 Advances in Pure Mathematics, 2013, 3, 409-414 http://dx.doi.org/10.4236/apm.2013.34059 Published Online July 2013 (http://www.scirp.org/journal/apm) A Certain Subclass of Analytic Functions with Bounded Positive Real Part Young Jae Sim, Oh Sang Kwon Department of Mathematics, Kyungsung University, Busan, South Korea Email: yjsim@ks.ac.kr, oskwon@ks.ac.kr Received March 6, 2013; revised April 11, 2013; accepted May 10, 2013 Copyright © 2013 Young Jae Sim, Oh Sang Kwon. This is an open access article distributed under the Creative Commons Attribu-tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT ,TFor real numbers  and  such that 01, we denote by  the class of normalized analytic func- tions which satisfy Re fzz,T , where denotes the open unit disk. We find some relationships involving functions in the class . And we estimate the bounds of coefficients and solve Fekete-Szegö problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or bi-uni- valent functions. Keywords: Functions of Bounded Positive Real Part; Fekete-Szegö Problem; Inverse Functions; Bi-Univalent Functions 1. Introduction Let A denote the class of analytic functions in the unit disk :1zz 01fS00f which is normalized by and . Also let denote the sub- class of A which is composed of functions which are univalent in . We say that f is subordinate to F in , written as fFz , if and only if fzFwz for some Schwarz function such that wz00w and 1wz . If zF is univalent in , then the subordination fF is equivalent to 00fF and .  fFDefinition 1.1. Let  and  be real numbers such that 01. The function fA belongs to the class ,T if f satisfies the following inequality: Re fz.z We remark that, for given real numbers  and  01 , ,fT if and only if f satis- fies each of the following two subordination relation- ships:  zz1121fz z and 1121zfz zz:p. Now, we define an analytic function by 1πi21e1logπ1izpz z pp. (1) The above function was introduced by Kuroki and Owa  and they proved maps onto a convex domain :Reww , conformally. Using this fact and the definition of subordination, we can ob- tain the following Lemma, directly. Lemma 1.1. Let fzA and 01. Then ,fT if and only if 12πi1e1ilogπ1zzfzp11nnnpz Bz (2) in . And we note that the function , defined by (1), has the form , where 12πii1 e.πnnBnn  (3) Copyright © 2013 SciRes. APM Y. J. SIM, O. S. KWON 410  and  such that For given real numbers 01,, we denote T the class of bi- univalent functions consisting the functions in A such that ,fT1fT and ,1, where f is the inverse function of f. In our present investigation, we first find some relation- ships for functions in bounded positive class ,TS,T. And we solve several coefficient problems including Fe- kete-Szegö problems for functions in the class. Further- more, we estimate the bounds of initial coefficients of in- verse functions and bi-univalent functions. For the coef- ficient bounds of functions in special subclasses of , the readers may be referred to the works [2-4]. 2. Relations Involving Bounds on the Real Parts In this section, we shall find some relations involving the functions in . And the following Lemma will be needed in finding the relations. Lemma 2.1 (see Miller and Mocanu ) Let  be a set in the complex plane and let b be a complex number such that . Suppose that a function satisfies the condition Re 0bi, ;z2: for all real 2,i  z2Rebb and all pz212bzbz . If the function defined by is analytic in and if pz b ,zp zRe 0pz, then in . pzTheorem 2.2. Let fA, 12 1 and Re fz.z (4) Then  21 .33z2Re f zz (5) Proof. We put 22133 and let  11pz fzzp01p. Then is analytic in and . And   11,,pzpz zzfzpzp z  ,1 1.rsr swhere   Using (4), we have ,::Re:.pzzp zzww ,  with 212 Rei ,Now, let . And we shall find the maximum value of  . Now, we put i,1 i 1:iuv u v, where and are real numbers. Then 221uv  and 21uv . Hence  2222 2111 2121:.2uEE  is increasing on the interval Since 2,1 2 212 , for , we have  2222121,EEGG  where 2111 .2G :F  Now we define a function by 2221.FG G    We note that F is continuous on and is even. Since 00F and F is decreasing on 0, for 12 1, 031FF for . Hence 2131.222uF Therefore, 31 .22uRei , And this shows that  , for all  Copyright © 2013 SciRes. APM Y. J. SIM, O. S. KWON 411031FF with 212 Re 0. By Lemma 2.1, we get pz in and this shows that the inequality (5) holds and the proof of Theorem 2.2 is completed. Theorem 2.3. Let fA1,  and Refz z. (6) Then  21 .33z2Re f zz (7) Proof. We put 221331 and note that  for 1. And let  11pz fzz  and  ,1rsr 1 .s And, we put  i,1 i 1  :i,uvu ve :, where and are real numbers. As in the proof of Theorem 2.2, we can get  ,: :Rpzzp zzww  by (6). And  22211121:.2uE2221 E Since  is decreasing on the interval 22,1 , for 212 , we have  2222121,EEGG where  211 .12G :F   Now we define a function by  2221.FG G   We note that for . Hence F is continuous on and is even. Since and 00FF is increasing on 0, for 1, 2131.222uF Therefore, 31.22uRei , And this shows that  , for all  212  with . By Lemma 2.1, we get Re 0pz  