We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u\circ f^{-1}\in EXP\left(\right)$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1+KlogK)\le \left(\right||u\circ f^{-1}{\left|\right|}_{EXP\left(\right)}{)/(\left|\right|u\left|\right|}_{EXP\left(\right)})\le 1+KlogK$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{\infty }$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1/K\le \left(dis{t}_{EXP\left(f\right(G\left)\right)}(u\circ f^{-1},L^{\infty }\left(f\left(G\right)\right))\right)/\left(dis{t}_{EXP\left(f\right(G\left)\right)}(u,L^{\infty }\left(G\right))\right)\le K$ for every u ∈ EXP(). We also prove that...