(Asking again in a new question because previous version had insufficient conditions as pointed out in answer there.)

Define the densities:

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

where $f(x)=g(x^2)$, and the following hold: $g$ is twice-differentiable, increasing, and convex or concave on $(0,\infty)$, $g''(x)$ is non-decreasing on $(0,\infty)$, and $f'(0^+) < \infty$. (Interestingly these densities indeed have the same measure for all $\theta$ whenever $g$ is concave or convex, which can be shown using an integral representation.)

Show that for all $r$, $p(\phi;\theta,r)$ has a monotone likelihood ratio (decreasing for concave $g$, increasing for convex $g$). I.e., for $0\le\theta_1 < \theta_2\le\pi/2$:

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

is monotonic on $[0,\pi/2]$.

Examples of functions are $f(x) = |x|^p$, $1\le p<2$, or for $p>2$. (For $p=2$, $p(\phi;\theta) = \sin^2(2\phi)$.) And the function $f(x) = \log( \cosh(x))$, which is concave in $x^2$ (i.e. $\log(\cosh(\sqrt{x}))$ is concave on $[0,\infty)$), and twice differentiable at $x=0$ (unlike $|x|^p$, $p<2$).

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 the Karlin-Rubin theorem. The result can be proved for $f(x)=x^4$ by simplifying the derivative expression. This corresponds to using kurtosis as the the cost function, and the uniqueness result is already known in this case. For $f(x) = |x|$, the likelihood ratio is non-increasing, constant around $\phi=0$ and $\phi=\pi/2$.