Table of Contents
ISRN Applied Mathematics
VolumeΒ 2011, Article IDΒ 263657, 8 pages
http://dx.doi.org/10.5402/2011/263657
Research Article

On the Asymptotic Behavior of p-Faber Polynomials for Domains with Piecewise Analytic Boundary

Department of Mathematics, Mersin University, 3343 Mersin, Turkey

Received 3 July 2011; Accepted 8 August 2011

Academic Editor: X.-L.Β Gao

Copyright Β© 2011 T. Tunc and M. Kucukaslan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let πΊβŠ‚β„‚ be a domain bounded by a piecewise analytic Jordan's curve L, and let 𝐹𝑛,𝑝 denote the p-Faber polynomials associated with G. We derive estimates of the form 𝐹𝑛,𝑝(𝑧)=𝑂(1/π‘›πœ‚), (π‘›β†’βˆž) for π‘§βˆˆπΊ, where πœ‚ 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 πΊβŠ‚β„‚ be a domain bounded by a Jordan curve 𝐿, and let Ξ¦ be the conformal mapping of the domain Ω∢=β„‚βˆžβ§΅πΊ, where β„‚βˆžβˆΆ=β„‚βˆͺ{∞}, onto Ξ”βˆΆ={π‘€βˆΆ|𝑀|>1} with the usual normalization at infinity: 𝑀=Ξ¦(𝑧)=𝛼𝑧+π‘Ž0+π‘Ž1𝑧+β‹―,𝛼>0,π‘§βˆˆΞ©.(1.1) Let Ψ∢=Ξ¦βˆ’1βˆΆΞ”β†’Ξ© denote the inverse conformal map. Then, 𝑧=Ξ¨(𝑀)=𝛽𝑀+𝑏0+𝑏1𝑀+β‹―,|𝑀|>1,(1.2) where 𝛽=1/𝛼 gives the capacity π‘π‘Žπ‘(𝐿) of 𝐿.

Let 0<𝑝<∞. The 𝑝-Faber polynomials 𝐹𝑛,𝑝 associated the domain 𝐺 are defined as the polynomial part of the expansion of Φ𝑛Φ(𝑧)ξ…žξ€Έ(𝑧)1/𝑝,𝑛=0,1,2,…,(1.3) in a neighborhood of the infinity. Therefore, from (1.1) we have Φ𝑛Φ(𝑧)ξ…žξ€Έ(𝑧)1/π‘βˆΆ=𝐹𝑛,𝑝(𝑧)+𝐸𝑛,𝑝(𝑧),π‘§βˆˆΞ©,(1.4) where 𝐹𝑛,𝑝(𝑧)∢=𝛼𝑛+1/𝑝𝑧𝑛+β‹―(1.5) is the 𝑝-Faber polynomial of degree 𝑛 and 𝐸𝑛,𝑝𝑐(𝑧)=0𝑧+𝑐1𝑧2+𝑐2𝑧3+β‹―(1.6) is the singular part of Φ𝑛(𝑧)(Ξ¦ξ…ž(𝑧))1/𝑝. Indeed, the number 𝑐0 vanishes, since 𝑧((Ξ¦ξ…ž(𝑧))1/π‘βˆ’π›Ό1/𝑝) tends to zero when 𝑧 tends to infinity. Also, the 𝑝-Faber polynomials associated with 𝐺 can be defined by the generating function Ψ𝑔(𝑀)∢=ξ…žξ€Έ(𝑀)1βˆ’1/𝑝=Ξ¨(𝑀)βˆ’π‘§βˆžξ“π‘›=0𝐹𝑛,𝑝(𝑧)𝑀𝑛+1,π‘§βˆˆπΊ,|𝑀|>1,(1.7) see [1, 2].

