Int. J. Open Problems Comput. Sci. Math., Vol. 1, No. 1, June 2008 On Some α-convex Functions Mugur Acu 1, Fatima Al-Oboudi 2, Maslina Darus 3, Shigeyoshi Owa 4 Yaşar Polato glu 5 and Emel Yavuz 5 1 Department of Mathematics, University Lucian Blaga of Sibiu, Sibiu, Romania 2 Department of Mathematics, Girls College of Education,Riyadh, Saudi Arabia 3 School of Mathematical Sciences, Universiti Kebangsaan Malaysia,Malaysia e-mail maslina@ukm.my 4 Department of Mathematics Kinki University Higashi-Osaka, Osaka 577-8502 Japan 5 Department of Mathematics and Computer Science,.Istanbul Kültür University, Turkey Abstract In this paper, we define a general class of α-convex functions, denoted by ML β,α q, with respect to a convex domain D qz H u U, q0 = 1, qu = D contained in the right half plane by using the linear operator D β defined by D β : A A, D β fz = z + 1 + j 1 β a j z j, where β, R, β 0, 0 and fz = z+ a j z j. Regarding the class ML β,α q, we give a inclusion theorem and a transforming theorem, from which we may obtain many particular results. Keywords: α-convex functions, generalized Libera integral operator, Briot- Bouquet differential subordination, modified Sălăgean operator. 1 Introduction Let HU be the set of functions which are regular in the unit disc U, A = {f HU : f0 = f 0 1 = 0}, H u U = {f HU : f is univalent in U} and S = {f A : f is univalent in U}.
2 Mugur Acu et al. Let D n be the Sălăgean differential operator see [14] defined as: D n : A A, D 1 fz = Dfz = zf z, Remark If f A, fz = z + n N and D 0 fz = fz D n fz = DD n 1 fz. a j z j, then D n fz = z + j n a j z j is also in A. Let n N and 0. Let denote with D n the Al-Oboudi operator see [7] defined by D n : A A, D 0 fz = fz, D 1 fz = 1 fz + zf z = D fz, D n fz = D D n 1 fz. We observe that D n is a linear operator and for fz = z + a j z j we have D n fz = z + 1 + j 1 n a j z j. The aim of this paper is to define a general class of α-convex functions with respect to a convex domain D, contained in the right half plane, by using a modified Sălăgean operator,which is also a modified Al-Oboudi operator, and to obtain some properties of this class. 2 Preliminary results We recall here the definitions of the well - known classes of starlike functions, convex functions and α-convex functions see [8] { S = f A : Re zf } z fz > 0, z U, S c = CV = K = { f A ; Re 1 + zf } z > 0, z U f z, M α = {f A ; Re Jα, f; z > 0, z U, α R}, where Jα, f; z = 1 α zf z fz + α 1 + zf z f z
On Some α-convex Functions 3 We observe that M 0 = S and M 1 = S c where S and S c are the class of starlike functions, respectively the class of convex functions. Remark By using the subordination relation, we may define the class M α thus if fz = z+a 2 z 2 +..., z U, then f M α if and only if Jα, f; z 1 + z 1 z, z U, where by we denote the subordination relation. Let consider the generalized Libera integral operator L a : A A defined as: fz = L a F z = 1 + a z F t t a 1 dt, a C, Re a 0. 1 z a 0 In the case a = 1 this operator was introduced by Libera [7] and it was studied by many authors in different general cases. In this general form a C, Re a 0 was used first time by Pascu in [13]. The next theorem is result of the so called admissible functions method introduced by Miller and Mocanu see [10], [11], [12]. Theorem Let q be convex in U and Re[βqz + γ] > 0, z U. If p HU with p0 = q0 and p satisfied the Briot-Bouquet differential subordination pz + zp z βpz + γ qz, then pz qz. Definition 2.1 [4] Let β, R, β 0, 0 and fz = z + denote by D β the linear operator defined by a j z j. We D β : A A, D β fz = z + 1 + j 1 β a j z j. Definition 2.2 [4] Let qz H u U, with q0 = 1 and qu = D, where D is a convex domain contained in the right half plane, β, R, β 0 and 0. We say that a function fz A is in the class SL β q if fz D β fz qz, z U. Theorem 2.1 [4] Let β, R, β 0 and 1. If F z SL β q then fz = L a F z SL β q, where L a is the integral operator defined by 1.
