estimating a particular analytic function on a bounded sector.

Question

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ~~~r>0,-\frac{\pi}{2}<\theta<\frac{\pi}{2}, $$ for some constant $C$ and $n>0$. Furthermore, on the positive real line, and for any fixed $a>0$, we have the following (stronger) estimates $$ |f(r,a)|\leq C \frac{r}{a^{n+1}},~~~0<r\leq a, $$ where $C$ is a constant independent with $a$. My question is can we also get the following improved estimates of $f$ $$ |f(re^{i\theta},a)|\leq C \frac{r}{a^{n+1}},~~~0<r< a,~~-\frac{\pi}{2}<\theta<\frac{\pi}{2}? $$ If not, can you give a counterexample to this? The motivation of this question comes from estimating the Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for small $|z|$ With $\Re z>0$.To be more precise, if we let $K(z,x,y)$ be its kernel, then $K$ satisfies the estimates above with $a=|x-y|$. Although we can compute the kernel explicitly (for all $\Re z>0$ ) by using Fourier transform ($K(z,x,y)=\frac{z}{(z^2+a^2)^{n+1}}$,a=|x-y|), I think one may also obtain this by combining the kernel estimates on the positive real line (the second estimate above) with a weaker estimates on the right half plane (the first estimate above) and some analytic function theory. I don't know theorems like Hadamard's Three line lemma or its variants would be useful here. Thanks in advance for any comment.