 Advances in Pure Mathematics, 2012, 2, 291-295 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  23z z33qz , which was investigated by Sokół . 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 00f1. The set of all func-tions fA 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KRe 1zf zfz. Robertson introduced in , the class  of con-vex functions of order , which is defined by 1 :Re 1,zKfA zfz 0, 10zf . If , then a function of this set is univalent and if 0 it may fail to be univalent. We denote KK. 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 fF, if and only if, fzFwz for some Schwarz function wz , and 00w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 11zz1zffzz . (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 . A function p that is not univalent and is not convex and maps unit circle onto a concave set was considered in . Now, we shall introduce the class of analytic functions used in the sequel. Definition 1.1. The function fA belongs to the class SQ, 3,1, if it satisfies the condition  2333fz qzzz q (2) Let the function  be given by (2). We note that  23133 331 33133qzz zzzz     Sokół investigated in  that the image of the unit cir-cle 1z q under the function  is a curve described by 222 1:02xax ykx, where Reand Imiixqe yqe, with 0, 2π and 29123a and *Corresponding author. Copyright © 2012 SciRes. APM Y. J. SIM, O. S. KWON 292   25433kiqe. Thus the curve is symmetric with respect to real axis and  satisfies 291 Re23iqe3,230, 2π (3) where . Especially, if 0, then  0qz , which maps onto the right of line 11 z12x2131. And we note that if  12qq, 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 gA such that  11gzzhA and a function such that  33 zhz  and fzgzhzSQ. Proof. Let f be in . Then there exists an ana-lytic function with wz00w and 1wz for such that z 313wzfz wz. (4) From (4) we have   21fz wzf zwz 3wzwz. Define g and h so that 21gzwzgzwz and  23wzwzhzhz , respectively. Then 11gzz,  33 zhz and 222fzgzhzfzgzhz . fzgzhzHence fSQ, which proves Theorem 2.1. Theorem 2.2. If , 3,1 and zr, 01r, then 2222111 3111 3fzrrrrfSQ (5) Proof. Suppose that . Then fzgzhz For some g and h such that g11zz and 33hz z, respectively. And above subordination equations imply that  221111gzrr and 221113 13hzrr, respectively. Since fzgzhz, the modulus of fz satisfies the inequality (5). Next, we shall solve some coefficient problem for a special function to be in the class SQ. nnTheorem 2.3. The function gzzcz belongs to the class SQ, whenever 2227 24443cn. 11nnzncz , if we put Proof. Since gnGzgz, then 211nGz nczz. , Hence for 21Gz nc. Re 1Gz ncSince , if Copyright © 2012 SciRes. APM Y. J. SIM, O. S. KWON 2933231nc gz SQ, (6) Then n and we can easily derive that the inequality (6) is equivalent to 2224 443n27c. 3. The Relations o f the Classes SQ and K It is well-known that the following implication holds:  12fz Re 10Rezf zfz . (7) More generally, the above implication (7) is can be generalized as following:    zfzzk zfzkz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:az and are such that 0fEf0panp. Lemma 3.1.  Let with and let Qna azqz Be analytic in with anda nEfmnqz . If q is not subordinate to p, then there exist points 00 andizre, and there exists a number for which ,qzrp00qz pm p and 00zq z11. Theorem 3.2. Let  . If a function f belongs to the class A and 2Re 13zf zfzforfzfSQ, then . Proof. Suppose that 0 and fSQ or equiva- lently, zqz0zf. Then by Lemma 3.1, there exist and 1,  and m such that 10fz q and 0zzzfz mq. Since 23 3Re 112 , For 11,     000Re 12Re 133 23312Re 13 123 31Re13233 2Re123 31Re133 2Re12zf zfzmqqmmmmm      .10 In case , since the inequality (3) induces the following inequality:   291 33Re ,132323  (8) 0002Re 12391312122391 312 123 22311,33zf zfzmmm      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,  000Re 123122333 3121233131133zfzfzmmm3312221,3m    which is a contradiction to the hypothesis, hence fzqz, and fSQ. If we put 12 in Theorem 3.2, we can get next Corollary. Corollary 3.1. For fA, the following implication holds:  2Re 15Rezf zfz 1 227 .49fSQfz Theorem 3.3. Let and let 1, 1fSQ. Then f is convex for 13z, if 0, and 2366CCz, where 23C3 0, if fSQ. Proof. Let . Then  2333zzfz  and there exists a Schwarz function wz with 00w and 1z s wuch that 2333fz wzw z . Then 24zw z wzw z231133zf zzw zfz wz  . Hence   23zf zzw zfz wz 413wzwz (9) Using the well-known estimate :  2211wzzwz, We have from (9) 2123413zwz wzzf zfz zwz  Hence if 21234 113zwzwzzwz , (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 221 11zwzz, (11) wz zAnd (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 22242342333 .TRrRrrr Rrr  Now 282340TRrRr rr  implies 212341 08rr rRR r . 00T is equivalent to And 203618:3rr 0rr0rr. That is, f need not be convex for . And for 0TR,  is equivalent to 2221623338BBr rRR r  . where 223 4Brrr . Put 242264 16162333.PrrBrrr Then 048 0P and Copyright © 2012 SciRes. APM Y. J. SIM, O. S. KWON Copyright © 2012 SciRes. APM 295REFERENCES   3 0.221 643236432P   J. Sokół, “A Certain Class of Starlike Functions,” Com-puters and Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 611-619. doi:10.1016/j.camwa.2011.05.041 110Pr10,r such that Hence there exists a  and for 10, . Hence for 1rr0Pr0rrRr , , attains its maximum at 2RrTR for . Now 1r0Rr 222013 23323 336 ,6TrrrrrCCr M. S. Rovertson, “Certain Classes of Starlike Functions,” Michigan Mathematical Journal, Vol. 76, No. 1, 1954, pp. 755-758.  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. 33030r   W. Janowski, “Some Extremal Problems for Certain Fa- milies of Analytic Functions,” Annales Polonici Mathe-matici, Vol. 28, 1973, pp. 297-326.  R. Jurasiska and J. Stankiewics, “Coefficients in Some Classes Defined by Subordination to Multivalent Majo-rants,” Annales Polonici Mathematici, Vol. 80, 2003, pp. 163-170. where 23C3, which proves Theorem 3.3. If we put 12 in Theorem 3.3, we can get next Corollary.  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 .  P. Duren, “Univalent functions, A Series of Comprehen-sive Studies in Mathematics,” Vol. 259, Springer-Verlag, New York, 1983. 4. Acknowledgements The research was supported by Kyungsung University Research Grants in 2012.