  In this paper, we present estimates for solutions and for the attraction
domain of the trivial solution for systems with delayed and nonlinear weighted
homogeneous right-hand side of positive degree. The results are achieved via a
generalization of the Lyapunov-Krasovskii functional construction presented
recently for homogeneous systems with standard dilation. Along with the
classical approach for the calculation of the estimates within the
Lyapunov-Krasovskii framework, we develop a novel approach which combines the
use of Lyapunov-Krasovskii functionals with ideas of the Razumikhin framework.
More precisely, a lower bound for the functional on a special set of functions
inspired by the Razumikhin condition is constructed, and an additional
condition imposed on the solution of the comparison equation ensures that this
bound can be used to estimate all solutions in a certain neighbourhood of the
trivial one. An example shows that this approach yields less conservative
estimates in comparison with the classical one.
