Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer pairs $(h,m)\in [H,2H]\times[M,2M]$ such that $$ \tag{1} m h^\alpha = m_0 h_0^\alpha. $$ If $\alpha$ is rational, say $\alpha = \frac{p}{q}$, then $m^q h^p = m_0^q h_0^p$, and so the number of solutions is at most the number of divisors of $m^q h^p$, which is $O((HM)^\varepsilon)$ for any $\epsilon > 0$, the implied constant depending on both $\alpha$ and $\varepsilon$.
When $\alpha$ is irrational, I expect that the same bound should hold, but I'm not sure how to show this. Thus my question,
Is the number of solutions to (1) bounded by $O((HM)^\varepsilon)$ when $\alpha$ is irrational?
For context, the application I have in mind is a bound for certain exponential sums in the spirit of the classical paper "Exponential sums with monomials" by Fouvry and Iwaniec.
Any feedback and references are most appreciated.
