Minimizing $f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}$ subject to the constraint

Question

Let $f(r)$ be a function defined as follows \begin{align} f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}} \end{align}

where $0 < A,c$ and $ t\in (0,1)$. I want to solve \begin{align} \min_{\frac{1}{t}<x} f(x) \end{align}

My approach:

Let $g(x)=\ln(f(x))$ then both $g(x)$ and $f(x)$ have the same minimum where \begin{align} g(x)=\frac{tx-1}{x-1} \ln(A)+\frac{(1-t)x}{x-1} \ln(c)+\frac{1-t}{x-1} \ln \left(\Gamma(0.5+x) \right)-\frac{1-t}{2(x-1)}\ln(\pi) \end{align}

and \begin{align} &g'(x)=\\ &=\frac{1-t}{(x-1)^2} \ln(A)-\frac{1-t}{(x-1)^2} \ln(c)+(1-t)\frac{(x-1)\psi^{(0)}(x+0.5)-\log(\Gamma(x+0.5))}{(x-1)^2}+\frac{1-t}{2(x-1)^2}\ln(\pi)\\ &=\frac{1-t}{(x-1)^2} \left(\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5)\right) \end{align}

where $\psi(x)$ is the so called digamma function.

So, this means we have to focus on \begin{align} h(x)=\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5) \end{align}

for $\frac{1}{t} <x$.

But how to solve $h(x)=0$ or say for what $x$ is $h(x)>0$ ???

If this impossible to do then the approximated solution is also fine?

I also feel that there might be a simpler approach with out using derivative.

Thank you for any help.

Answer 1

Hopefully the following can somehow help or give some idea.

Rewrite \begin{align} h(x)=\ln \left( \frac{ \sqrt{\pi} A}{c\Gamma(x+0.5)} \right)+ (x-1)\psi^{(0)}(x+0.5) \end{align} as \begin{align} h(x)= (x-1)\psi^{(0)}(x+0.5)- \ln \left( \Gamma(x+0.5) \right) +\alpha \end{align} where $\alpha =\ln \left( \frac{ \sqrt{\pi} A}{c} \right)$.

Plot using Mathematica

1. graph of $h(x)$ intersection with $0$-plane

   Plot3D[{[Alpha] + (x - 1) PolyGamma[0, 1/2 + x] - Log[Gamma[1/2 + x]], 0}, {x, 0, 3}, {[Alpha], -2, 2}]

2. Contour plot of $h(x)=0$

   ContourPlot[{[Alpha] + (x - 1) PolyGamma[0, 1/2 + x] - Log[Gamma[1/2 + x]], 0}, {x, 0, 5}, {[Alpha], -3, 4}]