Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}\_K$ and let $f(X)\in \mathcal{O}\_K[X]$. Denote the Mahler measure by $M$.


Is there any known result about the comparison of the values $M(\alpha)$ and $M(f(\alpha))$?



Mahler measures of values of polynomials
This is all standard stuff about height functions. More generally, if we use absolute heights, then for any $f(x)\in\overline{\mathbb{Q}}[x]$ of degree $n$ there are constants $C\_1(f)>0$ and $C\_2(f)>0$ so that for any $\alpha\in\overline{\mathbb{Q}}$, 
$$ C\_1(f)H(\alpha)^n \le H\bigl(f(\alpha)\bigr) \le C\_2(f)H(\alpha)^n.$$
For $C\_2(f)$ it's easy to get an explicit formula in terms of the coefficients of $f$ using the triangle inequality, as in Moreschi's answer. For $C\_1(f)$, one can get an explicit formula that depends on the coefficients and the resultant of $f$. See for example Lang's *Fundamentals of Diophantine Geometry*.


