Lower and upper bounds involving a logarithmic expression

Question

I would like to know what are the upper and lower bounds of the given expression for $\theta\in(0,1)$ \begin{align*} \rho(\theta) = \frac{6}{\theta^{2}}\left[-5\theta + (1+3\theta)\ln(1-\theta) + (3+\theta)\sqrt{\theta}\ln\left(\frac{1+\sqrt{\theta}}{1-\sqrt{\theta}}\right)\right] - 3 \end{align*}

Perhaps one could propose to study its monotonicity, but the resulting expression $\rho'(\theta)$ is not much inviting to deal with. Any suggestions?

Answer 1

Hint

Use the first few terms of the Taylor expansion of $\ln (1+x)$ as $$\ln (1+x)=\sum_{n=1}^{\infty}{(-1)^{n+1}\over n}x^n \quad,\quad -1<x<1$$