On Sharp Hölder Estimates of the Cauchy-Riemann Equation on Pseudoconvex Domains in Cn with One Degenerate Eigenvalue

Abstract

Let Ω be a smoothly bounded pseudoconvex domain in [superscript]Cn[/superscript] with one degenerate eigenvalue and assume that there is a smooth holomorphic curve V whose order of contact with bΩ at [subscript]z0[/subscript] ∈bΩ is larger than or equal to η. We show that the maximal gain in Hölder regularity for solutions of the ∂¯-equation is at most 1/η.

Sanghyun Cho 1 and Young Hwan You 2

Academic Editor:Chun-Gang Zhu

1, Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2, Department of Mathematics, Indiana University East, Richmond, IN 47374, USA

Received 8 October 2015; Accepted 5 November 2015

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.

1. Introduction

For any open set [figure omitted; refer to PDF] , we let [figure omitted; refer to PDF] denote the space of functions in Hölder class [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a smoothly bounded pseudoconvex domain in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Suppose that there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that, for all [figure omitted; refer to PDF] -closed forms [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] , we can solve [figure omitted; refer to PDF] in [figure omitted; refer to PDF] with a gain of regularity of the solution [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . In this event, we want to find a necessary condition and determine how large [figure omitted; refer to PDF] can be. When [figure omitted; refer to PDF] , it is well known that [figure omitted; refer to PDF] . However, when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] depends on the boundary geometry of [figure omitted; refer to PDF] near [figure omitted; refer to PDF] .

Note that the Hölder estimates of [figure omitted; refer to PDF] -equation are well known when [figure omitted; refer to PDF] is bounded strongly pseudoconvex domain in [figure omitted; refer to PDF] . However, for weakly pseudoconvex domains in [figure omitted; refer to PDF] , Hölder estimates are known only for special pseudoconvex domains, that is, pseudoconvex domains of finite type in [figure omitted; refer to PDF] , convex finite type domains in [figure omitted; refer to PDF] , and pseudoconvex domains of finite type with diagonal Levi-form in [figure omitted; refer to PDF] , and so forth. Proving Hölder estimates for general pseudoconvex domains in [figure omitted; refer to PDF] is one of big questions in several complex variables. Meanwhile, it is of great interest to find a necessary condition or optimal possible gain of the Hölder estimates for [figure omitted; refer to PDF] .

Several authors have obtained necessary conditions for Hölder regularity of [figure omitted; refer to PDF] on restricted classes of domains [1-4]. Let [figure omitted; refer to PDF] , the "Bloom-Graham" type, be the maximum order of contact of [figure omitted; refer to PDF] with any [figure omitted; refer to PDF] -dimensional complex analytic manifold at [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then Krantz [2] showed that [figure omitted; refer to PDF] . Krantz's result is sharp for [figure omitted; refer to PDF] and when [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -form. Also McNeal [3] proved sharp Hölder estimates for [figure omitted; refer to PDF] -form [figure omitted; refer to PDF] under the condition that [figure omitted; refer to PDF] has a holomorphic support function at [figure omitted; refer to PDF] . Note that the existence of holomorphic support function is satisfied for restricted domains and it is often the first step to prove the Hölder estimates for [figure omitted; refer to PDF] -equation [4].

Straube [5] proved necessary condition for Hölder regularity gain of Neumann operator [figure omitted; refer to PDF] . More specifically, if Neumann operator [figure omitted; refer to PDF] has Hölder regularity gain of [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is larger than or equal to order of contact of an analytic variety (possibly singular) [figure omitted; refer to PDF] at [figure omitted; refer to PDF] . However, it should be emphasized that there is no natural machinery to pass between necessary conditions for Hölder regularity of [figure omitted; refer to PDF] -Neumann operator and that of [figure omitted; refer to PDF] , in contrast to the case of [figure omitted; refer to PDF] -Sobolev topology.

Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a smooth defining function of [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] be a smooth 1-dimensional analytic variety passing through [figure omitted; refer to PDF] . We say [figure omitted; refer to PDF] has order of contact larger than or equal to [figure omitted; refer to PDF] with [figure omitted; refer to PDF] at [figure omitted; refer to PDF] if there is a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] sufficiently close to [figure omitted; refer to PDF] . Here smooth means that [figure omitted; refer to PDF] if [figure omitted; refer to PDF] represents a parametrization of [figure omitted; refer to PDF] . Recently, the second author, You [6], proved a necessary condition for Hölder estimates for bounded pseudoconvex domains of finite type in [figure omitted; refer to PDF] . That is, if there is a 1-dimensional smooth analytic variety [figure omitted; refer to PDF] passing through [figure omitted; refer to PDF] and the order of contact of [figure omitted; refer to PDF] with [figure omitted; refer to PDF] is larger than or equal to [figure omitted; refer to PDF] , then the gain of the regularity in Hölder norm should be less than or equal to [figure omitted; refer to PDF] . To get a necessary condition for Hölder estimates, we first need a complete analysis of boundary geometry near [figure omitted; refer to PDF] of finite type.