in and this shows that the inequality (7) holds and the proof of Theorem 2.3 is completed. By combining Theorem 2.2 and 2.3, we can get the following Theorem. Theorem 2.4. Let  and  be real numbers such that 121 and let f be a function in the class ,T. Then 2221 21Re .33 33fz zz 3. Coefficient Problems Involving Functions in T,,T In the present section, we will solve some coefficient problems involving functions in the class . And our first result on the coefficient estimates involves the function class ,T1nnnqz Bz and the following Lemma will be needed. Lemma 3.1. (see Rogosinski ) Let be analytic and univalent in and suppose that qz1nnnpz Az maps onto a convex domain. If is analytic in and satisfies the following subordina- tion:  .pz qzz Then 1.nAB n Theorem 3.2. Let  and  be real numbers such that 01 1,nnnfzzaz T. If the function  , then Copyright © 2013 SciRes. APM Y. J. SIM, O. S. KWON 412 1122nBann2,3,,Bn (8) where 1B is given by 12πB1sin π.  Proof. Let us define qzfz (9) and 12πi1eogπ1zpz z 1il . (10) Then, the subordination (2) can be written as follows: .z zpz11nnnBzqz p (11) Note that the function defined by (10) is con-vex in and has the form pz , where .Bn11,nnnAz12πii1eπnnn If we let qz then by Lemma 3.1, we see that the subordination (11) implies that 1,nnAB where 12πB1sin π. Now, the equality (9) implies that 2fzqzn. And if is even, the coefficient of in both sides lead to nz2212 ,nnA A12122nn nnaAA A  which is the sum of 2n terms. Hence, 12122111112222222,nnnnaAA ABB BBnB 2212nnA A  n which leads to the inequality (8). If is odd, 21211232122,nnn nn nnaAA AAAA   which is the sum of 12n terms in the bracket. Hence, we get  2121 12 3212222111 1112222,nnn nn nnaAAAA AABBB BBnB   which leads to the inequality (8). Therefore, the proof of Theorem 3.2 is completed. And now, we shall solve the Fekete-Szegö problem for ,fT and we will need the following Lemma: Lemma 3.3. (see Keogh and Merkers ) Let 2121pzcz cz  be a function with positive real part in . Then, for any complex number , 2212max1;12.cc  Now, the following result holds for the coefficient of ,fT01. Theorem 3.4. Let  and let the function zn given by 2nnzz az be in the class ff,T. Then, for a complex number , 23212πi41sin π3π11max 1;e,22aa   where i1 3.2πqz  Proof. Let us consider a function given by .qzfz (12) ,fTThen, since , we have  ,qz pzz where 12πi11e1ilogπ11nnnzpz zBz nB with is given by (3). Let 1212111.1pqzhzhzhzpqz Copyright © 2013 SciRes. APM Y. J. SIM, O. S. KWON 413Then is analytic and has positive real part in the open unit disk . We also have h312πi41sin π3π11max 1;e,22b 11hzphz. (13) qzWe find from the equations (12) and (13) that 21112aBh and 222221 111,12Bh Bh3121111 1366aBhBh which imply that 22121,Bhh3213aa where 2122BB11113.4 4BB  Applying Lemma 3.3, we can obtain 221211; 12.Bhh321132max3aaB (14) And substituting 12πii1e1πB (15) and 14πii1e0122πB (16) in (14), we can obtain the result as asserted. Using Theorem 3.4, we can get the following result. Corolla ry 3.1. Let  and let the function f, given by 2n,nnfzazT be in the class ,. Also let the function 1f1, defined by 1ffz zff z (17) be the inverse of f. If 12nnnfwwbw001;,4wrr  (18) then 22πb1sin π and where 5i2π22ba. Proof. The relations (17) and (18) give 23232.baa and Thus, we can get the estimate for 2b by 22121sin π,πbaB  2 immediately. Furthermore, an application of Theorem 3.4 (with ) gives the estimates for 3b,fT, hence the proof of Corollary 3.1 is completed. Finally, we shall estimate on some initial coefficients for the bi-univalent functions . Theorem 3.5. For given  and  such that f be given by 01 , let 2nnnzz az,Tf be in the class . Then  22sin1 sinπa (19) and  327sin1sinπ3a (20) with 1π. ,fTProof. If ,fT, then  and ,gT 12.nnn, where gzfzz bz  Hence  :zfzQpzz  and  ,:zgz zLpz pz is given by (1). Let where 12121111pQzhzhzhzpQz  Copyright © 2013 SciRes. APM Y. J. SIM, O. S. KWON Copyright © 2013 SciRes. APM 414 and  1111pLzkz kpLz2121.z kzh k Then and are analytic and have positive real part in . Also, we have 11hzphzQz and 1.1kzpkzLz By suitably comparing coefficient, we get 21112aBh (21) 222221 111 16 12Bh Bh312111136aBhBh (22) 21112bBk (23) and 222221 111,12Bk BkB B11.hk3121111 136 6bBkBk (24) where 1 and 2 are given by (15) and (16), respec- tively. Now, considering (21) and (23), we get  2121.hBB (25) Also, from (22),(23),(24) and (25), we find that 221224aBhk (26) Therefore, we have 22121hBB2122121444.aBhkBBB This gives the bound on 2 as asserted in (19). Now, further computations from (22), (24)-(26) lead to a23122121175.12 12aBhkhBB This equation, together with the well-known estimates : , and 12h22h22klead us to the inequality (20). Therefore, the proof of Theorem 3.5 is completed. 4. Acknowledgements The research was supported by Kyungsung University Re-search Grants in 2013. REFERENCES  K. Kuroki and S. Owa, “Notes on New Class for Certain Analytic Functions,” RIMS Kokyuroku 1772, 2011, pp. 21-25.  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