Decay estimates for the arithmetic means of coefficients connected with composition operators

Abstract

Let f∈ L∞(T) with ‖ f‖ ∞≤ 1. If f(0) ≠ 0 , n, k∈ Z, and bn , n - k= ∫ Ef(x) ne- 2 π i ( n - k ) xdx, E= { x∈ T: | f(x) | = 1 } , we prove that the arithmetic means 1N∑n=MM+N|bn,n-k|2 decay like {logNlog2N···logqN}-1 as N→ ∞, uniformly in k∈ Z. © 2018, Springer Nature Switzerland AG.