Higher order Glaeser inequalities and optimal regularity of roots of real functions

We prove a higher order generalization of the Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k + alpha)-th root of a function of class C-k whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.