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Bounded harmonic mappings related to starlike functions
Bounded harmonic mappings related to starlike functions Durdane Varol, Melike Aydoğan, and Yaşar Polatoğlu Citation: AIP Conference Proceedings 1602, 644 (2014); doi: 10.1063/1.4882553 View online: http://dx.doi.org/10.1063/1.4882553 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1602?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Harmonic maps relative to α-connections of statistical manifolds AIP Conf. Proc. 1641, 395 (2015); 10.1063/1.4906003 Quasiconformal harmonic mappings related to Janowski alpha-spirallike functions AIP Conf. Proc. 1602, 779 (2014); 10.1063/1.4882574 Coherent state maps related to the bounded positive operators J. Math. Phys. 48, 123514 (2007); 10.1063/1.2821615 On harmonic maps into gauge groups J. Math. Phys. 39, 6684 (1998); 10.1063/1.532649 Harmonic vortices in bounded plasmas Phys. Fluids 29, 339 (1986); 10.1063/1.865949 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 Bounded Harmonic Mappings Related to Starlike Functions Durdane Varola, Melike Aydoğana and Yaşar Polatoğlub a Department of Mathematics, Işık University, Meşrutiyet Köyü, Şile, İstanbul, Turkey Email: [email protected], [email protected] b Department of Mathematics and Computer Science, İstanbul Kültür Üniversitesi, İstanbul, Turkey Email: [email protected] Abstract. Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be a sense-preserving harmonic mapping in the open unit disc ࣞ ൌ ሼݖȁȁݖȁ ൏ ͳሽǤ If ݂ satisfies the conditionቤ ͳ ݃Ԣሺݖሻ ܾͳ ݄Ԣ ሺݖሻ ͳ െ ܯቤ ൏ ܯǡ ܯ ǡ then݂ is called bounded harmonic mapping. The main purpose of this ʹ paper is to give some properties of the class of bounded harmonic mapping. Keywords: bounded harmonic mapping, starlike functions, distortion theorem, growth