Abstract

Let be a domain bounded by a piecewise analytic Jordan's curve L, and let denote the p-Faber polynomials associated with G. We derive estimates of the form , for , where depends on geometric properties of L and the parameter p. Also, we show that O cannot be replaced by o in the relation given above.

1. Introduction and Main Results

Let be a domain bounded by a Jordan curve , and let be the conformal mapping of the domain , where , onto with the usual normalization at infinity: Let denote the inverse conformal map. Then, where gives the capacity of .

Let . The -Faber polynomials associated the domain are defined as the polynomial part of the expansion of in a neighborhood of the infinity. Therefore, from (1.1) we have where is the -Faber polynomial of degree and is the singular part of . Indeed, the number vanishes, since tends to zero when tends to infinity. Also, the -Faber polynomials associated with can be defined by the generating function see [1, 2].

If tends to infinity, the -Faber polynomials coincide with the usual Faber polynomials . If , then it follows immediately from (1.4) that where is the Faber polynomials of of degree .

The aim of this paper is to find asymptotic behaviors of -Faber polynomials in and in .

If is rectifiable, then belongs to the Smirnov class . In addition, both and have nontangential limits almost everywhere on boundary of the unit disc and ; respectively, and they are integrable with respect to the arc length measure; see [3] or [4].

Assume now that the boundary is rectifiable. For such a boundary, Cauchy’s integral formula yields the following representations for the -Faber polynomials and their associated singular parts: Our assumption for this work is that is a piecewise analytic Jordan’s curve. That is, , , where are a finite number of analytic Jordan’s arcs meeting at corners where has an exterior angle with . Cusps are, therefore, excluded, but may occur—a “smooth corner”.

Let . Gaier [5] proved that for , uniformly on compact subsets of . Stylianopoulos [6] indicated that one can obtain by using (1.8) and (1.13), where is an arbitrary compact subset of , and is a constant depending on and . Estimates of the form (1.13) have been given by Suetin [2, 7], for boundary has high degree of smoothness.

For simplicity of notation, we write It is clear that, if and , then .

The main results of the paper are the following.

Theorem 1.1. Let , and let be bounded by a piecewise analytic Jordan curve . If has exterior angles with and if , then for , uniformly on compact subsets of .

Remark 1.2. From the proof of Theorem 1.1, it follows that an estimate of the form (1.13) holds also all derivatives of the -Faber polynomials.

Remark 1.3. If has no smooth corners; that is, all , cannot be replaced by in (1.13).

Remark 1.4. If tends to infinity, we get the result (1.13) of Gaier.
Similar arguments in the proof of Theorem 1.1 provide the following theorem.

Theorem 1.5. Under the assumptions of Theorem 1.1, we have, for any, where depends on only.

Remark 1.6. The equalities (1.13) and (1.14) are also valid for under condition .

Remark 1.7. If , we get the result of Stylianopoulos [6]: We will prove Theorems 1.1, 1.5 and Remark 1.3 in section 3.

2. Auxiliary Results

We are going to follow the analog used by Gaier in [5]. Since the components of are assumed to be analytic Jordan’s arcs, the mapping from to can be continued analytically beyond each point of that is not corner; this follows from Schwarz's reflection principle. Since has no cusps, we can refine this statement. For each , there are (short) line segments and in the interior of meeting at and , respectively, at a (small) positive angle and a rectifiable arc lying entirely in the interior of and connecting the other two endpoints of and such that has analytic continuation beyond into the strip-like domain bounded by , and (see [5] or [6] for figure). Therefore, the arc is in while and are arcs in except for their endpoints and . More can be said: since reflection preserves angles and since the exterior angle at of opening is mapped onto a angle of opening at , we see that lies in a Stolz angle in with corner at : and similarly Furthermore, we have the following inequalities for on : The similar inequalities of (2.3)–(2.6) can be written by replacing , , and with , , and , respectively. The inequalities (2.3) and (2.4) emerge from Lehman's asymptotic expansions of conformal mappings near an analytic corner [8]. The inequality (2.5) follows from (2.3), because reflection preserves angles. Finally, (2.6) is a simple fact of conformal mapping geometry.

The last auxiliary facts are on the behavior of near obtained by using Lehman's asymptotic expansions of conformal mappings near an analytic corner [8]. Assume that has a corner at with exterior angle , and that . Let further be fixed. Similar arguments in [5] apply to the one can obtain that if , then If , then

3. Proofs of Theorems

From now on, denote positive constants, not necessarily the same at the different places and not depending on degree of the -Faber polynomial and the parameter . We recite that

Proof of Theorem 1.1. Here we use the integral representation (1.9) of the - Faber polynomials in . Given a compact subset of , we deform the path of integration by replacing each component of by some (), as mentioned before, with proper orientation and such that . This gives The contributions of the arcs to these integrals are obviously for some , so that typical integrals extended over the line segments (and ) remain to be estimated. If is arc length of measured from , by using (2.4) and (2.5), and, therefore, If , then the integral in the last line is convergent; that is, From this we obtain the Theorem 1.1.
In the case , similar arguments show that, under the condition , the inequality (3.6) is valid. This proves Remark 1.4.

Proof of Remark 1.3. We now show that in Theorem 1.1 cannot be replaced by . Let be any piecewise analytic Jordan’s curve that has no smooth corners; that is, all . We assume that one of them is at with exterior angle . Assume that    for some fixed .
(a) If and , we consider for Since for , where is Gamma Function (see [9, page 225]), then This contradicts (2.7).
(b) If , then, by using (3.8) and we get which is contradiction to (2.8).

Proof of Theorem 1.5. The proof goes along similar lines as those taken in the proof of Theorem 1.1 with one significant difference: here, lies in , instead of , and, thus, can tend to without need altering the curve . As a consequence, the set defined above does not depend on , and; thus, .