Infimum of $\left|\frac{p+q+p^{-1}+q^{-1}}{\ln(p)-\ln(q)}\right|$

Question

Let $A$ be a positive constant and $\textbf{D}$ the set $\left\{(p,q)\in\textbf{R}^2 \mid 0<q<p \right\}$. I am looking at the infimum of the expression $$\left|\frac{p+q+A\left(\frac{1}{p}+\frac{1}{q}\right)}{\ln(\frac{p}{q})}\right| \quad (\text{for } (p,q)\in \textbf{D}).$$ Does someone have an idea? Many thanks!

Answer 1

As either $p$ or $q$ tends to $\infty$, the expression increases without bound.

Thus the minimum will occur for some finite $p$ and $q$. Because also the expression is never zero in your domain, the minimum will be a local minimum of the function $f(p,q) = \frac{p+q+A(p^{-1}+q^{-1})}{\ln(p) - \ln (q) }$. This can be found using calculus.

You may not be able to solve the corresponding equations in terms of elementary functions, but wolframalpha can approximate it for you.

For $A = 1$, it gets an infimum of $\approx 3.01776$