theorem, coefficient inequality. PACS: 02.30.Fn; 02.30.Gp; 02.30.Px INTRODUCTION Let ȳ be the family of functions ߶ሺݖሻ regular in ࣞ and satisfying the conditions ߶ሺͲሻ ൌ Ͳǡ ȁ߶ሺݖሻȁ ൏ ͳ for everyࣞ א ݖǤ Next, denote by ܲ the family of functions ሺݖሻ ൌ ͳ ݖ ͳ ʹ ݖ ʹ ڮregular in ࣞ and such that ሺݖሻ is inܲ if and only if ͳ ߶ሺݖሻ ሺݖሻ ൌ ͳ െ ߶ሺݖሻ for some߶ሺݖሻ אȳand every ࣞ א ݖǤ Moreover, letܵ כdenote the family of functions݄ሺݖሻ ൌ ݖ ܿʹ ʹ ݖ ڮregular in ࣞ and such that ݄ሺݖሻis in ܵ כif and only if ݄Ԣሺݖሻ ൌ ሺݖሻ ݖ ݄ሺݖሻ ʹ for some ሺݖሻ ܲ אand ࣞ א ݖǤ Let ͳݏሺݖሻ ൌ ݖ ݀ʹ ݖ ڮand ʹݏሺݖሻ ൌ ݖ ݁ʹ ʹ ݖ ڮbe analytic functions in the open unit discࣞǤ If there exists a function߶ሺݖሻ אȳ such that ͳݏሺݖሻ ൌ ʹݏሺ߶ሺݖሻሻfor all ࣞ א ݖǡ then we say that ͳݏሺݖሻ is subordinate to ʹݏሺݖሻ and we write ͳݏሺݖሻ ʹݏ طሺݖሻǤ Specially if ʹݏሺݖሻ is univalent inࣞ, then ͳݏሺݖሻ ʹݏ طሺݖሻ if and only if ͳݏሺࣞሻ ʹݏ ؿሺࣞሻ implies ͳݏሺࣞ ݎሻ ʹݏ ؿሺࣞ ݎሻǡ whereࣞ ݎൌ ሼݖȁȁݖȁ ൏ ݎǡ Ͳ ൏ ݎ൏ ͳሽ. (Subordination and Lindelöf principle [2],[4]) Finally, a planar harmonic mapping in the open unit disc ࣞis a complex-valued harmonic function ݂ǡ which maps ࣞonto the some planar domain ݂ሺࣞሻǤ Since ࣞis a simply connected domain, the mapping ݂has a canonical decomposition ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ,where ݄ሺݖሻ and ݃ሺݖሻ are analytic inࣞ and have the following power series expansion, λ ݄ሺݖሻ ൌ ܽ݊ ݊ ݖǡ ݊ൌͲ λ ݃ሺݖሻ ൌ ܾ݊ ݊ ݖǡ ݊ൌͲ whereܽ݊ ǡ ܾ݊ אԧ, ݊ ൌ Ͳǡͳǡʹǡ ǥas usual we call ݄ሺݖሻ the analytic part of ݂and݃ሺݖሻ is co-analytic part of݂Ǥ An elegant and complete treatment theory of the harmonic mapping is given Duren’s monograph.