ON POLYHARMONIC UNIVALENT MAPPINGS

ON POLYHARMONIC UNIVALENT MAPPINGS J. CHEN, A. RASILA and X. WANG In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by HS pλ, and its subclass HS 0 pλ, where λ [0, ] is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in the 950s. We generalize the main results of Avci and Z lotkiewicz from the 990s to the classes HS pλ and HS 0 pλ, showing that the mappings in HS pλ are univalent and sense preserving. We also prove that the mappings in HS 0 pλ are starlike with respect to the origin, and characterize the extremal points of the above classes. AMS 200 Subject Classification: Primary 30C45; Secondary 30C20, 30C65. Key words: polyharmonic mapping, starlikeness, convexity, extremal point.. INTRODUCTION A complex-valued mapping F = u + iv, defined in a domain D C, is called polyharmonic or p-harmonic if F is 2p p times continuously differentiable, and it satisfies the polyharmonic equation p F = p F = 0, where := is the standard complex Laplacian operator = 4 2 z z := 2 x 2 + 2 y 2. It is well known [7, 20] that for a simply connected domain D, a mapping F is polyharmonic if and only if F has the following representation: F z = z 2k G k z, where G k are complex-valued harmonic mappings in D for k {,, p}. Furthermore, the mappings G k can be expressed as the form G k = h k + g k for k {,, p}, where all h k and g k are analytic in D [, 3]. Obviously, for p = resp. p = 2, F is a harmonic resp. biharmonic mapping. The biharmonic model arises from numerous problems in science and MATH. REPORTS 565, 4 203, 343 357

344 J. Chen, A. Rasila and X. Wang 2 engineering [6, 8, 9]. However, the investigation of biharmonic mappings in the context of the geometric function theory has started only recently [ 3, 5, 6, 8 0]. The reader is referred to [7, 20] for discussion on polyharmonic mappings, and [, 3] for the properties of harmonic mappings. In [4], Avci and Z lotkiewicz introduced the class HS of univalent harmonic mappings F with the series expansion:. F z = hz + gz = z + a n z n + b n z n such that n a n + b n b 0 b <, and the subclass HC of HS, where n 2 a n + b n b 0 b <. The corresponding subclasses of HS and HC with b = 0 are denoted by HS 0 and HC 0, respectively. These two classes constitute a harmonic counterpart of classes introduced by Goodman [5]. They are useful in studying questions of the so-called δ-neighborhoods Ruscheweyh [22], see also [20] and in constructing explicit k-quasiconformal extensions Fait et al. [4]. In this paper, we define polyharmonic analogues HS p λ and HSpλ, 0 where λ [0, ], to the above classes of mappings. Our aim is to generalize the main results of [4] to the mappings of the classes HS p λ and HSpλ. 0 This paper is organized as follows. In Section 3, we discuss the starlikeness and convexity of polyharmonic mappings in HSpλ. 0 Our main result, Theorem, is a generalization of [4], Theorem 4. In Section 4, we find the extremal points of the class HSpλ. 0 The main result of this section is Theorem 2, which is a generalization of [4], Theorem 6. Finally, we consider convolutions and existence of neighborhoods. The main results in this section are Theorems 3 and 4 which are generalizations of [4], Theorems 7 i and 8 respectively. Note that the bounds for convexity and starlikeness for harmonic mappings given in [4] have been recently improved by Kalaj et al. [7]. 2. PRELIMINARIES For r > 0, write D r = {z : z < r}, and let D := D, i.e., the unit disk. We use H p to denote the set of all polyharmonic mappings F in D with a series

