  An improved Amati correlation was constructed in (ApJ 931 (2022) 50) by us
recently. In this paper, we further study constraints on the $\Lambda$CDM and
$w$CDM models from the gamma ray bursts (GRBs) standardized with the standard
and improved Amati correlations, respectively. By using the Pantheon type Ia
supernova sample to calibrate the latest A220 GRB data set, the GRB Hubble
diagram is obtained model-independently. We find that at the high redshift
region ($z>1.4$) the GRB distance modulus from the improved Amati correlation
is larger apparently than that from the standard Amati one. The GRB data from
the standard Amati correlation only give a lower bound limit on the present
matter density parameter $\Omega_{\mathrm{m0}}$, while the GRBs from the
improved Amati correlation constrain the $\Omega_{\mathrm{m0}}$ with the $68\%$
confidence level to be $0.308^{+0.066}_{-0.230}$ and $0.307^{+0.057}_{-0.290}$
in the $\Lambda$CDM and $w$CDM models, respectively, which are consistent very
well with those given by other current popular observational data including
BAO, CMB and so on. Once the $H(z)$ data are added in our analysis, the
constraint on the Hubble constant $H_0$ can be achieved. We find that two
different correlations provide slightly different $H_0$ results but the
marginalized mean values seem to be close to that from the Planck 2018 CMB
observations.
