We investigate some estimates of the integral $\int_0^{2\pi}\text{Log}\,|P(e^{i\th})|\df{d\th}{2\pi}$, if the polynomial $P(z)$ has a concentration at low degrees measured by the $l_p$-norm, $1\le p\le 2$. We also obtain better estimates for some concentrations than those obtained in [1].