Advances in Pure Mathematics, 2012, 2, 291295 http://dx.doi.org/10.4236/apm.2012.24038 Published Online July 2012 (http://www.SciRP.org/journal/apm) A Certain Subclass of Analytic Functions Young Jae Sim, Oh Sang Kwon* Department of Mathematics, Kyungsung University, Busan, Korea (South) Email: {yjsim, *oskwon}@ks.ac.kr Received February 20, 2012; revised April 20, 2012; accepted April 28, 2012 ABSTRACT In the present paper, we introduce a class of analytic functions in the open unit disc by using the analytic function 2 3z z 33qz , which was investigated by Sokół [1]. We find some properties including the growth theorem or the coefficient problem of this class and we find some relation with this new class and the class of convex functions. Keywords: Univalent Functions; Convex Functions; Subordination; Order of Convexity 1. Introduction Let H denote the class of analytic functions in the unit disc :1zz 0f on the complex plane . Let A denote the subclass of H consisting of functions normal ized by and 00f1 . The set of all func tions A that are convex univalent in by K. Re call that a set E is said to be convex if and only if the linear segment joining any two points of E lies entirely in E. Let the function f be analytic univalent in the unit disc on the complex plane with the normalization. Then f maps onto a convex domain E if and only if 0zU K Re 1zf z fz . Robertson introduced in [2], the class of con vex functions of order , which is defined by 1 :Re 1, z KfA z fz 0, 1 0 zf . If , then a function of this set is univalent and if 0 it may fail to be univalent. We denote K . Let S be denote the subset of A which is composed of univalent functions. We say that f is subor dinate to F in , written as F , if and only if, zFwz for some Schwarz function wz , and 00w 1wz , z . The class of convex functions K can be defined in several ways, for example we say that f is convex if it satisfies the condition 1 1 zz 1zf zz . (1) Many subclass of K have been defined by the condi tion (1) with a convex univalent function p, given arbi trary, instead of the functions 11zz. Janowski considered the function p, which maps the unit disc onto a disc in [3,4]. An interesting case when the function p is convex but is not univalent was considered in [5]. A function p that is not univalent and is not convex and maps unit circle onto a concave set was considered in [1]. Now, we shall introduce the class of analytic functions used in the sequel. Definition 1.1. The function A belongs to the class SQ , 3,1 , if it satisfies the condition 2 3 33 fz qzzz q (2) Let the function be given by (2). We note that 2 31 33 331 3 3 133 qzz zzz z Sokół investigated in [1] that the image of the unit cir cle 1z q under the function is a curve described by 2 22 1 :0 2 xax ykx , where Reand Im ii xqe yqe , with 0, 2π and 2 91 23 a and *Corresponding author. C opyright © 2012 SciRes. APM
Y. J. SIM, O. S. KWON 292 2 54 33 k i qe . Thus the curve is symmetric with respect to real axis and satisfies 2 91 Re 23 i qe 3, 23 0, 2π (3) where . Especially, if 0 , then 0 qz , which maps onto the right of line 11 z 12x 21 31 . And we note that if 12 qq , then . 2. Some Properties of Functions in SQ Now we shall find some properties of functions in the class SQ . Theorem 2.1. If a function f belongs to the class SQ , 3,1 , then there exists a function A such that 11 zz hA and a function such that 33 zhz and zgzhz SQ . Proof. Let f be in . Then there exists an ana lytic function with wz 00w and 1wz for such that z 3 13wz fz wz . (4) From (4) we have 21 fz wz f zwz 3 wz wz . Define g and h so that 21 zwz zwz and 23 wz wz hz hz , respectively. Then 11 zz , 33 zhz and 222 zgzhz zgzhz . zgzhz Hence fSQ , which proves Theorem 2.1. Theorem 2.2. If , 3,1 and zr, 01r , then 2 2 2 2 1 11 3 1 11 3 fz rr rr fSQ (5) Proof. Suppose that . Then zgzhz For some g and h such that 11zz and 33hz z , respectively. And above subordination equations imply that 22 11 11 gz rr and 22 11 13 13 hz rr , respectively. Since zgzhz , the modulus of z satisfies the inequality (5). Next, we shall solve some coefficient problem for a special function to be in the class SQ . n n Theorem 2.3. The function zzcz belongs to the class SQ , whenever 2 2 27 244 43 cn . 1 1n nzncz , if we put Proof. Since n Gzgz , then 21 1n Gz ncz z . , Hence for 21Gz nc . Re 1Gz nc Since , if Copyright © 2012 SciRes. APM
