This paper follows the recent discussion on the sparse solution recovery with quasi-norms $\ell_q,~q\in(0,1)$ when the sensing matrix possesses a Restricted Isometry Constant $δ_{2k}$ (RIC). Our key tool is an improvement on a version of "the converse of a generalized Cauchy-Schwarz inequality" extended to the setting of quasi-norm. We show that, if $δ_{2k}\le 1/2$, any minimizer of the $l_q$ minimization, at least for those $q\in(0,0.9181]$, is the sparse solution of the corresponding underdetermined linear system. Moreover, if $δ_{2k}\le0.4931$, the sparse solution can be recovered by any $l_q, q\in(0,1)$ minimization. The values $0.9181$ and $0.4931$ improves those reported previously in the literature.