A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $li{m}_{x\mapsto x_0}\left(\right|f\left(x\right)-f\left(x_0\right)\left|\right)/\left(\right|\varphi \left(x\right)-\varphi \left(x_0\right)\left|\right)=0$ for every $x_0$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $li{m}_{t\mapsto t_0}\left(\right|t-t_0\left|\right)/\left(\right|\varphi \left(t\right)-\varphi \left(t_0\right)\left|\right)=0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

We prove some quantitatively sharp estimates concerning the Δ₂ and ∇₂ conditions for functions which generalize known ones. The sharp forms arise in the connection between Orlicz space theory and the theory of elliptic partial differential equations.