 - r}}} \leq 2 \] for almost all \(\zeta \in \mathbb{S}^{n - 1}\), modulus the Hausdorff measure \({\mathcal{H}}_{n-1}\). The proof relies on an exponential estimate for \(C^2\) functions satisfying (2).