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:

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

From the derivative expression, after changing variables and simplifying, I believe the result will be true if for $y>x\ge 0$ :

$$\bigg(\frac{y^2-x^2}{\cosh^2(x)}-x\tanh(x)-y\tanh(y)\!\bigg)\log\!\bigg(\frac{\cosh(y)}{\cosh(x)}\bigg) + \tanh^2(x)\big(y^2-x^2\big) > 0 $$

Edit with progress(?): The bound:

$$\log(\cosh(y))-\log(\cosh(x)) \le \frac{\tanh(x)}{2x}\big(y^2-x^2\big) $$

follows from the concavity of $\log(\cosh(\!\sqrt{x}))$. Using this bound on just the positive $(y^2-x^2)/\cosh^2(x)$ term, we can isolate the $\log\cosh$ term, and produce the following inequality: for $y>x\ge 0$,

$$\frac{\tanh(x)\big(y^2-x^2\big){}^2+2\hspace{0.5pt}x\sinh^2(x)\big(y^2-x^2\big)}{2\hspace{0.5pt}x\cosh^2(x)\big(x\tanh(x)+y\tanh(y)\big)} -\log\bigg(\frac{\cosh(y)}{\cosh(x)}\bigg) \;>\;0$$

This function is actually increasing in $y$, so taking the derivative with respect to $y$ eliminates $\log\cosh$ entirely, and it would suffice to show that the resulting expression is positive. However the expression does not seem to simplify in any convenient way, though it only contains powers of $x,y,\tanh(x)$, and $\tanh(y)$.

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.