Maximum of the modulus of kernels in Gauss-Turán quadratures

We study the kernels Kn,s(z) in the remainder terms Rn,s(f) of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at ±1, when the weight ω 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 |Kn,s(z)| attains its maximum on the real axis (positive real semi-axis) for each n ≥ n0, n0 = n0(ρ, 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 ≥ n0, n0 = n0(ρ, s). Numerical examples are included.