Functional inequalities for the error function

Summary.

Let erf be the error function. We determine the best possible constants $ \alpha_n $ and $ \beta_n $ such that the functional inequalities $ \alpha_n \, \text{\rm erf}\Bigl(\sum_{i=1}^{n}x_i\Bigr) \leq{\sum_{i=1}^{n}\text{\rm erf}(x_i) -\prod_{i=1}^{n}\text{\rm erf}(x_i)}\leq{\beta_n \, \text{\rm erf}\Bigl(\sum_{i=1}^{n}x_i\Bigr)} $ hold for all $ x{_i} \geq 0 (i = 1,..., n) $. Furthermore, we determine the sharp constants $ \alpha_n $ and $ \beta_n $ under the assumption that $ x{_i} \leq 0 (i = 1,..., n) $. Our results extend and complement an inequality due to Mitrinović-Esseen.