Some remarks on rotation theorems for complex polynomials

For any complex polynomial $P(z)=c_0+c_1z+...+c_nz^n, c_n\not=0,$ having all its zeros in the unit disk $|z|\le 1,$ we consider the behavior of the function (arg$P(e^{i\theta}))'_{\theta}$ when the real argument $\theta$ changes. We give some sharp estimates of this function involving of the values of $P(e^{i\theta}),$ arg$P(e^{i\theta})$ or the coefficients $c_k, k=0,1,n-1,n.$