If 𝑝 tends to infinity, the 𝑝-Faber polynomials coincide with the usual Faber polynomials 𝐹𝑛. If 𝑝=1, then it follows immediately from (1.4) that 𝐹𝑛,1𝐹(𝑧)=ξ…žπ‘›+1(𝑧)𝑛+1,(1.8) where 𝐹𝑛 is the Faber polynomials of G of degree 𝑛.

The aim of this paper is to find asymptotic behaviors of 𝑝-Faber polynomials 𝐹𝑛,𝑝 in 𝐺 and 𝐸𝑛,𝑝 in Ξ©.

If 𝐿 is rectifiable, then Ξ¦ξ…ž belongs to the Smirnov class 𝐸1(Ξ©{∞}). In addition, both Ξ¨ξ…ž and Ξ¦ξ…ž have nontangential limits almost everywhere on 𝕋 boundary of the unit disc π”»βˆΆ={π‘€βˆˆβ„‚βˆΆ|𝑀|<1} and 𝐿; respectively, and they are integrable with respect to the arc length measure; see [3] or [4].

Assume now that the boundary 𝐿 is rectifiable. For such a boundary, Cauchy’s integral formula yields the following representations for the 𝑝-Faber polynomials and their associated singular parts: 𝐹𝑛,𝑝1(𝑧)=ξ€œ2πœ‹π‘–πΏΞ¦π‘›ξ€·Ξ¦(𝜁)ξ…žξ€Έ(𝜁)1/π‘πΈπœβˆ’π‘§π‘‘πœ,π‘§βˆˆπΊ,𝑛,𝑝1(𝑧)=ξ€œ2πœ‹π‘–πΏΞ¦π‘›ξ€·Ξ¦(𝜁)ξ…žξ€Έ(𝜁)1/π‘πœβˆ’π‘§π‘‘πœ,π‘§βˆˆπΊ.(1.9) Our assumption for this work is that 𝐿 is a piecewise analytic Jordan’s curve. That is, ⋃𝐿=π‘šπ‘—=1𝐿𝑗, π‘šβˆˆβ„•, where 𝐿𝑗 are a finite number of analytic Jordan’s arcs meeting at corners 𝑧𝑗 where 𝐿 has an exterior angle πœ†π‘—πœ‹ with 0<πœ†π‘—<2. Cusps are, therefore, excluded, but πœ†π‘—=1 may occurβ€”a β€œsmooth corner”.

Let πœ†βˆΆ=min𝑗=1,…,π‘š{πœ†π‘—}. Gaier [5] proved that 𝐹𝑛1(𝑧)=π‘‚π‘›πœ†ξ‚(π‘›β†’βˆž),(1.10) for π‘§βˆˆπΊ, uniformly on compact subsets of 𝐺. Stylianopoulos [6] indicated that one can obtain 𝐹𝑛,1(𝑧)≀𝑐(𝐿,𝐹)π‘›πœ†+1,π‘§βˆˆπΉ(1.11) by using (1.8) and (1.13), where 𝐹 is an arbitrary compact subset of 𝐺, and 𝑐(𝐿,𝐹) is a constant depending on 𝐿 and 𝐹. Estimates of the form (1.13) have been given by Suetin [2, 7], for boundary 𝐿 has high degree of smoothness.

For simplicity of notation, we write ξ‚΅1πœ‚βˆΆ=πœ‚(πœ†,𝑝)∢=(1βˆ’πœ†)1βˆ’π‘ξ‚Ά.(1.12) It is clear that, if 0<πœ†<2 and 1/2β‰€π‘β‰€βˆž, then |πœ‚|<1.

The main results of the paper are the following.

Theorem 1.1. Let 1/2β‰€π‘β‰€βˆž, and let πΊβŠ‚β„‚ be bounded by a piecewise analytic Jordan curve 𝐿. If 𝐿 has exterior angles πœ†π‘—πœ‹ with 0<πœ†π‘—<2 and if πœ†βˆΆ=minπ‘—πœ†π‘—, then 𝐹𝑛,𝑝1(𝑧)=𝑂𝑛1βˆ’πœ‚ξ‚(π‘›β†’βˆž),(1.13) for π‘§βˆˆπΊ, uniformly on compact subsets of 𝐺.