In this paper we prove a necessary condition for the sharp Hölder estimates of [figure omitted; refer to PDF] -equation near [figure omitted; refer to PDF] when [figure omitted; refer to PDF] is a smoothly bounded pseudoconvex domain in [figure omitted; refer to PDF] and the Levi-form of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] has [figure omitted; refer to PDF] -positive eigenvalues. Our method used to prove the following main theorem will be useful for a study of necessary conditions of Hölder estimates of [figure omitted; refer to PDF] -equation for other kinds of finite type domains.

Theorem 1.

Let [figure omitted; refer to PDF] be a smoothly bounded pseudoconvex domain in [figure omitted; refer to PDF] and assume that the Levi-form of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] has [figure omitted; refer to PDF] -positive eigenvalues. Assume that there is a smooth holomorphic curve [figure omitted; refer to PDF] whose order of contact with [figure omitted; refer to PDF] at [figure omitted; refer to PDF] is larger than or equal to [figure omitted; refer to PDF] . If there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and a constant [figure omitted; refer to PDF] so that, for each [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , there is a [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] then [figure omitted; refer to PDF] .

To prove Theorem 1 we use the analysis of the local geometry near [figure omitted; refer to PDF] in [7] and use the method developed in [6]. In particular Proposition 4 is a key coordinate change which shows that [figure omitted; refer to PDF] which represents the smooth variety [figure omitted; refer to PDF] and the terms mixed with [figure omitted; refer to PDF] and strongly pseudoconvex directions vanishes up to order [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the largest integer less than or equal to [figure omitted; refer to PDF] .

Remark 2.

In general, we note that [figure omitted; refer to PDF] . Thus we have [figure omitted; refer to PDF] in (3). We also note that [figure omitted; refer to PDF] is a positive integer.

2. Special Coordinates

Let [figure omitted; refer to PDF] be as in the statement of Theorem 1 and let [figure omitted; refer to PDF] be a smooth defining function of [figure omitted; refer to PDF] near [figure omitted; refer to PDF] . We may assume that there is a coordinate system [figure omitted; refer to PDF] about [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for some constant [figure omitted; refer to PDF] , in a small neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] . In this section, we construct special coordinates [figure omitted; refer to PDF] near [figure omitted; refer to PDF] which change the given smooth holomorphic curve [figure omitted; refer to PDF] into the [figure omitted; refer to PDF] -axis. We will exclude the trivial case, [figure omitted; refer to PDF] , and hence we assume that [figure omitted; refer to PDF] is a positive integer. Set [figure omitted; refer to PDF] .

As in the proof of Proposition 2.2 in [7], after a linear change of coordinates followed by standard holomorphic changes of coordinates, we can remove inductively the pure terms such as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] terms as well as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] terms, [figure omitted; refer to PDF] , in the Taylor series expansion of [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the smooth 1-dimensional variety satisfying (2). Without loss of generality, we may assume that (2) is satisfied in [figure omitted; refer to PDF] -coordinates defined in (4). Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , be a local parametrization of [figure omitted; refer to PDF] . We may assume that [figure omitted; refer to PDF] , and, hence, after reparametrization, we can write [figure omitted; refer to PDF] and it satisfies [figure omitted; refer to PDF]

Lemma 3.

[figure omitted; refer to PDF] vanishes to order at least [figure omitted; refer to PDF] .

Proof.

The proof is similar to the proof of Lemma 2.3 in [6]. Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] vanishes to order [figure omitted; refer to PDF] . Suppose that [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . In terms of [figure omitted; refer to PDF] coordinates in (4), we can write [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] vanishes to order at least [figure omitted; refer to PDF] , there must be some cancelation between the parenthesis part and summation part. However, this is impossible because parenthesis part consists only of pure terms while summation part consists of mixed power terms.

Proposition 4.

There is a holomorphic coordinate system [figure omitted; refer to PDF] with [figure omitted; refer to PDF] such that, in terms of [figure omitted; refer to PDF] coordinates, [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] and it satisfies [figure omitted; refer to PDF]

Proof.

