I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$) \begin{equation} f(\theta)= \frac{h(t,\beta)}{h(\beta, \beta)} = \frac{1-t-\frac{\varphi_{\theta}(t)}{\varphi_{\theta}^{\prime}(\beta)}}{1-\beta-\frac{\varphi_{\theta}(\beta)}{\varphi_{\theta}^{\prime}(\beta)}} \end{equation} where $0<\beta<t<1$. The function $\varphi_{\theta}(t)$ is the generator of the Ali-Mikhail-Haq copula, \begin{equation} \varphi_{\theta}(t)=\log \left( \frac{1-\theta(1-t)}{t}\right)\\ \varphi_{\theta}^{\prime}(\beta)=\frac{\theta-1}{\beta(1-\theta(1-\beta))} \end{equation}

However, $f^{\prime}(\theta)$ seems to get me nowhere. I would appreciate any hint. Thank you so much.

I have been trying to solve the following function is non-increasing (non-decreasing) with respect $\theta$ where $\theta \in (0,1)$ (resp. $\theta \in (-1,0)$) \begin{equation} f(\theta)= \frac{h(t,\beta)}{h(\beta, \beta)} = \frac{1-t+\frac{\varphi_{\theta}(t)}{\varphi_{\theta}^{\prime}(\beta)}}{1-\beta+\frac{\varphi_{\theta}(\beta)}{\varphi_{\theta}^{\prime}(\beta)}}=\frac{1-t+\frac{\beta (1-\theta(1-\beta))}{\theta-1}\log ( \frac{1-\theta(1-t)}{t})}{1-\beta+\frac{\beta (1-\theta(1-\beta))}{\theta-1}\log ( \frac{1-\theta(1-\beta)}{\beta})} \end{equation} where $0<\beta<t<1$. The function $\varphi_{\theta}(t)$ is the generator of the Ali-Mikhail-Haq copula, \begin{equation} \varphi_{\theta}(t)=\log \left( \frac{1-\theta(1-t)}{t}\right)\\ \varphi_{\theta}^{\prime}(\beta)=\frac{\theta-1}{\beta(1-\theta(1-\beta))} \end{equation}

However, $f^{\prime}(\theta)$ seems to get me nowhere. I would appreciate any hint. Thank you so much.