This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ ��,$ where $ ��>2$ is real and non-integral. For fixed non-zero real numbers $ ��_i $ not all of the same sign we write \begin{equation*} \mathcal F (\textbf{x}) = ��_1 x_1^��+ \cdots + ��_s x_s^��. \end{equation*} For a fixed positive real number $ ��$ we give an asymptotic formula for the number of positive integer solutions of the inequality $ | \mathcal F (\textbf{x}) | < ��$ inside a box of side length $P.$ Moreover, we investigate the problem of representing a large positive real number by a positive definite generalized polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.

Accepted for publication in Mathematika