Maximum of the modulus of kernels in Gauss-Turán quadratures

We study the kernels $K_{n,s}(z)$ in the remainder terms $R_{n,s}(f)$ of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at $\pm 1$, when the weight $\omega$ is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel $\vert K_{n,s}(z)\vert$ attains its maximum on the real axis (positive real semi-axis) for each $n\geq n_0, n_0=n_0(\rho,s)$. It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes $n$ in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each $n\geq n_0, n_0=n_0(\rho,s)$. Numerical examples are included.