A note on strong asymptotics of weighted Chebyshev polynomials

Denote by Tn(.,w) the nth degree Chebyshev polynomial with respect to the positive weight w. It is shown in this note that if w is C1+ then Tn(cosϕ,w)=ℜ{e−inϕπ2(eiϕ)}+o(1) where π(z) stands for the Szegő function corresponding to w. This extends an earlier result proved by Kroó and Peherstorfer who verified this asymptotic relation for C2+α,α>0 weights.