 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 denote the class of analytic functions in the unit disk :1zz 01fS 00f which is normalized by and . Also let denote the sub- class of which is composed of functions which are univalent in . We say that is subordinate to in , written as F z , if and only if zFwz for some Schwarz function such that wz 00w and 1 wz . If z is univalent in , then the subordination F is equivalent to 00fF and . fF Definition 1.1. Let and be real numbers such that 01 . The function A belongs to the class , T if satisfies the following inequality: Re fz .z We remark that, for given real numbers and 01 , ,fT if and only if satis- fies each of the following two subordination relation- ships: zz 112 1 fz z and 112 1 z fz z z :p . Now, we define an analytic function by 1 πi2 1e 1log π1 iz pz z pp . (1) The above function was introduced by Kuroki and Owa [1] 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 zA and 01 . Then ,fT if and only if 1 2πi 1e 1ilog π1 z z fz p 1 1n n n pz Bz (2) in . And we note that the function , defined by (1), has the form , where 1 2πi i1 e. π n n Bn n (3) C opyright © 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 such that , fT 1 fT and , 1 , where is the inverse function of . 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 [5]) 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 pz 2 12 bzbz . If the function defined by is analytic in and if pz b ,zp z Re 0 pz , then in . pz Theorem 2.2. Let A, 12 1 and Re fz .z (4) Then 21 . 33 z 2 Re f z z (5) Proof. We put 2 21 33 and let 1 1 pz fz z p 01p . Then is analytic in and . And 11 ,, pz pz zz fz p zp z ,1 1.rsr s where Using (4), we have ,::Re:.pzzp zzww , with 2 12 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 2 222 2 111 21 2 1 :. 2 u E E is increasing on the interval Since 2 ,1 2 2 12 , for , we have 2 2 22 12 1, E E GG where 2 111 . 2 G :F Now we define a function by 2 22 1.FG G We note that is continuous on and is even. Since 00F and is decreasing on 0, for 12 1, 031FF for . Hence 2131 . 222 uF Therefore, 31 . 22 u Rei , And this shows that , for all Copyright © 2013 SciRes. APM  Y. J. SIM, O. S. KWON 411 031FF with 2 12 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 A1 , and Re fz z . (6) Then 21 . 33 z 2 Re f z z (7) Proof. We put 2 21 33 1 and note that for 1 . And let 1 1 pz fz z 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 2 22 111 2 1 :. 2 u E 22 21 E Since is decreasing on the interval 22 ,1 , for 2 12 , we have 2 2 22 12 1, EE GG where 2 11 . 1 2 G :F Now we define a function by 2 22 1.FG G We note that for . Hence is continuous on and is even. Since and 00F is increasing on 0, for 1 , 2131 . 222 uF Therefore, 31. 22 u Rei , And this shows that , for all 2 12 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 be a function in the class ,T . Then 22 21 21 Re . 33 33 fz z z 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 1 n n n qz Bz and the following Lemma will be needed. Lemma 3.1. (see Rogosinski [6]) Let be analytic and univalent in and suppose that qz 1 n n n pz Az maps onto a convex domain. If is analytic in and satisfies the following subordina- tion: .pz qzz Then 1. n AB n Theorem 3.2. Let and be real numbers such that 01 1 , n n n fzzaz T . If the function , then Copyright © 2013 SciRes. APM  Y. J. SIM, O. S. KWON 412 1 1 22 n B an n 2,3,, Bn (8) where 1 B is given by 1 2 π B 1 sin π. Proof. Let us define qz z (9) and 1 2πi 1e og π1 z pz z 1i l . (10) Then, the subordination (2) can be written as follows: .z z pz 1 1n n nBz qz p (11) Note that the function defined by (10) is con- vex in and has the form pz , where .Bn 1 1, n n n Az 1 2πi i1e π n nn If we let qz then by Lemma 3.1, we see that the subordination (11) implies that 1,n n AB where 1 2 π B 1 sin π. Now, the equality (9) implies that 2 zqz n. And if is even, the coefficient of in both sides lead to n z 2 21 2 , n n A A 121 22 nn n naAA A which is the sum of 2n terms. Hence, 121 22 111 11 22 222 22, nnn naAA A BB B BnB 2 21 2 n n A A n which leads to the inequality (8). If is odd, 2 121123212 2, nnn nn n naAA AAAA which is the sum of 12n terms in the bracket. Hence, we get 2 121 12 3212 222 111 1 11 2 2 22, n nn nn n na AAAA AA BBB B BnB 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 [7]) Let 2 12 1pzcz cz be a function with positive real part in . Then, for any complex number , 2 21 2max1;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 2n n zz az be in the class ,T . Then, for a complex number , 2 32 1 2πi 41 sin π 3π 11 max 1;e, 22 aa where i1 3. 