  This paper is concerned with the study of diagonal Diophantine inequalities
of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral.
For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write
\begin{equation*} \mathcal F (\textbf{x}) = \lambda_1 x_1^\theta + \cdots +
\lambda_s x_s^\theta. \end{equation*} For a fixed positive real number $ \tau $
we give an asymptotic formula for the number of positive integer solutions of
the inequality $ | \mathcal F (\textbf{x}) | < \tau $ 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.
