My question is this:


If $\frac{\sqrt[n]{\prod\_{i=1}^n(p\_i + 1)}}{\sqrt[n]{\prod\_{i=1}^n(m\_i + 1)}} = e ^\beta$


can I find an expression (either exact or approximate) for $\frac{\sqrt[n]{\prod\_{i=1}^np\_i}}{\sqrt[n]{\prod\_{i=1}^nm\_i}}$ as a function of $\beta$


If that's not possible then how about finding an expression (again, either exact or approximate) for $\frac{p}{m}$ as a function of $\beta$ if we know that $\frac{p+1}{m+1} = e ^\beta$


Note: p and m are real positives



An algebraic equation question
This is not possible. Suppose e.g. that $e^\beta=2$. Then we have $p+1=2(m+1)$, i.e. $\frac{p}{m}=2+\frac{1}{m}$. Hence $\frac{p}{m}$ can attain any real number $>2$, depending on what $m$ is. In general the only thing one can say is $\frac{p}{m}>e^\beta$, if $\beta>0$, and $\frac{p}{m}<e^\beta$, if $\beta<0$.


