A Carlitz type result for linearized polynomials

Abstract: For an arbitrary $q$-polynomial $f$ over $\mathbb{F}{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$.