Two inequalities between cardinal invariants

We prove two $\mathrm{Z}\mathrm{F}\mathrm{C}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{$\mathrm{c}\mathrm{o}\mathrm{v}$}}^{\ast}({$\mathcal{Z}$}{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when $\kappa =\omega$, that if $\kappa$ is any regular uncountable cardinal, then ${\mathfrak{s}}{\kappa} \leq {\mathfrak{b}}_{\kappa}$.