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 𝑝-Faber polynomials.

Remark 1.3. If 𝐿 has no smooth corners; that is, all πœ†π‘—β‰ 1, 𝑂 cannot be replaced by π‘œ in (1.13).

Remark 1.4. If 𝑝 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 anyπ‘›βˆˆβ„•, ||𝐸𝑛,𝑝||≀(𝑧)𝑐(𝐿)1dist(𝑧,𝐿)𝑛1βˆ’πœ‚,π‘§βˆˆΞ©,(1.14) where 𝑐(𝐿) depends on 𝐿 only.

Remark 1.6. The equalities (1.13) and (1.14) are also valid for 0<𝑝<1/2 under condition max𝑗=1,…,π‘šπœ†π‘—<1/(1βˆ’π‘).

Remark 1.7. If 𝑝=1, we get the result of Stylianopoulos [6]: ||𝐸𝑛,1||≀(𝑧)𝑐(𝐿)1dist(𝑧,𝐿)𝑛,π‘§βˆˆΞ©.(1.15) 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 𝐿𝑗 of 𝐿 are assumed to be analytic Jordan’s arcs, the mapping Ξ¦ from Ξ© to Ξ” can be continued analytically beyond each point of 𝐿 that is not corner; this follows from Schwarz's reflection principle. Since 𝐿 has no cusps, we can refine this statement. For each 𝑗=1,2,…,π‘š, there are (short) line segments 𝑠𝑗 and π‘ ξ…žπ‘— in the interior of 𝐿 meeting 𝐿𝑗 at 𝑧𝑗 and 𝑧𝑗+1, respectively, at a (small) positive angle and a rectifiable arc 𝛾𝑗 lying entirely in the interior of 𝐿 and connecting the other two endpoints of 𝑠𝑗 and π‘ ξ…žπ‘— such that Ξ¦ has analytic continuation beyond 𝐿𝑗 into the strip-like domain 𝑔𝑗 bounded by 𝐿𝑗,𝑠𝑗,π‘ ξ…žπ‘—, and 𝛾𝑗 (see [5] or [6] for figure). Therefore, the arc Ξ¦(𝛾𝑗) is in 𝔻 while Ξ¦(𝑠𝑗) and Ξ¦(π‘ ξ…žπ‘—) are arcs in 𝔻 except for their endpoints Ξ¦(𝑧𝑗) and Ξ¦(𝑧𝑗+1). More can be said: since reflection preserves angles and since the exterior angle at 𝑧𝑗 of opening πœ†π‘—πœ‹ is mapped onto a angle of opening πœ‹ at Ξ¦(𝑧𝑗), we see that Ξ¦(𝑠𝑗) lies in a Stolz angle in 𝔻 with corner at Ξ¦(𝑧𝑗): ||Φ𝑧(𝑧)βˆ’Ξ¦π‘—ξ€Έ||||||1βˆ’Ξ¦(𝑧)≀𝑐1forπ‘§βˆˆπ‘ π‘—,𝑗=1,2,…,π‘š,(2.1) and similarly ||Φ𝑧(𝑧)βˆ’Ξ¦π‘—+1ξ€Έ||||||1βˆ’Ξ¦(𝑧)β‰€π‘ξ…ž1forπ‘§βˆˆπ‘ ξ…žπ‘—,𝑗=1,2,…,π‘š.(2.2) Furthermore, we have the following inequalities for 𝑧 on 𝑠𝑗(𝑗=1,2,…,π‘š): ||𝑧Φ(𝑧)βˆ’Ξ¦π‘—ξ€Έ||||β‰₯constβ‹…π‘§βˆ’π‘§π‘—||1/πœ†π‘—,||Ξ¦(2.3)ξ…ž||||(𝑧)≀constβ‹…π‘§βˆ’π‘§π‘—||1/πœ†π‘—βˆ’1||||||,(2.4)Ξ¦(𝑧)≀1βˆ’constβ‹…π‘§βˆ’π‘§π‘—||1/πœ†π‘—||,(2.5)dist(𝑧,𝐿)β‰₯constβ‹…π‘§βˆ’π‘§π‘—||.(2.6) The similar inequalities of (2.3)–(2.6) can be written by replacing 𝑠𝑗, 𝑧𝑗, and πœ†π‘— with π‘ ξ…žπ‘—, 𝑧𝑗+1, and πœ†π‘—+1, 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 𝑔 near 𝕋 obtained by using Lehman's asymptotic expansions of conformal mappings near an analytic corner [8]. Assume that 𝐿 has a corner at 𝜁 with exterior angle πœ†πœ‹, 0<πœ†<2 and that 𝜁=Ξ¨(1). Let further π‘§βˆˆπΊ be fixed. Similar arguments in [5] apply to the 𝑔(𝑀)=(Ξ¨β€²(𝑀))1βˆ’1/𝑝/(Ξ¨(𝑀)βˆ’π‘§) one can obtain that if 0<πœ†<1, then 𝐢𝑔(𝑀)β‰…(π‘€βˆ’1)πœ‚,as𝑀→1+,with𝐢≠0.(2.7) If 1<πœ†<2, then π‘”ξ…žπΆ(𝑀)β‰…(π‘€βˆ’1)πœ‚+1,as𝑀→1+,with𝐢≠0.(2.8)

