Suppose that \(1 < p < \infty \), \(q=p/(p-1)\), and for non-negative \(f\in L^p(-\infty\! ,\infty )\) and any real x we let \(F(x)-F(0)=\int _0^xf(t)\ dt\); suppose in addition that \(\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0\). Moser's second one-dimensional inequality states that there is a constant \(C_p\), such that \(\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p\) for each f with \(||f||_p\le 1\) and every \(a\le 1\). Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.