Y. J. SIM, O. S. KWON 293 3 231nc gz SQ , (6) Then n and we can easily derive that the inequality (6) is equivalent to 2 2 24 4 43n 27 c . 3. The Relations o f the Classes SQ and K It is wellknown that the following implication holds: 1 2 fz Re 10Re zf z fz . (7) More generally, the above implication (7) is can be generalized as following: zfzzk z zkz fz kz . Evidently, the implication (7) implies the relation 012KSQ. In this chapter, we find some general relation between the classes K and SQ . Let us denote by Q the class of functions f that are analytic and injective on Ef ndlimfz , where Ef :a z and are such that 0 Ef 0pa n p . Lemma 3.1. [6] Let with and let Q n a azqz Be analytic in with anda n Ef mn qz . If q is not subordinate to p, then there exist points 00 and i zre , and there exists a number for which ,qzrp 00 qz p m p and 00 zq z 11 . Theorem 3.2. Let . If a function f belongs to the class A and 2 Re 13 zf z fz forfz fSQ , then . Proof. Suppose that 0 and fSQ or equiva lently, zqz 0 z . Then by Lemma 3.1, there exist and 1 , and m such that 1 0 fz q and 0 zz zfz mq . Since 23 3 Re 1 12 , For 11 , 00 0 Re 1 2 Re 1 33 233 12Re 13 1 23 3 1Re 13 233 2Re1 23 3 1Re 13 3 2Re12 zf z fz mq q m m m m m . 10 In case , since the inequality (3) induces the following inequality: 2 91 33 Re , 1323 23 (8) 00 0 2 Re 1 23913 121 2 23 91 3 12 1 23 2 231 1, 33 zf z fz mm m Copyright © 2012 SciRes. APM
Y. J. SIM, O. S. KWON 294 which is a contradiction to the hypothesis. In case 01 , using the inequality (8) again, 00 0 Re 1 23 12 23 33 3 121 23 313 11 33 zfz fz m m m 33 1 2 2 21, 3 m which is a contradiction to the hypothesis, hence zqz , and fSQ . If we put 12 in Theorem 3.2, we can get next Corollary. Corollary 3.1. For A, the following implication holds: 2 Re 15 Re zf z fz 1 2 27 . 49 fSQ fz Theorem 3.3. Let and let 1, 1 fSQ . Then f is convex for 13z, if 0 , and 236 6 CC z , where 23C3 0, if fSQ. Proof. Let . Then 2 3 33zz fz and there exists a Schwarz function wz with 00w and 1z s wuch that 2 3 33 fz wzw z . Then 2 4zw z wz w z 23 11 33 zf zzw z fz wz . Hence 23 zf zzw z fz wz 4 13 wz wz (9) Using the wellknown estimate [7]: 2 2 1 1 wz z wz , We have from (9) 2 1234 13 zwz wz zf z fz zwz Hence if 2 1234 1 13 zwzwz zwz , (10) Then f is convex. So it is enough to find the condition of z0 to satisfy the inequality (10). In case , then inequality (10) reduces to 2 21 1 1 zwz z , (11) wz z And (11) is satisfied for 13z, since . Hence we can conclude that f is convex for 13z 0 , in case . Now we suppose that 0 and let wz R and zr. And let us put 22 2 4234 2333 . TRrRrrr R rr Now 2 82340TRrRr rr implies 2 1 2341 0 8 rr r RR r . 00T is equivalent to And 2 0 3618 : 3 rr 0 rr 0 rr . That is, f need not be convex for . And for 0TR , is equivalent to 22 2 162333 8 BBr r RR r . where 2 23 4Brrr . Put 242 2 64 16 162333. PrrBr rr Then 048 0P and Copyright © 2012 SciRes. APM
Y. J. SIM, O. S. KWON Copyright © 2012 SciRes. APM 295 REFERENCES 3 0. 22 1 643236432P [1] J. Sokół, “A Certain Class of Starlike Functions,” Com puters and Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 611619. doi:10.1016/j.camwa.2011.05.041 1 10Pr 10,r such that Hence there exists a and for 1 0, . Hence for 1 rr 0Pr0rr Rr , , attains its maximum at 2 Rr TR for . Now 1 r0Rr 2 2 2 0 13 23 323 3 36 , 6 Tr rr rr CC r [2] M. S. Rovertson, “Certain Classes of Starlike Functions,” Michigan Mathematical Journal, Vol. 76, No. 1, 1954, pp. 755758. [3] W. Janowski, “Extremal Problems for a Family of Func tions with Positive Real Part and Some Related Families,” Annales Polonici Mathematici, Vol. 23, 1970, pp. 159 177. 330 30 r [4] W. Janowski, “Some Extremal Problems for Certain Fa milies of Analytic Functions,” Annales Polonici Mathe matici, Vol. 28, 1973, pp. 297326. [5] R. Jurasiska and J. Stankiewics, “Coefficients in Some Classes Defined by Subordination to Multivalent Majo rants,” Annales Polonici Mathematici, Vol. 80, 2003, pp. 163170. where 23C3 , which proves Theorem 3.3. If we put 12 in Theorem 3.3, we can get next Corollary. [6] S. S. Miller and P. T. Mocanu, “Differential Subordina tions, Theory and Applications,” Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York, 2000. Corollary 3.2. Let 12fSQ. Then f is convex for 173896z 0.453847 . [7] P. Duren, “Univalent functions, A Series of Comprehen sive Studies in Mathematics,” Vol. 259, SpringerVerlag, New York, 1983. 4. Acknowledgements The research was supported by Kyungsung University Research Grants in 2012.
