We show how to improve constraints on Omega_m, sigma_8, and the dark-energy equation-of-state parameter, w, obtained by Mantz et al. (2008) from measurements of the X-ray luminosity function of galaxy clusters, namely MACS, the local BCS and the REFLEX galaxy cluster samples with luminosities L> 3 times 10^{44} erg/s in the 0.1--2.4 keV band. To this aim, we use Tinker et al. (2008) mass function instead of Jenkins et al. (2001) and the M-L relationship obtained from Del Popolo (2002) and Del Popolo et al. (2005). Using the same methods and priors of Mantz et al. (2008), we find, for a Lambda$CDM universe, Omega_m=0.28^{+0.05}_{-0.04} and sigma_8=0.78^{+0.04}_{-0.05}$ while the result of Mantz et al. (2008) gives less tight constraints $Omega_m=0.28^{+0.11}_{-0.07}$ and sigma_8=0.78^{+0.11}_{-0.13}. In the case of a wCDM model, we find Omega_m=0.27^{+0.07}_{-0.06}, $sigma_8=0.81^{+0.05}_{-0.06}$ and $w=-1.3^{+0.3}_{-0.4}$, while in Mantz et al. (2008) they are again less tight Omega_m=0.24^{+0.15}_{-0.07}, sigma_8=0.85^{+0.13}_{-0.20} and w=-1.4^{+0.4}_{-0.7}. Combining the XLF analysis with the f_{gas}+CMB+SNIa data set results in the constraint Omega_m=0.269 pm 0.012, sigma_8=0.81 pm 0.021 and w=-1.02 pm 0.04, to be compared with Mantz et al. (2008), Omega_m=0.269 pm 0.016, sigma_8=0.82 pm 0.03 and w=-1.02 pm 0.06. The tightness of the last constraints obtained by Mantz et al. (2008), are fundamentally due to the tightness of the $f_{gas}$+CMB+SNIa constraints and not to their XLF analysis. Our findings, consistent with w=-1, lend additional support to the cosmological-constant model.
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