3 On polyharmonic univalent mappings 345 expansion of the following form: 2. F z = z 2k h k z + g k z = z 2k a n,k z n + b n,k z n with a, = and b, <. Let Hp 0 denote the subclass of H p for b, = 0 and a,k = b,k = 0 for k {2,, p}. In [20], J. Qiao and X. Wang introduced the class HS p of polyharmonic mappings F with the form 2. satisfying the conditions 2.2 2k + n a n,k + b n,k b, 2k a,k + b,k, 0 b, + 0 b, + a,k + b,k <, and the subclass HC p of HS p, where 2.3 2k + n 2 a n,k + b n,k b, 2k a,k + b,k, a,k + b,k <. The classess of all mappings F in Hp 0 which are of the form 2., and subject to the conditions 2.2, 2.3, are denoted by HSp, 0 HCp, 0 respectively. Now, we introduce a new class of polyharmonic mappings, denoted by HS p λ, as follows: A mapping F H p with the form 2. is said to be in HS p λ if 2.4 2k +nλn+ λ an,k + b n,k 2 2k a,k + b,k, 2k a,k + b,k < 2, where λ [0, ]. We denote by HSpλ 0 the class consisting of all mappings F in Hp, 0 with the form 2., and subject to the condition 2.4. Obviously, if λ = 0 or λ =, then the class HS p λ reduces to HS p or HC p, respectively. Similarly, if p = λ = 0 or p = λ =, then HS p λ reduces to HS or HC. If F z = z 2k an,k z n + b n,k z n

346 J. Chen, A. Rasila and X. Wang 4 and Gz = z 2k An,k z n + B n,k z n, then the convolution F G of F and G is defined to be the mapping F Gz = z 2k an,k A n,k z n + b n,k B n,k z n, while the integral convolution is defined by F Gz = z 2k a n,k A n,k n z n + b n,kb n,k z n n. See [2] for similar operators defined on the class of analytic functions. Following the notation of J. Qiao and X. Wang [20], we denote the δ- neighborhood of F the set by { N δ F z = Gz : 2k + n an,k A n,k + b n,k B n,k + } 2k a,k A,k + b,k B,k + b, B, δ, where Gz = p Ruscheweyh [22]. z 2k An,k z n + B n,k z n and A, = see also 3. STARLIKENESS AND CONVEXITY We say that a univalent polyharmonic mapping F with F 0 = 0 is starlike with respect to the origin if the curve F re iθ is starlike with respect to the origin for each 0 < r <. Proposition [2]. If F is univalent, F 0 = 0 and d dθ arg F re iθ > 0 for z = re iθ 0, then F is starlike with respect to the origin. A univalent polyharmonic mapping F with F 0 = 0 and d dθ F reiθ 0 whenever 0 < r <, is said to be convex if the curve F re iθ is convex for each 0 < r <. Proposition 2 [2]. If F is univalent, F 0 = 0 and [ θ arg θ F reiθ ] > 0 for z = re iθ 0, then F is convex. Let X be a topological vector space over the field of complex numbers, and let D be a set of X. A point x D is called an extremal point of D if it

5 On polyharmonic univalent mappings 347 has no representation of the form x = ty + tz 0 < t < as a proper convex combination of two distinct points y and z in D. Now, we are ready to prove results concerning the geometric properties of mappings in HS 0 pλ. Theorem A [20], Theorems 3., 3.2 and 3.3. Suppose that F HS p. Then F is univalent and sense preserving in D. In particular, each member of HS 0 p or HC 0 p maps D onto a domain starlike w.r.t. the origin, and a convex domain, respectively. Theorem. Each mapping in HSpλ 0 maps the disk D r, where r max{ 2, λ}, onto a convex domain. Proof. Let F HS 0 pλ, and let r 0, be fixed. Then r F rz HS 0 pλ by 2.4, and we have 2k + n 2 a n,k + b n,k r 2k+n 3 2k + nλn + λ an,k + b n,k provided that 2k + n 2 r 2k+n 3 2k + nλn + λ for k {,, p}, n 2 and 0 λ, which is true if r max{ 2, λ}. Then the result follows from Theorem A. Immediately from Theorem A, we get the following. Corollary. Let F HS p λ. Then F is a univalent, sense preserving polyharmonic mapping. In particular, if F HS 0 pλ, then F maps D onto a domain starlike w.r.t. the origin. Example. Let F z = z + 0 z2 + 5 z2. Then F HS 0 2 3 is a univalent, sense preserving polyharmonic mapping. In particular, F maps D onto a domain starlike w.r.t. the origin, and it maps the disk D r, where r 2 3, onto a convex domain. See Figure. This example shows that the class HS 0 pλ of polyharmonic mappings is more general than the class HS 0 which is studied in [4] even in the case of harmonic mappings i.e. p=. Example 2. Let F 2 z = z + 0 z2 + 49 0 z2. Then F 2 HS 0 00 is a univalent, sense preserving polyharmonic mapping. In particular, F 2 maps D

