Sapirovskii 18 proved that \(|X|\leq\pi\chi(X)^{c(X)\psi(X)}\), for a regular space \(X\). We introduce the \(\theta\)-pseudocharacter of a Urysohn space \(X\), denoted by \(\psi_\theta (X)\), and prove that the previous inequality holds for Urysohn spaces replacing the bounds on celluarity \(c(X)\leq\kappa\) and on pseudocharacter \(\psi(X)\leq\kappa\) with a bound on Urysohn cellularity \(Uc(X)\leq\kappa\) (which is a weaker conditon because \(Uc(X)\leq c(X)\)) and on \(\theta\)-pseudocharacter \(\psi_\theta (X)\leq\kappa\) respectivly (note that in general \(\psi(\cdot)\leq\psi_\theta (\cdot)\) and in the class of regular spaces \(\psi(\cdot)=\psi_\theta(\cdot)\)). Further, in 6 the authors generalized the Dissanayake and Willard's inequality: \(|X|\leq 2^{aL_{c}(X)\chi(X)}\), for Hausdorff spaces \(X\) 25, in the class of \(n\)-Hausdorff spaces and de Groot's result: \(|X|\leq 2^{hL(X)}\), for Hausdorff spaces 11, in the class of \(T_1\) spaces (see Theorems 2.22 and 2.23 in 6). In this paper we restate Theorem 2.22 in 6 in the class of \(n\)-Urysohn spaces and give a variation of Theorem 2.23 in 6 using new cardinal functions, denoted by \(UW(X)\), \(\psi w_\theta(X)\), \(\theta\hbox{-}aL(X)\), \(h\theta\hbox{-}aL(X)\), \(\theta\hbox{-}aL_c(X)\) and \(\theta\hbox{-}aL_{\theta}(X)\). In 5 the authors introduced the Hausdorff point separating weight of a space \(X\) denoted by \(Hpsw(X)\) and proved a Hausdorff version of Charlesworth's inequality \(|X|\leq psw(X)^{L(X)\psi(X)}\) 7. In this paper, we introduce the Urysohn point separating weight of a space \(X\), denoted by \(Upsw(X)\), and prove that \(|X|\leq Upsw(X)^{\theta\hbox{-}aL_{c}(X)\psi(X)}\), for a Urysohn space \(X\).