Abstract: We examine properties of the minimizer u* of a class of differentiable functionals where both the data-term and the regularization term are symmetric and nearly affine beyond a small neighborhood of the origin. Customarily, such functions are used to regularize a quadratic data-fidelity term in order to produce solutions where edges are preserved. The functionals we consider in this paper behave quite differently. They were recently successfully applied to provide a strict order for the pixels of digital (quantized) images f thus enabling exact histogram specification. We give upper and lower bounds for the error $\|u* - f\|_\infty$, where the upper bound is independent of the input image f. Interestingly, in the numerical experiments with natural digital images f, the estimated upper bound is easily reached up to a small error. To explain this phenomenon we give simple statistical estimates for the behavior of neighboring pixels. We apply our estimates to specify the parameters of the model.