  Stability of linear systems with uncertain bounded time-varying delays is
studied under assumption that the nominal delay values are not equal to zero.
An input-output approach to stability of such systems is known to be based on
the bound of the $L_2$-norm of a certain integral operator. There exists a
bound on this operator in two cases: in the case where the delay derivative is
not greater than 1 and in the case without any constraints on the delay
derivative. In the present note we fill the gap between the two cases by
deriving an operator bound which is an increasing and continuous function of
the delay derivative upper bound $d \geq 1$. For $d\to \infty$ the new bound
corresponds to the second case and improves the existing bound. As a result,
improved frequency-domain and time-domain stability criteria are erived for
systems with the delay derivative greater than 1.