2π qz Proof. Let us consider a function given by .qz z (12) ,fT Then, since , we have ,qz pzz where 1 2πi 1 1e 1ilog π1 1n n n z pz z Bz n B with is given by (3). Let 1 2 12 1 11. 1 pqz hzhzhz pqz 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 3 1 2πi 41 sin π 3π 11 max 1;e, 22 b 1 1 hz phz . (13) qz We find from the equations (12) and (13) that 211 1 2 aBh and 2222 21 11 1 , 12 Bh Bh 31 211 11 1 366 aBhBh which imply that 22 121 , Bhh 32 1 3 aa where 2 1 22 B B11 113 . 4 4 BB Applying Lemma 3.3, we can obtain 22 121 1; 12. Bhh 32 1 1 3 2max 3 aa B (14) And substituting 1 2πi i1e 1π B (15) and 1 4πi i1e 01 22π B (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 , given by 2 n , n n zaz T be in the class , . Also let the function 1 1 , defined by 1 fz z ff z (17) be the inverse of . If 1 2 n n n fwwbw 00 1 ;, 4 wrr (18) then 2 2 π b 1 sin π and where 5i 2π 22 ba . Proof. The relations (17) and (18) give 2 323 2.baa and Thus, we can get the estimate for 2 b by 221 21 sin π, π baB 2 immediately. Furthermore, an application of Theorem 3.4 (with ) gives the estimates for 3 b ,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 be given by 01 , let 2 n n n zz az ,T be in the class . Then 2 2sin1 sin π a (19) and 3 27 sin1sin π3 a (20) with 1π . ,fT Proof. If ,fT , then and ,gT 1 2 . n n n , where zfzz bz Hence :zfzQpzz and ,:zgz zLpz pz is given by (1). Let where 1 2 12 1 11 1 pQz hzhzhz pQz Copyright © 2013 SciRes. APM  Y. J. SIM, O. S. KWON Copyright © 2013 SciRes. APM 414 and 1 1 1 1 pLz kz k pLz 2 12 1. z kz h k Then and are analytic and have positive real part in . Also, we have 1 1 hz phz Qz and 1. 1 kz pkz Lz By suitably comparing coefficient, we get 211 1 2 aBh (21) 2222 21 11 1 1 6 12 Bh Bh 31 211 11 36 aBhBh (22) 211 1 2 bBk (23) and 2222 21 11 1 , 12 Bk Bk B B 11 .hk 31 211 11 1 36 6 bBkBk (24) where 1 and 2 are given by (15) and (16), respec- tively. Now, considering (21) and (23), we get 2 121 .hBB (25) Also, from (22),(23),(24) and (25), we find that 2 2122 4aBhk (26) Therefore, we have 22 121 hBB 21 22 121 4 44. aBhk BBB This gives the bound on 2 as asserted in (19). Now, further computations from (22), (24)-(26) lead to a 2 3122121 17 5. 12 12 aBhkhBB This equation, together with the well-known estimates [8]: , 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 [1] K. Kuroki and S. Owa, “Notes on New Class for Certain Analytic Functions,” RIMS Kokyuroku 1772, 2011, pp. 21-25. [2] H. M. Srivastava, A. K. Mishra and P. Gochhayat, “Cer- tain Subclasses of Analytic and Bi-Univalent Functions,” Applied Mathematics Letters, Vol. 23, No. 10, 2010, pp. 1188-1192. doi:10.1016/j.aml.2010.05.009 [3] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, “Coefficient Estimates for a Certain Subclass of Analytic and Bi-Uni- valent Functions,” Applied Mathematics Letters, Vol. 25, No. 6, 2012, pp. 990-994. doi:10.1016/j.aml.2011.11.013 [4] R. M. Ali, K. Lee, V. Ravichandran and S. Supramaniam, “Coefficient Estimates for Bi-Univalent Ma-Minda Star- like and Convex Functions,” Applied Mathematics Letters, Vol. 25, No. 3, 2012, pp. 344-351. [5] S. S. Miller and P. T. Mocanu, “Differential Subordina- tions, Theory and Applications,” Marcel Dekker, 2000. [6] W. Rogosinski, “On the Coefficients of Subordinate Func- tions,” Proceeding of the London Mathematical Society, Vol. 2, No. 48, 1943, pp. 48-62. [7] F. Keogh and E. Merkers, “A Coefficient Inequality for Certain Classes of Analytic Functions,” Proceedings of the American Mathematical Society, Vol. 20, No. 1, 1969, pp. 8-12. doi:10.1090/S0002-9939-1969-0232926-9 [8] P. Duren, “Univalent Functions,” Springer-Verlag, New York, 1983.
|
