Let $f(x) = \log(\cosh(x))$, and define the kernel density:

$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)},\quad 0\le \phi,\theta \le \mbox{$\large\frac{\pi}{2}$},\;\,r>0$$

Show that for all $r>0$, $p_r(\phi;\theta)$ has a decreasing likelihood ratio. I.e., for $0\le\theta_1 < \theta_2\le\pi/2$, the function:

$$ h(\phi) = \frac{f\big(r\cos(\phi-\theta_2)\big) - f\big(r\cos(\phi+\theta_2)\big) }{f\big(r\cos(\phi-\theta_1)\big) - f\big(r\cos(\phi+\theta_1)\big)}$$

is decreasing on $[0,\pi/2]$, and:

$$ \frac{\partial}{\partial \phi} \frac{\partial}{\partial \theta}\hspace{2pt} \log p_r(\phi;\theta) \le 0,\quad 0 \le \phi,\theta\le \mbox{$\large\frac{\pi}{2}$}$$

If we define: $$g_1\hspace{-1pt}(r,x) = x\tanh(x)-(r^2-x^2)\big(1-\tanh^2(x)\big) $$ and: $$g_2\hspace{-0.5pt}(r,x) = (r^2-x^2)\tanh^2(x) $$

then the result will be true if, for $r>0$ and $u>v\ge 0$:

$$\big(g_1\hspace{-1pt}(r,u)+g_1\hspace{-1pt}(r,v)\big)\Big(\hspace{-1pt}\log\hspace{-1pt}\big(\hspace{-1.5pt}\cosh(u)\big)-\log\hspace{-1pt}\big(\hspace{-1.5pt}\cosh(v)\big)\hspace{-1pt}\Big) +g_2\hspace{-0.5pt}(r,u)-g_2\hspace{-0.5pt}(r,v) \;<\; 0$$

This result is important to prove uniqueness of stable optima in the unmixing and deconvolution of linear mixtures of independent random variables with strongly sub- and super-gaussian densities, using variation diminishing property of MLR densities.