348 J. Chen, A. Rasila and X. Wang 6 onto a domain starlike w.r.t. the origin, and it maps the disk D r, where r 2, onto a convex domain. See Figure. Fig.. The images of D under the mappings F z = z + 0 z2 + 5 z2 left and F 2z = z + 0 z2 + 49 0 z2 right.

7 On polyharmonic univalent mappings 349 4. EXTREMAL POINTS First, we determine the distortion bounds for mappings in HS p λ. Lemma. Suppose that F HS p λ. hold: For 0 λ 2, Then, the following statements b, z b, 2 + λ z 2 F z + b, z + b, 2 + λ z 2. Equalities are obtained by the mappings for properly chosen real µ and ν; 2 For 2 < λ, F z = z + b, e iµ z + b, 2 + λ eiν z 2, F z + b, z + b, 3 a,2 + b,2 z 2 + a,2 + b,2 z 3 2 + λ and F z b, z b, 3 a,2 + b,2 z 2 a,2 + b,2 z 3. 2 + λ Equalities are obtained by the mappings F z = z + b, e iη z + b, 3 a,2 + b,2 e iϕ z 2 + a,2 + b,2 e iψ z z 2, 2 + λ for properly chosen real η, ϕ and ψ. Proof. Let F HS p λ, where λ [0, ]. By 2., we have F z + b, z + a n,k + b n,k + a,k + b,k z 2. For 0 λ 2, we have 4. 2 + λ 2k, where k {2,, p}, and 4.2 2 + λ 2k + nλn + λ,

350 J. Chen, A. Rasila and X. Wang 8 where k {,, p} and n 2. Then 4., 4.2 and 2.4 give so a n,k + b n,k + a,k + b,k b, 2 + λ 2 + λ a n,k + b n,k 2k + nλn + λ 2k 2 + λ a,k + b,k, b, z b, 2 + λ z 2 F z + b, z + b, 2 + λ z 2. By 2., we obtain F z + b, z + + a,2 + b,2 z 3. a n,k + b n,k + a,k + b,k z 2 k=3 For 2 < λ, we have 4.3 2 + λ 2k, where k {3,, p}, and 4.4 2 + λ 2k + nλn + λ, where k {,, p}, n 2. Then 4.3, 4.4 and 2.4 imply Then a n,k + b n,k + a,k + b,k k=3 b, 2 + λ 2k + nλn + λ 2 + λ a n,k + b n,k 2k 2 + λ a,k + b,k 3 a,2 + b,2. k=3 F z b, z b, 3 a,2 + b,2 z 2 a,2 + b,2 z 3 2 + λ

9 On polyharmonic univalent mappings 35 and F z + b, z + b, 3 a,2 + b,2 z 2 + a,2 + b,2 z 3. 2 + λ The proof of this lemma is complete. Remark. Suppose that F HS p λ is of the form F z = z 2k G k z = z 2k an,k z n + b n,k z n. Then for each k {,, p}, G k z a,k + b,k z + b, 2 + λ z 2. Lemma 2. The family HS p λ is closed under convex combinations. Proof. Suppose F i HS p λ and t i [0, ] with i= t i =. Let F i z = z 2k a i n,k zn + b i n,k zn. By Lemma, there exists a constant M such that F i z M, for all i =,, p. It follows that i= t if i z is absolutely and uniformly convergent, and by Remark, the mapping i= t if i z is polyharmonic. Since i= t if i z is absolutely and uniformly convergent, we have t i F i z = t i z 2k a i n,k zn + i= = i= i= z 2k t i a i n,k zn + i= b i n,k zn t i b i n,k zn By 2.4, we get 4.5 2k + nλn + λ t i a i n,k + t i b i n,k i= i= i t i 2k + nλn + λ a n,k + bi n,k 2. i= It follows from 2k t i a i,k + t i b i,k < 2 i= and 4.5 that i= t if i HS p λ. i=.

