An algebraic equation question [closed]

Question

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

Answer 1

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 $\beta<0$.