I found this inequality from Duren's book Theory of Hp-spaces(2000) page 66. According to the book inequality should be true for $\frac{1}{2}\leq r < 1$ and $p>\frac{1}{2}$. I have tried to prove that
$$\frac{1}{[(1-r)^2 + 2\pi^{-2}\theta^2]^p} < \frac{1}{(1-r)^{2p-1}[1+2\pi^{-2}\theta^2]^p}$$
to get the inequality but I could not find the way. I was able to get
$$\frac{1}{[(1-r)^2 + 2\pi^{-2}\theta^2]^p} = \frac{1}{(1-r)^{2p}\left[1 + \frac{2\pi^{-2}\theta^2}{(1-r)^2}\right]^p} < \frac{1}{(1-r)^{2p}\left[1 + 2\pi^{-2}\theta^2\right]^p}$$ but I got stuck here.