Let function $f(z),\,z=re^{i\theta}\in\mathbb C$ for $r,\theta\in\mathbb R$ on the complex plane $\mathbb C$, be analytic in the interior and continuous on the whole closed domain of the wedge $|\theta|\le \frac{\pi}{2\alpha},\,\alpha>1$.
Does $$\limsup_{r\to\infty}\frac{\ln\ln(f(re^{i\theta}))}{\ln r}=\rho \tag{1}$$ for some postive $\rho$ point-wise in $\theta$ implies uniformly in $\theta$?
Suppose Equation (1) holds
2.a point-wise;
2.b uniformly,
does $$\limsup_{r\to\infty}\frac{\ln(f(re^{i\theta}))}{r^\rho}=\tau \tag{2}$$ for some $\tau\in\mathbb R$ point-wise in $\theta$ imply uniformly in $\theta$?