Let me define first
(1) A convex function $\Phi \colon \mathbb{R}\to \mathbb{R}^+$ which satisfies the conditions,
(a) $\Phi(0)=0$
(b)$\Phi(-x)=\Phi(x)$
(c) $\lim_{x \to \infty}\Phi(x)=+\infty$, is called the Young function.
(2) A Young function $\Phi \colon \mathbb{R}\to \mathbb{R}^+$ is said to satisfy the $\Delta_{2}$ condition if, $\Phi(2x)\leq K\Phi(x)$ for $x\geq x_{0}\geq 0$ for some absolute constant $K>0$
Can someone give some example that does not satisfy $\Delta_{2}$ condition?Thanks