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Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$. Conjecture: if $p,q \in \mathcal{C}$, then …

I would like to show that the function, $$ f(x) = \frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c\,;x\big)} $$ is concave for $0 < x < 1 …

It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density: $$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big( \hspace{-1pt}\c …

Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function: $$f(\phi;\theta) = \mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+1} …

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) …

(Asking again in a new question because the previous version had insufficient conditions, as pointed out in the answer there.) Define the densities: $$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\theta)\ …

(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.) Define the densities: $$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\ …

Define the family of densities: $$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta)\big)\Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}, \qua …

Prove that: $$ f(x) = \log\big( {}_2F_1(a,\,b\,;\,c\,;\,x^{-1})\big),\;\;a,b,c>0 $$ is convex (and decreasing) on $(1,\infty)$. It actually seems that the stronger result that $f\big((x+1)^{\beta}\bi …