[3] Proceedings of the 3rd International Conference on Mathematical Sciences AIP Conf. Proc. 1602, 644-649 (2014); doi: 10.1063/1.4882553 © 2014 AIP Publishing LLC 978-0-7354-1236-1/$30.00 644 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 Lewy [7] proved in 1936 that the harmonic mapping of ݂ is locally univalent inࣞ if and only if its Jacobien ݂ܬൌ ȁ݄Ԣሺݖሻȁʹ െ ȁ݃Ԣ ሺݖሻȁʹ is different from zero inࣞǤ In the view of this result, locally univalent harmonic mappings in the open unit disc ࣞare either sense-preserving ifȁ݄Ԣሺݖሻȁ ȁ݃Ԣሺݖሻȁ in ࣞor sense-reversing if ȁ݃Ԣሺݖሻȁ ȁ݄Ԣሺݖሻȁin ࣞǤ Throughout this paper, we will restrict ourselves to the study of sense-preserving harmonic mappings. We also note that݂ ൌ ݄ሺݖሻ ݃ሺݖሻis sense-preserving in ࣞ if and only if݄Ԣሺݖሻ does not vanish in ࣞ and the second dilatation ݃Ԣ ሺݖሻ has the property ȁ߱ሺݖሻȁ ൏ ͳfor all ࣞ א ݖǤ Therefore, the class of all sense-preserving harmonic ߱ሺݖሻ ൌ ݄Ԣ ሺݖሻ mappings in the open unit discࣞ withܽͲ ൌ ܾͲ ൌ Ͳ and ܽͳ ൌ ͳwill be denoted by ܵ ܪǤ Thus,ܵ ܪcontains standard class ܵ of univalent functions. The family of all mappings݂ ܪܵ אwith the additional property ݃Ԣ ሺͲሻ ൌ Ͳ, i.e., ܾͳ ൌ Ͳis denoted byܵ Ͳܪ. Hence it is clear thatܵ ܪܵ ؿ Ͳܪܵ ؿǤ The main purpose of this paper is to investigate the class of harmonic mappings ͳ Ԣ ݃ ሺݖሻ ͳ ܾͳ כሺܯሻ ൌ ቐ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ ܪܵ אȁ ቮ Ԣ ܵܪ െ ܯቮ ൏ ܯǡ ܯ ǡ ݄ሺݖሻ כܵ אሺܯሻቑ ʹ ݄ ሺݖሻ For this investigation we will need the following theorem and lemma. Theorem 1 ([2], [4])Let ݄ሺݖሻ be an element of ܵ כǡ then ݎ ݎ ȁ݄ሺݖሻȁ ሺͳ ݎሻʹ ሺͳ െ ݎሻʹ and ͳݎ ͳെݎ ȁ݄Ԣ ሺݖሻȁ Ǥ ͵ ሺͳ െ ݎሻ͵ ሺͳ ݎሻ Theorem 2 ([2])If ܨሺݖሻand ܩሺݖሻ are regular in ࣞ, ܨሺͲሻ ൌ ܩሺͲሻǡ ܩሺݖሻ mapsࣞ onto a many-sheeted region which is ܨԢ ሺݖሻ ܨሺݖሻ ܲ אand ܲ אǤ starlike with respect to the origin and ܩԢ ሺݖሻ ܩሺݖሻ Lemma 1 ([5]) Let ߶ሺݖሻ be regular in the unit diskࣞ with߶ሺͲሻ ൌ ͲǤ Then if ȁ߶ሺݖሻȁ attains its maximum value on the circle ȁݖȁ ൌ ݎat the point ͳݖ, one has ߶ ͳݖԢ ሺ ͳݖሻ ൌ ݇߶ሺ ͳݖሻǡ for ݇ ͳǤ MAIN RESULTS Theorem 3 Let݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element ofܵ כܪሺܯሻǡ then ݃Ԣ ሺݖሻ ݄Ԣ ሺݖሻ ͳ ͳܾ ط ͳݖ ͳߙݖ (1) whereߙ ൌ ͳ െ Ǥ ܯ Proof. Since ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ݄ሺݖሻ ൌ ݖ ܽʹ ʹ ݖ ݄ ฺ ڮԢ ሺݖሻ ൌ ͳ ʹܽʹ ݖ ڮǡ ݃ሺݖሻ ൌ ܾͳ ݖ ܾʹ ʹ ݖ ฺ ڮ Thus, ͳ ܾͳ ݃Ԣ ሺݖሻ ݄Ԣ ሺݖሻ ൌ ܾʹ ͳ Ԣ ݃ ሺݖሻ ൌ ͳ ʹ ݖ ڮǤ ܾͳ ܾͳ ܩԢሺݖሻ ܩԢሺݖሻ ͳ ܩԢሺݖሻ ฺቤ Ԣ െ ܯቤ ൏ ฺ ܯቤ ή Ԣ െ ͳቤ ൏ ͳ ݄Ԣሺݖሻ ݄ ሺݖሻ ݄ ܯሺݖሻ 645 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 ߮ሺݖሻ ൌ ͳ ܯ ή ܩԢሺݖሻ ݄Ԣ ሺݖሻ െ ͳ, ߮ሺݖሻis analytic and߮ሺͲሻ ൌ ߶ሺݖሻ ൌ ͳ ܯ െ ͳǡ therefore we consider the function ͳ ߮ሺݖሻ െ ߮ሺͲሻ ൌ ͳ െ ߮ሺͲሻ߮ሺݖሻ ܩԢ ሺݖሻ ሺ െ ͳሻ ݄ ܯԢ ሺݖሻ ͳ ͳ ܩԢ ሺݖሻ ͳ െ ሺͳ െ ሻሺ ݄ ܯԢ ሺݖሻ ܯ െ ͳሻ Ǥ This function satisfies the conditions of Schwarz lemma. Then we can write ܩԢ ሺݖሻ ݄Ԣ ሺݖሻ The equality (2) shows that ൌ ͳ߶ ሺݖሻ ͳߙ߶ ሺݖሻ ǡ ͳ ߙൌ ܯ െ ͳǤ (2) ݃Ԣሺݖሻ ͳݖ ͳܾ ط Ǥ ݄Ԣሺݖሻ ͳ ߙݖ Corollary 1 Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ܨሺȁܾͳ ȁǡ ߙǡ െݎሻ ȁ݃Ԣ ሺݖሻȁ ܨሺȁܾͳ ȁǡ ߙǡ ݎሻ where ܨሺȁܾͳ ȁǡ ߙǡ ݎሻ ൌ Proof. Since the transformation߱ሺݖሻ ൌ radius ሺݎሻ ൌ ሺͳെߙ ሻݎ ͳെߙ ʹ ʹ ݎ ͳݖ ͳߙݖ (3) ͳ ݎȁܾͳ ȁሺͳ ݎሻ ή Ǥ ሺͳ െ ݎሻ͵ ͳ ߙݎ maps ȁݖȁ ൌ ݎonto the disc with the centre ܥሺݎሻ ൌ ቀ ͳെߙʹ ݎ ͳെߙ ʹ ʹ ݎ ǡ Ͳቁand the , then using Subordination or Lindelöf principle we can write ቚ ܩԢ ሺݖሻ ݄Ԣ ሺݖሻ െ ͳെߙ ʹ ݎ ͳെߙ ʹ ʹ ݎ ቚ ሺͳെߙ ሻݎ ͳെߙ ʹ ʹ ݎ Ǥ (4) After the simple calculations from (4) and using Theorem 1 we obtain (3). Theorem 4 Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ݃ሺݖሻ ͳݖ ͳܾ ط Ǥ ݄ሺݖሻ ͳ ߙݖ Proof. Using Corollary 2.2 we have ͳ ݃Ԣሺݖሻ ܾͳ ቮ ݄Ԣሺݖሻ െ ሺͳ െ ߙሻݎ ͳ െ ߙʹݎ ቮ ʹ ʹ ͳ െ ߙʹ ʹ ݎ ͳെߙ ݎ ݃Ԣሺݖሻ ܾͳ ሺͳ െ ߙ ʹݎሻ ȁܾͳ ȁሺͳ െ ߙሻݎ െ ቤ ݄Ԣሺݖሻ ͳ െ ߙʹ ʹ ݎ ͳ െ ߙʹ ʹ ݎ ฺቤ Therefore we can write ݃Ԣሺݖሻ ߱ሺࣞ ݎሻ ൌ ቄݖȁ ቚ݄Ԣ ሺݖሻ െ ܾͳ ሺͳെߙ ʹ ݎሻ ȁܾͳ ȁሺͳെߙሻݎ ͳെߙ ʹ ʹ ݎ ͳെߙ ʹ ʹ ݎ ቚ ቅ (5) Now we define the function ߶ሺݖሻby ݃ሺݖሻ ݄ ሺݖሻ ൌ ܾͳ ͳ߶ ሺݖሻ ͳߙ߶ ሺݖሻ ǡ (6) 646 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 then we have ͳ ߶ሺͲሻ ͳ ߶ሺͲሻ ݃ሺͲሻ ൌ ܾͳ ൌ ܾͳ ฺͳൌ ฺ ߶ሺͲሻ ൌ ͲǤ ͳ ߙ߶ሺͲሻ ͳ ߙ߶ሺͲሻ ݄ሺͲሻ ߶ሺݖሻis analytic. Now we show thatȁ߶ሺݖሻȁ ൏ ͳfor all ࣞ א ݖǤ Indeed assume the contrary; there exists a point ݖonȁݖȁ ൌ ݎsuch thatȁ߶ሺ ͳݖሻȁ ൌ ͳǤ Taking the derivative from (6) and using Jack lemma (Lemma 1) we obtain ͳ ߶ሺݖሻ ሺͳ െ ߙሻ߶ݖԢሺݖሻ ݄ሺݖሻ ݃Ԣሺݖሻ ൌ ܾͳ ܾͳ ή ͳ ߙ߶ሺݖሻ ሺͳ ߙ߶ሺݖሻሻʹ ݄ݖԢሺݖሻ ݄Ԣሺݖሻ ฺ ߱ሺ ͳݖሻ ൌ ሺͳ െ ߙሻ ߶ݖԢ ሺ ͳݖሻ ݃Ԣ ሺ ͳݖሻ ͳ ߶ሺ ͳݖሻ ͳ െ ʹݎ ൌ ܾ ቆ ή ቇ ߱ בሺࣞ ݎሻǤ ͳ ݄Ԣ ሺ ͳݖሻ ͳ ߙ߶ሺ ͳݖሻ ሺͳ ߙ߶ሺ ͳݖሻሻʹ ሺͳ ʹ ݎሻ ʹߠ݅ ݁ݎ because ȁ߶ሺ ͳݖሻȁ ൌ ͳ and ݇ ͳǤ But this contradicts the condition (5) and so our assumption is wrong, i.e., ȁ߶ሺݖሻȁ ൏ ͳ for allࣞ א ݖǤ Corollary 2 Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ݎ ሺͳݎሻʹ ͳܨሺȁܾͳ ȁǡ ߙǡ െݎሻ ȁ݃ሺݖሻȁ where ͳܨሺȁܾͳ ȁǡ ߙǡ ݎሻ ൌ Proof. Since ݃ሺݖሻ ݄ሺݖሻ ͳܾ ط ͳݖ ͳߙݖ ݎ ሺͳെݎሻʹ ͳܨሺȁܾͳ ȁǡ ߙǡ ݎሻ (7) ȁܾͳ ȁሺͳ ݎሻ Ǥ ͳ ߙݎ ,then we have ฬ ݃ሺݖሻ ܾͳ ሺͳെߙʹݎሻ െ ฬ ݄ሺݖሻ ͳെߙʹʹݎ ͳ หሺͳെߙ ݎ หܾͳെߙ ʹ ʹݎǤ ሻ (8) Using Theorem 1 and the inequality (8) we obtain (7). Corollary 3Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ሺͳെݎሻʹ ሺͳݎሻʹ ሺͳ െ ሺ ͳܨሺȁܾͳ ȁǡ ߙǡ ݎሻሻʹ ሻ ݂ܬ ሺͳ െ ሺ ͳܨሺȁܾͳ ȁǡ ߙǡ െݎሻሻʹ ሻǤ ሺͳݎሻ ሺͳെݎሻ Proof. Since (9) ݂ܬൌ ȁ݄Ԣሺݖሻȁʹ െ ȁ݃Ԣ ሺݖሻȁʹ ൌ ȁ݄Ԣ ሺݖሻȁʹ ሺͳ െ ȁ߱ሺݖሻȁʹ ሻǡ then using (5) we get (9). Corollary 4 Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then න Proof. Using (5) we obtain Therefore, we have ȁܾͳ ȁሺͳ ݎሻ ͳ ݎ ȁܾͳ ȁሺͳ െ ݎሻ ͳ െ ݎ ή ݀ ݎ ȁ݂ȁ න ή ݀ݎǤ ͵ ͳ െ ߙݎ ͳ ߙݎ ሺͳ ݎሻ ሺͳ െ ݎሻ͵ ȁܾͳ ȁሺͳ െ ݎሻ ȁܾͳ ȁሺͳ ݎሻ ȁ߱ሺݖሻȁ Ǥ ͳ െ ߙݎ ͳ ߙݎ ሺͳߙ ݎሻെȁܾͳ ȁሺͳݎሻ ͳߙݎ ͳ െ ȁ߱ሺݖሻȁ ሺͳെߙ ݎሻെȁܾͳ ȁሺͳെݎሻ ͳെߙݎ ǡ (10) 647 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 ሺͳെߙ ݎሻȁܾͳ ȁሺͳെݎሻ ͳെߙݎ ሺͳߙ ݎሻȁܾͳ ȁሺͳݎሻ ͳ ȁ߱ሺݖሻȁ Ǥ ͳߙݎ (11) On the other hand, we have ሺͳ െ ȁ߱ሺݖሻȁሻȁ݄Ԣ ሺݖሻȁȁ݀ݖȁ ȁ݂݀ȁ ሺͳ ȁ߱ሺݖሻȁሻȁ݄Ԣ ሺݖሻȁȁ݀ݖȁǤ (12) Considering (10), (11), (12) and Theorem 1, we get the desired result. Theorem 5 Let ݂ ൌ ݄ሺݖሻ ݃ሺݖሻ be an element of ܵ כܪሺܯሻǡ then ܾ ݇ͳ σ݊݇ൌͳሺ݇ ͳሻʹ ቚ ܾͳ Proof. Since ʹ ͳ ͳ ܾ ݇ͳ ܯ ܯ ܾͳ ʹ െ ܽ݇ ቚ ሺʹ െ ሻʹ σ݊െͳ ݇ൌͳ ሺ݇ ͳሻ ቚܽ݇ ሺͳ െ ሻ ቚǤ (13) ݃Ԣሺݖሻ ݃Ԣ ሺݖሻ ݃Ԣ ሺݖሻ ͳݖ ͳݖ ͳ ߶ሺݖሻ ͳܾ ط ฺ Ԣ ͳܾ ط ฺ ൌ ܾͳ Ǥ Ԣ ሺݖሻ ͳ ͳ ݄Ԣሺݖሻ ݄ ሺݖሻ ݄ ͳ ߙݖ ͳ ቀ െ ͳቁ ݖ ͳ െ ቀͳ െ ቁ ߶ሺݖሻ ܯ ͳ Ԣ ݃ ሺݖሻ ܾͳ ݄ Ԣ ሺݖሻ ൌ ܯ ͳ߶ሺݖሻ ͳ ܯ Ǥ (14) ͳെቀͳെ ቁ߶ ሺݖሻ The equality (14) can be written in the following form ͳ ܾͳ ݃Ԣ ሺݖሻ ݄Ԣ ሺݖሻ ൌ ܩሺݖሻ ͳ ߶ሺݖሻ ܩሺݖሻ ͳ ߶ሺݖሻ ͳ ൌ ฺ ൌ ǡߙ ൌ ൬ͳ െ ൰ ܪሺݖሻ ͳ െ ቀͳ െ ͳ ቁ ߶ሺݖሻ ܪሺݖሻ ͳ െ ߙ߶ሺݖሻ ܯ ܯ Therefore, we have ܩሺݖሻ െ ܪሺݖሻ ൌ ൫ܪሺݖሻ ܽܩሺݖሻ൯߶ሺݖሻ ݊ λ ݇ λ λ ൌ ሺ݀݇ െ ݁݇ ሻ ݖ ሺ݀݇ െ ݁݇ ሻ ݖെ ൭ሺ݁݇ ܽ݀݇ ሻ ݖ൱ ൭ ܿ݇ ݇ ݖ൱ ݇ൌͳ ݇ ݇ ݇ൌ݊ͳ ݇ൌ݊ ݇ൌͳ ݇ ൌ ሾሺܽ ͳሻ σ݊െͳ ݇ൌͳ ሺ݁݇ ܽ݀݇ ሻ ݖሿ where ݀݇ ൌ ሺ݇ͳሻܾ ݇ͳ ܾͳ (15) ǡ ݁݇ ൌ ሺ݇ ͳሻܽ݇ͳ ǡ ݇ ൌ ͳǡʹǡ ǥ . Equality (15) can be written in the following manner ݊െͳ ݇ ݇ σ݊݇ൌͳሺ݀݇ െ ݁݇ ሻ ݇ ݖ σλ ݇ൌ݊ͳ ݖ ݇ݏൌ ሾሺܽ ͳሻ σ݇ൌͳ ሺ݁݇ ܽ݀݇ ሻ ݖሿ߶ሺݖሻ (16) where the coefficients ݇ݏhave been chosen suitably and the equality (16) can be written in the form ͳܨሺݖሻ ൌ ʹܨሺݖሻ߶ሺݖሻǡ then we have ȁ߶ሺݖሻȁ ൏ ͳ ȁ ͳܨሺݖሻȁʹ ൌ ȁ ʹܨሺݖሻ߶ሺݖሻȁʹ ൌ ȁ ʹܨሺݖሻȁʹ ȁ߶ሺݖሻȁʹ ൏ ȁ ʹܨሺݖሻȁʹ ฺ ͳ ʹߨ ͳ ʹߨ ʹ ʹ න ห ͳܨ൫ ߠ݅ ݁ݎ൯ห ݀ߠ න ห ʹܨ൫ ߠ݅ ݁ݎ൯ห ݀ߠ ʹߨ Ͳ ʹߨ Ͳ 648 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 193.255.121.155 On: Sat, 15 Aug 2015 00:44:20 ݊ ฺ ȁ݀݇ െ ݁݇ ݎ ݇ൌͳ ݊െͳ λ ȁʹ ʹ݇ ȁʹ ʹ݇ ȁݎ ݇ݏ ʹ ʹ݇ ሺܽ ͳሻ ݎ ݇ൌ݊ͳ ሺȁ݁݇ ܽ݀݇ ȁሻʹ ݇ʹ ݎ൩ ݇ൌͳ passing to the limit as ݎ՜ ͳ we obtain (13). The method of this proof has been based on the Clunie method[1] CONCLUSION In the present paper we have given the basic characterization which is analogue to the Libera Theorem ([4]). This characterization is used for the investigation of the class of bounded harmonic mappings related to the starlike functions. 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, 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, 297326 (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). 649 to the terms at: http://scitation.aip.org/termsconditions. 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