3. Proofs of Theorems

From now on, 𝑐 denote positive constants, not necessarily the same at the different places and not depending on 𝑛 degree of the 𝑝-Faber polynomial and the parameter 𝑝. We recite that ξ‚΅1πœ‚βˆΆ=(1βˆ’πœ†)1βˆ’π‘ξ‚Ά.(3.1)

Proof of Theorem 1.1. Here we use the integral representation (1.9) of the 𝑝- Faber polynomials 𝐹𝑛,𝑝 in 𝐺. Given a compact subset 𝐹 of 𝐺, we deform the path of integration by replacing each component 𝐿𝑗 of 𝐿 by some πΏξ…žπ‘—βˆΆ=𝑠𝑗βˆͺ𝐿𝑗βˆͺ𝑠′𝑗 (𝑗=1,2,…,π‘š), as mentioned before, with proper orientation and such that β‹ƒπΉβŠ‚πΏβ€²βˆΆ=π‘šπ‘—=1πΏξ…žπ‘—. This gives ||𝐹𝑛,𝑝||≀1(𝑧)2πœ‹π‘šξ“π‘—=1ξ€œπΏβ€²π‘—||||Ξ¦(𝜁)𝑛||||Ξ¦β€²(𝜁)1/𝑝||||||||πœβˆ’π‘§π‘‘πœβ‰€πΆ(𝐿,𝐹)π‘šξ“π‘—=1ξ€œπΏβ€²π‘—||||Ξ¦(𝜁)𝑛||||Ξ¦β€²(𝜁)1/𝑝||||π‘‘πœ=𝐢(𝐿,𝐹)π‘šξ“π‘—=1ξƒ©ξ€œπ‘ π‘—+ξ€œπ‘ β€²π‘—+ξ€œπΏπ‘—ξƒͺ||||Ξ¦(𝜁)𝑛||||Ξ¦β€²(𝜁)1/𝑝||||.π‘‘πœ(3.2) The contributions of the arcs 𝐿𝑗 to these integrals are obviously O(πœŒπ‘›) for some 𝜌<1, so that typical integrals 𝐼𝑗=ξ€œπ‘ π‘—||||Ξ¦(𝜁)𝑛||Ξ¦ξ…ž(||𝜁)1/𝑝||||π‘‘πœ(3.3) extended over the line segments 𝑠𝑗 (and π‘ ξ…žπ‘—) remain to be estimated. If 𝑠 is arc length of 𝑠𝑗 measured from 𝑧𝑗, by using (2.4) and (2.5), ||||Ξ¦(𝜁)≀1βˆ’π‘π‘ 1/πœ†π‘—ξ€·<expβˆ’π‘π‘ 1/πœ†π‘—ξ€Έ,||||Ξ¦β€²(𝜁)≀𝑐𝑠1/πœ†π‘—βˆ’1,(3.4) and, therefore, πΌπ‘—ξ€œ<π‘βˆž0ξ€·expβˆ’π‘π‘›π‘ 1/πœ†π‘—ξ€Έπ‘ (1/πœ†π‘—βˆ’1)/𝑝=𝑐𝑑𝑠𝑛(1βˆ’πœ†π‘—)/𝑝+πœ†π‘—ξ€œβˆž0π‘’βˆ’π‘‘π‘‘(1/π‘βˆ’1)(1βˆ’πœ†π‘—)𝑑𝑠.(3.5) If 1/2β‰€π‘β‰€βˆž, then the integral in the last line is convergent; that is, 𝐼𝑗<𝑐𝑛(1βˆ’πœ†π‘—)/𝑝+πœ†π‘—.(3.6) From this we obtain the Theorem 1.1.
In the case 0<𝑝<1/2, similar arguments show that, under the condition πœ†π‘—<1/(1βˆ’π‘), the inequality (3.6) is valid. This proves Remark 1.4.

