  The Jensen's inequality plays a crucial role in the analysis of time-delay
and sampled-data systems. Its conservatism is studied through the use of the
Gr\"{u}ss Inequality. It has been reported in the literature that fragmentation
(or partitioning) schemes allow to empirically improve the results. We prove
here that the Jensen's gap can be made arbitrarily small provided that the
order of uniform fragmentation is chosen sufficiently large. Non-uniform
fragmentation schemes are also shown to speed up the convergence in certain
cases. Finally, a family of bounds is characterized and a comparison with other
bounds of the literature is provided. It is shown that the other bounds are
equivalent to Jensen's and that they exhibit interesting well-posedness and
linearity properties which can be exploited to obtain better numerical results.