352 J. Chen, A. Rasila and X. Wang 0 From Lemma, we see that the class HS p λ is uniformly bounded, and hence, normal. Lemma 2 implies that HSpλ 0 is also compact and convex. Then there exists a non-empty set of extremal points in HSpλ. 0 Theorem 2. The extremal points of HS 0 pλ are the mappings of the following form: where and F k z = z + z 2k a n,k z n or F k z = z + z 2k b m,k z m, a n,k = b m,k =, for n 2, k {,, p}, 2k + nλn + λ, for m 2, k {,, p}. 2k + mλm + λ Proof. Assume that F is an extremal point of HSpλ, 0 of the form 2.. Suppose that the coefficients of F satisfy the following: 2k + nλn + λ an,k + b n,k <. If all coefficients a n,k n 2 and b n,k n 2 are equal to 0, we let F z = z + 2 + λ z2 and F 2 z = z 2 + λ z2. Then F and F 2 are in HSpλ 0 and F = 2 F + F 2. This is a contradiction, showing that there is a coefficient, say a n0,k 0 or b n0,k 0, of F which is nonzero. Without loss of generality, we may further assume that a n0,k 0 0. For γ > 0 small enough, choosing x C with x = properly and replacing a n0,k 0 by a n0,k 0 γx and a n0,k 0 + γx, respectively, we obtain two mappings F 3 and F 4 such that both F 3 and F 4 are in HSpλ. 0 Obviously, F = 2 F 3+F 4. Hence, the coefficients of F must satisfy the following equality: 2k + nλn + λ an,k + b n,k =. Suppose that there exists at least two coefficients, say, a q,k and b q2,k 2 or a q,k and a q2,k 2 or b q,k and b q2,k 2, which are not equal to 0, where q, q 2 2. Without loss of generality, we assume the first case. Choosing γ > 0 small enough and x C, y C with x = y = properly, leaving all coefficients of F but a q,k and b q2,k 2 unchanged and replacing a q,k, b q2,k 2 by γx a q,k + 2k + q λq + λ and b γy q 2,k 2 2k 2 + q 2 λq 2 + λ,

On polyharmonic univalent mappings 353 or a q,k γx 2k + q λq + λ and b γy q 2,k 2 + 2k 2 + q 2 λq 2 + λ, respectively, we obtain two mappings F 5 and F 6 such that F 5 and F 6 are in HS 0 pλ. Obviously, F = 2 F 5 + F 6. This shows that any extremal point F HS 0 pλ must have the form F k z = z + z 2k a n,k z n or F k z = z + z 2k b m,k z m, where and a n,k = b m,k =, for n 2, k {,, p}, 2k + nλn + λ, for m 2, k {,, p}. 2k + mλm + λ Now, we are ready to prove that for any F HSpλ 0 with the above form must be an extremal point of HSpλ. 0 It suffices to prove the case of F k, since the proof for the case of Fk is similar. Suppose there exist two mappings F 7 and F 8 HSpλ 0 such that F k = tf 7 + tf 8 0 < t <. For q = 7, 8, let F q z = z 2k a q n,k zn + b q Then 4.6 ta 7 n,k + ta8 n,k = a n,k = n,k zn. 2k + nλn + λ. Since all coefficients of F q q = 7, 8 satisfy, for n 2 and k {,, p}, a q n,k 2k + nλn + λ, bq n,k 2k + nλn + λ, 4.6 implies a 7 n,k = a8 n,k, and all other coefficients of F 7 and F 8 are equal to 0. Thus, F k =F 7 =F 8, which shows that F k is an extremal point of HSpλ. 0 5. CONVOLUTIONS AND NEIGHBORHOODS Let CH 0 denote the class of harmonic univalent, convex mappings F of the form. with b = 0. It is known [] that the below sharp inequalities hold: 2 a n n +, 2 b n n. It follows from [], Theorems 5.4 that if H and G are in C 0 H, then H G or H G is sometime not convex, but it may be univalent or even