Proof of Remark 1.3. We now show that 𝑂 in Theorem 1.1 cannot be replaced by π‘œ. Let 𝐿 be any piecewise analytic Jordan’s curve that has no smooth corners; that is, all πœ†π‘—β‰ 1. We assume that one of them is at Ξ¨(1) with exterior angle πœ†. Assume that 𝐹𝑛,𝑝(𝑧)=π‘œ(1/𝑛1βˆ’πœ‚)  (π‘›β†’βˆž) for some fixed π‘§βˆˆπΊ.
(a) If 0<πœ†<1 and 𝑝>1, we consider for|𝑀|>1Ψ𝑔(𝑀)=ξ…žξ€Έ(𝑀)1βˆ’1/𝑝=Ξ¨(𝑀)βˆ’π‘§βˆžξ“π‘›=0𝐹𝑛,𝑝(𝑧)𝑀𝑛+1.(3.7) Since βˆžξ“π‘›=1π‘₯π‘›π‘›π›ΌβˆΌΞ“(1βˆ’π›Ό)(1βˆ’π‘₯)1βˆ’π›Ό,whenπ‘₯β†’1βˆ’(3.8) for 𝛼<1, where Ξ“ is Gamma Function (see [9, page 225]), then ||||𝑔(𝑀)=π‘œβˆžξ“π‘›=11𝑛1βˆ’πœ‚π‘€π‘›ξƒͺ1=π‘œ(1)(π‘€βˆ’1)πœ‚,as𝑀→1+.(3.9) This contradicts (2.7).
(b) If 1<πœ†<2, then, by using (3.8) and π‘”ξ…ž(𝑀)=βˆ’βˆžξ“π‘›=0(𝑛+1)𝐹𝑛,𝑝(𝑧)𝑀𝑛+2,(3.10) we get ||||𝑔′(𝑀)=π‘œβˆžξ“π‘›=1𝑛𝑛1βˆ’πœ‚π‘€π‘›ξƒͺ1=π‘œ(1)(π‘€βˆ’1)πœ‚+1,as𝑀→1+(3.11) 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, 𝑧 lies in Ξ©, instead of 𝐺, and, thus, can tend to 𝐿 without need altering the curve 𝐿′. As a consequence, the set 𝐹 defined above does not depend on 𝑧, and; thus, dist(𝑧,𝐹)>dist(𝐿,𝐹)=𝑐(𝐿).

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