Quasiconformal Harmonic Mappings Related to Janowski Alpha-Spirallike Functions Melike Aydoğana and Yaşar Polatoğlub aDepartment of Mathematics, Işık University, Meşrutiyet Köyü, Şile, İstanbul, Turkey Email: melike.aydogan@isikun.edu.tr bDepartment of Mathematics and Computer Science, İstanbul Kültür Üniversitesi, İstanbul, Turkey Email: y.polatoglu@iku.edu.tr Abstract. Let be a univalent sense-preserving harmonic mapping of the open unit disc If satisfies the condition then is called k-quasiconformal harmonic mapping in . In the present paper we will give some properties of the class of k-quasiconformal mappings related to Janowski alpha-spirallike functions. Keywords: k-quasiconformal mapping, Distortion theorem, Growth theorem, Coefficient inequality. PACS: 02.30.Fn; 02.30.Gp; 02.30.Px INTRODUCTION Let be the family of functions regular in and satisfying the conditions for every Next, for arbitrary fixed numbers denote by the family of functions regular in and such that is in if and only if (1) for some and every Moreover, let denote the family of functions regular in and such that is in if and only if there is a real number for which (2) is true for every Let and be analytic functions in If there exists a function such that for all then we say that is subordinate to and we write Specially if is univalent in , then if and only if implies where . (Subordination and Lindelöf principle [2],[4]) Finally, a planar harmonic mapping in the open unit disc is a complex-valued harmonic function which maps onto the some planar domain Since is a simply connected domain, the mapping has a canonical decomposition , where and are analytic in and have the following power series expansion, Proceedings of the 3rd International Conference on Mathematical Sciences AIP Conf. Proc. 1602, 779-784 (2014); doi: 10.1063/1.4882574 © 2014 AIP Publishing LLC 978-0-7354-1236-1/$30.00 This article is copyrighted as indicated in the article. Reuse of AIP content is s7u7b9ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29 where , as usual we call the analytic part of and is co-analytic part of An elegant and complete treatment theory of the harmonic mapping is given Duren’s monograph.[3] Lewy [7] proved in 1936 that the harmonic function is locally univalent in if and only if its Jacobien is different from zero in In the view of this result, locally univalent harmonic mappings in the open unit disc are either sense-preserving if in or sense-reversing if in Throughout this paper, we will restrict ourselves to the study of sense-preserving harmonic mappings. We also note that is sense-preserving in if and only if does not vanish in and the second dilatation has the property for all Therefore, the class of all sense-preserving harmonic mappings in the open unit disc with and will be denoted by Thus contains standard class of univalent functions. The family of all mappings with the additional property , i.e., is denoted by . Hence, it is clear that A univalent harmonic mapping is called k-quasiconformal if For the general definition of quasiconformal mapping, see[3]. The main purpose of this paper is to give some properties of the k-quasiconformal harmonic mappings For this investigation we will need the following theorem and lemma. Theorem 1 ([6]) Let be an element of then where and , where Lemma 1 ([5]) Let be regular in the unit disk with Then if attains its maximum value on the circle at the point , one has for MAIN RESULTS Theorem 2 Let be an element of then (3) Proof. We consider the linear transformation this transformation maps onto itself. On the other hand, we have This article is copyrighted as indicated in the article. Reuse of AIP content is s7u8b0ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29 Therefore, the function satisfies the conditions of Schwarz lemma, then we have (4) On the other hand, the linear transformation maps onto the disc with the centre and the radius Then, we write (5) Now we define the function by (6) Then, is analytic and . If we take the derivative from (6) and after the brief calcula tions we get (7) On the other hand, since then the boundary value of is Therefore, the equality (7) can be written in the following form on (8) In this step, if we use Jack lemma, This article is copyrighted as indicated in the article. Reuse of AIP content is s7u8b1ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29 But, this contradicts to (5) because so our assumption is wrong, i.e., for all Therefore (6) shows that Corollary 1 Let be an element of then This corollary is a simple consequence of the inequality (5). Corollary 2 Let be an element of then where where Proof. Using Theorem 2 and inequality (5) we can write (9) (10) If we use Theorem 1 in the inequalities (8) and (9) we obtain the desired result. Corollary 3 Let be an element of ,then This article is copyrighted as indicated in the article. Reuse of AIP content is s7u8b2ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29 Proof. Since then if we use Theorem 1 and the inequality (5), in this step we obtain the desired result. Corollary 4 Theorem 3 Let be an element of then (11) Proof. Using Theorem 2, then we can write Therefore, we have (12) where the coefficients have been chosen suitably and the equality (12) can be written in the form then we have passing to the limit as we obtain (11). The method of this proof was based on the Clunie method [1]. This article is copyrighted as indicated in the article. Reuse of AIP content is s7u8b3ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29 CONCLUSION In the present paper we have given the basic characterization of the class of quasiconformal harmonic mappings related to the Janowski alpha-spirallike functions. This characterization is analogue to the Libera Theorem [4] and is used for the investigation of the mentioned class. REFERENCES 1. Clunie, J., “On Meromorphic Schlicht Functions”, J. London Math. Soc. 34, 215-216 (1959). 2. Duren, P., Univalent Functions, Springer Verlag, 1983. 3. Duren, P., “Harmonic Mappings in the Plane” in Cambridge Tracts in Mathematics, Cambridge UK: Cambridge University Press, Vol. 156, 2004. 4. Goodman A. W., Univalent Functions, Tampa Florida: Mariner publishing Company INC, Volume I, 1983. 5. Jack, I. S., “Functions starlike and convex of order alpha”, J. London Math. Soc. 3, 369-374 (1971). 6. Janowski, W., “Some extremal problems for certain families of analytic functions I”, Annales Policini Mathematici 28, 297- 326 (1973). 7. Lewys, H., “On the non-vanishing of the Jacobian in certain one-to-one mappings”, Bull. Amer. Math. Soc. 42, 689-692 (1936). This article is copyrighted as indicated in the article. Reuse of AIP content is s7u8b4ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:30:29