Starlike Functions of Complex Order

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1 Applied Mathematical Sciences, Vol. 3, 2009, no. 12, Starlike Functions of Complex Order Aini Janteng School of Science and Technology Universiti Malaysia Sabah, Locked Bag No Kota Kinabalu, Sabah, Malaysia aini Abstract Let H denote the class of functions f which are harmonic and univalent in the open unit disc D {z : z < 1}. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in D and are related to the functions starlike of complex order. The author obtain coefficient conditions, growth result, extreme points, convolution and convex combinations. Mathematics Subject Classification: 30C45 Keywords: harmonic functions, starlike of complex order, coefficient estimates 1 Introduction A continuous complex-valued function f u+iv defined in a simply connected complex domain E is said to be harmonic in E if both u and v are real harmonic in E. There is a close inter-relation between analytic functions and harmonic functions. For example, for real harmonic functions u and v there exist analytic functions U and V so that u Re U and v Im V. Then fz hz+gz where h and g are, respectively, the analytic functions U+V /2 and U V /2. In this case, the Jacobian of f h + g is given by J f h z 2 g z 2. The mapping z fz is orientation preserving and locally one-to-one in E if and only if J f > 0inE. The function f h + g is said to be harmonic univalent in E if the mapping z fz is orientation preserving, harmonic

2 558 A. Janteng and one-to-one in E. We call h the analytic part and g the co-analytic part of f h + g. Let H denote the class of functions f h+g which are harmonic and univalent in D the unit disc with the normalization hz z + a n z n, gz b n z n. 1 There have been rigorous works conducted on the class of complex harmonic functions H. In [2], Sheil-Small and Clunie obtained properties for functions in this class. Since then, there have been other subclasses of H that were developed. These include the class of harmonic functions starlike in the unit disc D see Jahangiri [4] and convex harmonic functions in D see Kim et al. [5]. Other related works also appear in [1], [3] and [8]. Silverman in [7] formed the class H, a subclass of H which consists harmonic functions with negative coefficients. The class H is defined below. Let H be the subclass of H consisting of functions f h + g so that the functions h and g take the form hz z a n z n, gz b n z n. 2 Another interesting class of functions is the class of functions starlike of order bb C\{0}, first introduced by Nasr and Aouf in [6]. Wiatrowski [9] introduced the class of functions which are convex of order bb C\{0}. The authors, by combining defined the new class of functions as follows : Definition 1.1 Let f H. Then f HS b, β if and only if it satisfies 1 b [ zf ] z z fz 1 <β. 3 for b C\{0}, 0 <β 1, z θ z reiθ,f z θ fz freiθ, 0 r<1 and 0 θ<2π, Also, let HS b, β HS b, β H. In this paper, the author is motivated to determine properties of this new class which include coefficients results, growth bounds, extreme points, convolution properties and convex combinations.

3 Starlike functions of complex order Results The results begin with a necessary and sufficient condition for functions in HS b, β. Theorem 2.1 Let f h+g with h and g of the form 2. Then f HS b, β if and only if [n 1 + ] a n + [n +1+] b n. 4 Proof. Suppose that f HS b, β. Let wz be defined by wz zf z z fz z zg z 1zh 1. hz+gz Then from 3, we obtain the following inequality: wz n 1 a n z n + n +1 b n z n z a n z n b n z n <, z D. Since z r0 r<1 and f HS b, β, we obtain n 1 a n r n 1 + n +1 b n r n 1 1 a n r n 1 b n r n 1 <. 5 Now letting r 1 through real values in 5, we then have n 1 a n + n +1 b n 1 a n b n. 6 Thus, 6leads us to the desired assertion 4 of Theorem 2.1. Conversely, by applying the hypothesis 4 and letting z r0 r<1, we find from 3 that zh z zg z hz+gz hz+gz n 1 a n z n + n +1 b n z n z a n z n b n z n < [n 1 + ] a n + [n +1+] b n 0, by 4.

4 560 A. Janteng Hence, f HS b, β which completes the proof of Theorem 2.1. The harmonic functions fz z n 1+ x nz n n +1+ y nz n, 7 where x n + y n 1, show that the coefficient bound given in Theorem 2.1 is sharp. The functions of the form 7 are in HS b, β since n 1+ a n + n +1+ b n x n + y n 1. The growth result for functions in HS b, β is discussed in the following theorem. Theorem 2.2 If f HS b, β then fz r + 1+ r2, z r<1 and fz r 1+ r2, z r<1. Proof. Let f HS b, β. Taking the absolute value of f we have fz r + a n + b n r n r + a n + b n r 2 r + 1+ a n + 1+ b n r 2 1+ n 1+ r + 1+ r + 1+ r2 a n + n +1+ b n r 2