With [figure omitted; refer to PDF] -coordinates defined in (4), define [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , by [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] . In terms of [figure omitted; refer to PDF] coordinates, [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] vanishes to order [figure omitted; refer to PDF] , it follows from (5), (9), and (10) that [figure omitted; refer to PDF] and hence (8) is proved. Also we note that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , because of (8). This fact together with (10) proves that the first summation part in (7) is homogeneous polynomial of order [figure omitted; refer to PDF] .

Now we want to show that [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , in the third summation part in (7). On the contrary, let [figure omitted; refer to PDF] be the least integer such that [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In order to show that this is a contradiction, we use variants of the methods in Lemma 4.1 and Proposition 4.4 in [8]. For [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , define a scaling map [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] and then set [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] , and hence the first summation part in (7) will be disappeared in this limiting process. Also note that [figure omitted; refer to PDF] is the limit in the [figure omitted; refer to PDF] -topology of [figure omitted; refer to PDF] which, for each [figure omitted; refer to PDF] , is a defining function of a pseudoconvex domain [figure omitted; refer to PDF] , and hence [figure omitted; refer to PDF] is a defining function of a pseudoconvex domain [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a plurisubharmonic, nonholomorphic, polynomial of order [figure omitted; refer to PDF] provided it is nontrivial. Therefore the Hessian matrix [figure omitted; refer to PDF] is semidefinite Hermitian matrix and hence [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] Assume [figure omitted; refer to PDF] is nontrivial for some [figure omitted; refer to PDF] ; say, [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , take an appropriate argument of [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] . By (15), it follows that [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , and hence [figure omitted; refer to PDF] is holomorphic function of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . This is a contradiction proving our proposition.

3. A Construction of Special Functions

Let us take the coordinates [figure omitted; refer to PDF] defined in Proposition 4 near [figure omitted; refer to PDF] . In this section, we construct a family of uniformly bounded holomorphic functions [figure omitted; refer to PDF] with large derivatives in [figure omitted; refer to PDF] -direction along some curve [figure omitted; refer to PDF] defined in (39).

In the sequel, we set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . We will consider slices of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] -direction. From (7), [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and where [figure omitted; refer to PDF] 's are fixed constants in (7). Note that [figure omitted; refer to PDF] . Define [figure omitted; refer to PDF] and write [figure omitted; refer to PDF] for a convenience. Then [figure omitted; refer to PDF] term is absorbed in the expression of (16).

Let [figure omitted; refer to PDF] be the projection onto [figure omitted; refer to PDF] along [figure omitted; refer to PDF] -direction. Set [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] . Define a biholomorphism [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , by [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] , and, in terms of [figure omitted; refer to PDF] coordinates, [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF]

Set [figure omitted; refer to PDF] , the [figure omitted; refer to PDF] slice of [figure omitted; refer to PDF] , and set [figure omitted; refer to PDF] . Also set [figure omitted; refer to PDF] , and set [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is pseudoconvex domain in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is uniformly strongly pseudoconvex, independent of [figure omitted; refer to PDF] , provided [figure omitted; refer to PDF] is sufficiently small. In the same manner as in Proposition 4.1 in [9] or Proposition 2.5 in [10] (our case is much simpler because [figure omitted; refer to PDF] is uniformly strongly pseudoconvex independent of [figure omitted; refer to PDF] ), we can push out [figure omitted; refer to PDF] near [figure omitted; refer to PDF] uniformly independent of [figure omitted; refer to PDF] : For each small [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] and for each small [figure omitted; refer to PDF] we set [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is chosen so that [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is the maximally pushed out domain of [figure omitted; refer to PDF] near [figure omitted; refer to PDF] reflecting strong pseudoconvexity.

To connect the pushed out part [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we use a bumping family [figure omitted; refer to PDF] with front [figure omitted; refer to PDF] as in Theorem 2.3 in [11] or Theorem 2.6 in [10] (again the construction of a bumping family is much simpler because [figure omitted; refer to PDF] is uniformly strongly pseudoconvex). Set [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] becomes a pseudoconvex domain in [figure omitted; refer to PDF] which is pushed out near the origin provided [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are sufficiently small. In the sequel, we fix these [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and we note that these choices of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are independent of [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] .

According to Section 3 of [10], or by a method similar to dimension two case of [9], there exists [figure omitted; refer to PDF] holomorphic function [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] independent of [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is taken so that [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] .

Recall that the domains [figure omitted; refer to PDF] or [figure omitted; refer to PDF] are the domains in [figure omitted; refer to PDF] obtained by fixing [figure omitted; refer to PDF] . Define a biholomorphism [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and set [figure omitted; refer to PDF] . For a small constant [figure omitted; refer to PDF] to be determined, set [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . In terms of [figure omitted; refer to PDF] coordinates, for each [figure omitted; refer to PDF] , and for each [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] which is obtained by moving [figure omitted; refer to PDF] along [figure omitted; refer to PDF] direction, and set [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are small neighborhoods of [figure omitted; refer to PDF] including [figure omitted; refer to PDF] direction.

Lemma 5.

For sufficiently small [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] , or, equivalently, [figure omitted; refer to PDF]

Proof.

Assume [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , it follows from (7) and (24) that [figure omitted; refer to PDF] because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Combining (29) and (30), we obtain (28) provided [figure omitted; refer to PDF] is sufficiently small.

For each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] is holomorphic on [figure omitted; refer to PDF] . We will show that [figure omitted; refer to PDF] is bounded uniformly on [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] to be determined. For each [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , and define a nonisotropic polydisc [figure omitted; refer to PDF] by [figure omitted; refer to PDF] In order to proceed as in Section 7 of [9], we first show the following lemma which is similar to Lemma 4.3 in [9].

Lemma 6.

There is an independent constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Proof.

Assume [figure omitted; refer to PDF] . Then we have [figure omitted; refer to PDF] If we take [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] . This shows that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . By the same argument, we have [figure omitted; refer to PDF] provided [figure omitted; refer to PDF] . Therefore, if [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , it follows from (7) that [figure omitted; refer to PDF] Combining (34) and (35), one obtains [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the gradient of [figure omitted; refer to PDF] variables.

Now we prove (32). Assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , we can write [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . Combining (34), (36), and (37), we obtain that [figure omitted; refer to PDF] provided [figure omitted; refer to PDF] . This proves (32).

Let [figure omitted; refer to PDF] be the number in (16), and define [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and where [figure omitted; refer to PDF] is defined in (24), and [figure omitted; refer to PDF] is the number in (23). Note that [figure omitted; refer to PDF] for all sufficiently small [figure omitted; refer to PDF] provided [figure omitted; refer to PDF] is sufficiently small.

Remark 7.

In the above discussion, [figure omitted; refer to PDF] is any number such that [figure omitted; refer to PDF] . Thus, in particular, we can fix [figure omitted; refer to PDF] .

Theorem 8.

[figure omitted; refer to PDF] is bounded holomorphic function in [figure omitted; refer to PDF] and, along [figure omitted; refer to PDF] , [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] independent of [figure omitted; refer to PDF] .

Proof.

By (23) and (24), we already know that there is a [figure omitted; refer to PDF] holomorphic function [figure omitted; refer to PDF] on [figure omitted; refer to PDF] satisfying estimate (40). We only need to show that [figure omitted; refer to PDF] is bounded in [figure omitted; refer to PDF] . Assume [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] by Lemma 6. Now if we use the mean value theorem on polydisc [figure omitted; refer to PDF] and the fact that [figure omitted; refer to PDF] is holomorphic we will get the boundedness of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] .

4. Proof of Theorem 1

Without loss of generality, we may assume that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and where [figure omitted; refer to PDF] is given in (24). Let [figure omitted; refer to PDF] be the bounded holomorphic function in [figure omitted; refer to PDF] defined in Theorem 8, and set [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and where [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] Now set [figure omitted; refer to PDF] where [figure omitted; refer to PDF] solves [figure omitted; refer to PDF] as in the statement of Theorem 1, and hence [figure omitted; refer to PDF] is holomorphic. Set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Let us estimate the lower and upper bounds of the integral [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] , and it follows from (3) and (43) that [figure omitted; refer to PDF]

For the lower bound estimate, we start with an estimate of the holomorphic function [figure omitted; refer to PDF] with a large nontangential derivative constructed in Theorem 8. For each sufficiently small [figure omitted; refer to PDF] , set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then Taylor's theorem of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] variable shows that [figure omitted; refer to PDF] Now we take [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] for all sufficiently small [figure omitted; refer to PDF] . Returning to the lower bound estimate of [figure omitted; refer to PDF] , the mean value property, (3), (43), and (48) give us [figure omitted; refer to PDF] because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If we combine (46) and (49), we obtain that [figure omitted; refer to PDF] If we assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , (50) will be a contradiction. Therefore, [figure omitted; refer to PDF]

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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