Let \(n\geq 2\) and let \(A: \mathbb{R}^n\to[0,\infty]\) be any convex function satisfying the following properties: \[ A(0)= 0\quad\text{and}\quad A(\xi)= A(-\xi)\quad\text{for }\xi\in \mathbb{R}^n; \]
\[ \text{for every }t>0,\;\{\xi\in\mathbb{R}^n: A(\xi)\leq t\}\text{ is a compact set whose interior contains }0. \] We find an optimal Young function \(B:[0,\infty)\to [0,\infty]\) for which there exists a constant \(K\), depending only on \(n\), such that \[ \int_{\mathbb{R}^n} B\Biggl({|u(x)|\over K(\int_{\mathbb{R}^n} A(\nabla u) dy)^{1/n}}\Biggr) dx\leq \int_{\mathbb{R}^n} A(\nabla u) dx \] for every real-valued weakly differentiable function \(u\) on \(\mathbb{R}^n\) decaying to \(0\) at infinity. Here, \(\nabla u\) denotes the gradient of \(u\).