354 J. Chen, A. Rasila and X. Wang 2 convex if one of the mappings H and F satisfies some additional conditions. In this section, we consider convolutions of harmonic mappings F HS 0 λ and H C 0 H. Theorem 3. Suppose that Hz = z + A nz n + B n z n CH 0 and F HS 0λ. Then for 2 λ, the convolution F H is univalent and starlike, and the integral convolution F H is convex. Proof. If F z = z + a nz n + b n z n HS 0 λ, then for F H, we obtain n + n a n A n + b n B n n a n + n b n 2 2 nλn + λ a n + b n. Hence, F H HS 0. The transformations 0 F z Htz dt = F Hz t now show that F H HC 0. By Theorem A, the result follows. Remark 2. The proof of Theorem 3 does not generalize to polyharmonic mappings, when p 2. For example, let p = 2, and write and Hz = z + F z = z + 2 2 z 2k z 2k A n,k z n + B n,k z n a n,k z n + b n,k z n. Suppose that A n,k n+ 2, B n,k n 2 and F HS2 0 λ. Then for λ =, the convolution F H is univalent and starlike but it is not clear if this is true for 2 λ <. However, the integral convolution F H is convex for 2 λ. Example 3. Let Hz=Re { } { z z + iim z } z C 0 2 H. Then Hz maps D onto the half-plane Re{w} > 2, and let F z = z + 0 z2 + 5 z2 HS 0 2 3. Then the convolution F H is univalent and starlike, and the integral convolution F H is convex see Figure 2.

3 On polyharmonic univalent mappings 355 Fig. 2. The images of D under the mappings F Hz = z + 3 20 z2 0 z2 left and F Hz = z + 3 40 z2 20 z2 right. Finally, we are going to prove the existence of neighborhoods for mappings in the class HS p λ.

356 J. Chen, A. Rasila and X. Wang 4 Theorem 4. Assume that λ 0, ] and F HS p λ. If δ λ 2 2k a,k + b,k, p + λ then N δ F HS p. Proof. Let Hz = p z 2k A n,kz n +B n,k z n N δ F. Then 2k + n An,k + B n,k + 2k A,k + B,k + B, + + δ + δ + + 2k + n An,k a n,k + B n,k b n,k 2k A,k a,k + B,k b,k + B, b, 2k + n an,k + b n,k + p p + λ 2k + n an,k + b n,k + 2k a,k + b,k + b, 2k a,k + b,k + b, 2k + nλn + λ an,k + b n,k 2k a,k + b,k + b, δ+ p λ p+λ + λ p+λ Hence, H HS p. 2k a,k + b,k <. Acknowledgments. The research was partly supported by NSF of China No.07063 and Hunan Provincial Innovation Foundation for Postgraduates No.25000-4242. REFERENCES [] Z. Abdulhadi and Y. Abu Muhanna, Landau s theorem for biharmonic mappings. J. Math. Anal. Appl. 338 2008, 705 709. [2] Z. Abdulhadi, Y. Abu Muhanna and S. Khuri, On univalent solutions of the biharmonic equation. J. Inequal. Appl. 5 2005, 469 478. [3] Z. Abdulhadi, Y. Abu Muhanna and S. Khuri, On some properties of solutions of the biharmonic equation. Appl. Math. Comput. 7 2006, 346 35. [4] Y. Avci and E. Z lotkiewicz, On harmonic univalent mappings. Ann. Univ. Mariae Curie-Sk lodowska Sect. A 44 990, 7. [5] Sh. Chen, S. Ponnusamy and X. Wang, Landau s theorem for certain biharmonic mappings. Appl. Math. Comput. 208 2009, 427 433. [6] Sh. Chen, S. Ponnusamy and X. Wang, Compositions of harmonic mappings and biharmonic mappings. Bull. Belg. Math. Soc. Simon Stevin. 7 200, 693 704.

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