5 Starlike functions of complex order 561 and fz r a n + b n r n r a n + b n r 2 r 1+ a n + 1+ b n r 2 1+ n 1+ r 1+ r 1+ r2. a n + n +1+ b n r 2 Next, we determine the extreme points of closed hulls of HS b, β denoted by clcohs b, β. Theorem 2.3 f clcohs b, β if and only if fz n1 X n h n + Y n g n where h 1 z z, h n z z n 1+ zn n 2, 3,..., g n z z n1 X n + Y n 1,X n 0 and Y n 0. n +1+ z n n 2, 3,..., Proof. For h n and g n as given above, we may write fz X n h n + Y n g n n1 X n + Y n z n1 z n 1+ X nz n n 1+ X nz n n +1+ Y n z n. n +1+ Y n z n Then n 1+ a n + n +1+ b n n 1+ n 1+ X n n n +1+ Y n

6 562 A. Janteng Therefore f clcohs b, β. X n + Y n 1 X 1 Y 1 1. Conversely, suppose that f clcohs b, β. Set X n n 1+ a n, n 2, 3, 4,..., and Y n n +1+ b n, n 2, 3, 4,..., where n1 X n + Y n 1. Then fz hz+gz z a n z n b n z n z n 1+ X nz n n +1+ Y n z n z + h n z zx n + g n z zy n X n h n + Y n g n. n1 For harmonic functions fz z a n z n b n z n and F z z A n z n B n z n, we define the convolution of f and F as f Fz z a n A n z n b n B n z n. 8 In the next theorem, we examine the convolution properties of the class HS b, β. Theorem 2.4 For 0 <α β 1, let f HS b, β and F HS b, α. Then f F HS b, β HS b, α. Proof. Write fz z a n z n b n z n and F z z A n z n B n z n. Then the convolution of f and F is given by 8. Note that A n 1 and B n 1 since F HS b, α. Then we have [n 1+] a n A n + [n +1+] b n B n

7 Starlike functions of complex order 563 [n 1+] a n + [n +1+] b n. Therefore, f F HS b, β HS b, α since the right hand side of the above inequality is bounded by while α b. Now, we determine the convex combination properties of the members of HS b, β. Theorem 2.5 The class HS b, β is closed under convex combination. Proof. For i 1, 2, 3,..., suppose that f i HS b, β where f i is given by f i z z a n,i z n b n,i z n. For i1 c i 1, 0 c i 1, the convex combinations of f i may be written as c i f i z c 1 z c 1 a n,1 z n c 1 b n,1 z n c 2 z c 2 a n,2 z n c 2 b n,2 z n... i1 z c i c i a n,i z n c i b n,i z n i1 i1 i1 z c i a n,i z n c i b n,i z n. i1 Next, consider [n 1+] c i a n,i + [n +1+] c i b n,i i1 i1 c 1 [n 1+] a n, c m [n 1+] a n,m c 1 [n +1+] b n, c m [n +1+] b n,m +... { } c i [n 1+] a n,i + [n +1+] b n,i. i1 Now, f i HS b, β, therefore from Theorem 2.1, we have i1 Hence [n 1+] a n,i + [n +1+] b n,i.

8 564 A. Janteng [n 1+] i1 c i a n,i + [n +1+] i1 c i b n,i i1 c i. By using Theorem 2.1 again, we have i1 c i f i HS b, β. Acknowledgement The author is partially supported by FRG0118-ST-1/2007 Grant, Malaysia. References [1] Ahuja, O.P. and Jahangiri, J.M A subclass of harmonic univalent functions. J. of Natural Geometry, [2] Clunie, J. and Sheil Small, T Harmonic univalent functions. Ann. Acad. Aci. Fenn. Ser. A. I. Math., [3] Jahangiri, J.M Coefficient bounds and univalence criteria for harmonic functions with negative coefficients. Ann. Univ. Marie Curie. Sklodowska. Sec. A, [4] Jahangiri, J.M Harmonic functions starlike in the unit disk. J. Math. Anal. Appl., [5] Kim, Y.C., Jahangiri, J.M. and Choi, J.H Certain convex harmonic functions. Int. J. Math. Math. Sci., [6] Nasr, M.A. and Aouf, M.K Starlike functions of complex order. J. Natural Sci. Math., [7] Silverman, H Harmonic univalent functions with negative coefficients. J. Math. Anal. Appl., [8] Silverman, H. and Silvia, E.M Subclasses of harmonic univalent functions. New Zeal. J. Math., [9] Wiatrowski, P The coefficients of a certain family of holomorphic functions. Zeszyty Nauk. Univ. Lódz Nauk. Math. Przyrod Ser. II, Received: July, 2008