Let Φ be a function analytic in the disk and continuous up to the boundary, and let its modulus of continuity satisfy the Hölder condition of order α, 0 < α < 2, at a single boundary point. Under standard assumptions on the zeros of Φ, this function must be then at least α/2-Hölder (in a certain integral sense) at the same point. There are generalizations to not necessarily power-type Hölder smoothness. §0. Introduction 0.1. Consider a function Φ analytic in the unit disk and continuous up to the boundary. What is the relationship between the smoothness of Φ and of ϕ = |Φ|? Surely, it suffices to study this question for the restrictions of Φ and ϕ to the unit circle. The answer is well known: under some natural assumptions, Φ must be at least one half as smooth as ϕ, and this is best possible.This natural assumptions should be imposed on the zeros of Φ. Consider the canonical factorization (see [8] for the details) Φ = F θB, where F is the outer function constructed by ϕ, θ is a singular inner function, and B is the Blaschke product over the zeros of Φ. We remind the reader that for the boundary values of F (also denoted by the same letter F ) we have F = ϕe iH(log ϕ) , where H is the operator of harmonic conjugation. Next, θ is generated by a certain positive singular measure on the circle, and the boundary values of θ coincide a.e. with the function e −iHμ . It is well known (see [8]) that if Φ is continuous up to the boundary, then the support of μ is included in the set {t ∈ T : ϕ(t) = 0}; moreover, the zeros of B also may accumulate only to points of this set.No lower bound for the smoothness drop is available without further assumptions about the zeros of Φ (see an explanation in [6]), and the simplest way out is to forbid them radically in the disk, i.e., to assume that B = θ = 1. In this case, in the 1950s, Carleson and Jakobs proved that if ϕ ∈ Lip α , 0 < α < 1, then Φ = F ∈ Lip α/2 (T). The proof was not published, and later the result was rediscovered by Havin and Shamoyan (see [7]), who also included the case of α = 1. The story goes that Carleson extended the result to an arbitrary positive power-like smoothness, but the proof also did not appear in print. The only available proof of the fact that F ∈ Lip α/2 for all positive α is due to Shirokov, see [11].