The order of L 1 -approximation by elements of the disc algebra

We prove that the order of L 1 -approximation by elements of the disc algebra given by Khavinson, Pérez-González and Shapiro is precise. Let (cid:2) be the unit disc, T its boundary and consider the disk algebra A of those continuous functions on (cid:2) that are holomorphic in (cid:2) . In A the norm is the supremum norm

In connection with approximation in L 1 -norm by elements of a uniform algebra D. Khavinson, F. Pérez-González and H. Shapiro proved the following theorem (see [2,Theorem 3.3]).
Theorem Let f be a continuous function on T with f ∞ = 1. Assume there exists an H 1 -function G such that Then there exists a function G * in the disk algebra A such that G * ∞ ≤ 1 and See [2] for the motivation of this result and its connection to a theorem of Hoffman and Wermer on homomorphisms of uniform algebras.
The authors of [2] also verified that in (1) the bound Cε log 1 ε cannot be replaced by Cε ( [2, Theorem 3.4]), but the problem if the order O(ε log 1 ε ) in (1) can be improved at all, i.e., if it is precise or not, remained open and stated explicitly in Remark (i) in [2]. That problem was communicated to us by D. Khavinson [1]. In this note we prove that the stated order is, indeed, precise.

Theorem 1
There is a constant c > 0 with the property that for every sufficiently small Proof It will be convenient to verify the claim with ε replaced by ε 2 . Let where u(z) and v(z) are real. Using that for z = e it we have 1 Indeed, these are easy consequences of the inequality For example, for z = e it , ε ≤ |t| ≤ π (0 < ε ≤ 1), we obtain as was claimed above.
Let f ε = f = F/|F|, for which we have for small ε where we used that u ≤ e u − 1 ≤ 2u provided 0 ≤ u ≤ 1/2 (cf. (2)). Note that F is in the disk algebra and f is a continuous function with f ∞ = 1. Now let G * ∈ A, G * ∞ ≤ 1, be any function. We are going to show that with some c > 0 independent of ε, and that will prove the theorem (with ε replaced by ε 2 and f αε resp. F αε replacing f resp. G in it, where α is a constant for which f αε − F αε 1 ≤ ε 2 ; see (4)).
For the L 1 distance of F and G * we have The real part of is clearly nonnegative. Now to the pairs g 1 (z) and g 2 (z) := F(z) − 1 with nonnegative real part in and with imaginary part = 0 at the origin we can apply the "reverse triangle inequality" proved in [2, Lemma 3.5], where C 0 is an absolute constant. This yields On the right so, in view of (6), follows. Since on the left where, for the ∼ relation we used (3)), the inequality (5) follows from (7) for all sufficiently small ε.
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. 2 Funding Open access funding provided by University of Szeged.

Conflict of interest
There is no conflict of interest.
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