The paper is devoted to the analysis of an error term of Gauss-Turán quadrature formula \[ \int_{-1}^1f(x)w(x)\,dx=\sum_{i=1}^n\sum_{j=0}^{2s}A_{j,i}f^{(j)}(t_i)+R_{n,s}(f) \] for functions \(f\) analytic in the region \(\bigl\{z\in{\mathbb C}:| z| <1+2\varepsilon\bigr\}\). For the Gori-Micchelli weight functions \(w\) it is proved that \(R_{n,s}(f)\geq R_{n,s+1}(f)\), \(s=0,1,2,\dots\) in three cases \(n=1,2,3\). It is also proved that \(R_{n,1}(f)\geq R_{n+1,1}(f)\), \(n=0,1,2,\dots\) for \(w(t)=1/\sqrt{1-t^2}\). The same conclusion is announced for \(w(t)=(1-t^2)^{1/2+s}\). Some \(\ell^2\)-error bounds are also presented.