4 Mugur Acu et al. Definition 2.3 [5] Let qz H u U, with q0 = 1 and qu = D, where D is a convex domain contained in the right half plane, β, R, β 0 and 0. We say that a function fz A is in the class SL c β q if D β+2 fz qz, z U. fz Theorem 2.2 [5] Let β, R, β 0 and 1. If F z SL c β q then fz = L a F z SL c β q, where L a is the integral operator defined by 1. 3 Main results Definition 3.1 Let qz H u U, with q0 = 1, qu = D, where D is a convex domain contained in the right half plane, β 0, 0 and α [0, 1]. We say that a function fz A is in the class ML β,α q if. J β, α, f; z = 1 α Dβ+1 fz D β fz + α Dβ+2 fz fz qz, z U Remark 3.1 Geometric interpretation: fz ML β,α q if and only if J β, α, f : z take all values in the convex domain D contained in the right half-plane. Remark 3.2 We have ML β,0 q = SL β q and ML β,1q = SL c β q. Remark 3.3 It is easy to observe that if we choose different function qz we obtain variously classes of α-convex functions, such as for example, for = 1 and β = 0, the class of α-convex functions, the class of α-uniform convex functions with respect to a convex domain see [3], and, for = 1 and β = n N, the class UD n,α b, γ, b 0, γ [ 1, 1, b + γ 0 see [2], the class of α-n-uniformly convex functions with respect to a convex domain see [3]. Remark 3.4 For q 1 z q 2 z we have ML β,α q 1 ML β,α q 2. From the 1 + z above we obtain ML β,α q ML β,α. 1 z Remark 3.5 It is easy to observe that for every β 0, α [0, 1] and 0 we have idz ML β,α q, where idz = z, z U. Theorem 3.1 Let qz H u U, with q0 = 1, qu = D, where D is a convex domain contained in the right half plane, β 0 and 0. For all α, α [0, 1], with α < α, we have ML β,α q ML β,α q.
On Some α-convex Functions 5 Proof. From fz ML β,α q we have J β, α, f; z = 1 α Dβ+1 fz D β fz + α Dβ+2 fz fz qz, 2 where qz is univalent in U with q0 = 1 and maps the unit disc U into the convex domain D contained in the right half-plane. Define the function for fz A with Note that z pz = Dβ+1 fz D β fz = 1 + p 1 z + fz = z + a j z j. fz = z + j 1 + j 1 β+1 a j z j = z + = z + = 1 + j 1 β+1 a j z j + = z + fz z + = = fz + 1 fz + 1 j 1 + 1 1 + j 1 β+1 a j z j j 1 1 + j 1 β+1 a j z j j 1 1 + j 1 β+1 a j z j j 1 1 + j 1 β+1 a j z j 1 + j 1 1 1 + j 1 β+1 a j z j fz 1 1 + j 1 β+1 a j z j + 1 1 + j 1 β+2 a j z j = fz 1 = fz 1 Dβ+1 = 1 = 1 Dβ+1 1 fz z + 1 D β+2 fz z fz + z + 1 Dβ+2 fz z fz + 1 Dβ+2 fz fz + D β+2 fz.
6 Mugur Acu et al. Similarly we have z D β fz = z + j 1 + j 1 β a j z j = 1 1D β fz + Dβ+1 fz. Thus we see that pz + α zp z pz = Dβ fz fz +α 1 fz + D β+2 fz 1Dβ fz D β fz = 1 α Dβ+1 fz D β fz = J β, α, f; z. + α Dβ+2 fz fz fz + Dβ+1 From 2 we have pz + zp z qz, 1 α pz with p0 = q0, Re qz > 0, z U, α > 0 and 0. In this conditions from Theorem 2 we obtain pz qz or pz take all values in D. If we consider the function g : [0, α ] C, gu = pz + u zp z pz with g0 = pz D and gα = J β, α, f; z D, it easy to see that, fz Thus we have or gα = pz + α zp z pz J β, α, f; z qz fz ML β,α q. D, 0 α < α. From the above theorem we have Corollary 3.1 For every β 0, 0 and α [0, 1], we have ML β,α q ML β,0 q = SL βq.
On Some α-convex Functions 7 Theorem 3.2 Let qz H u U, with q0 = 1, qu = D, where D is a convex domain contained in the right half plane, β 0, α [0, 1] and 1. If F z ML β,α q then fz = L a F z SL β q, where L a is the integral operator defined by 1. Proof. From 1 we have Note that or and = a fz + 1 1 + af z = afz + zf z. 1 + a F z = a fz + z 1 fz + D β+2 fz fz, 1 + a F z = a + 1 1 fz + D β+2 fz 1 + ad β F z = a + 1 1 Dβ With the following definition for pz, fz D β fz = pz, p0 = 1, fz + Dβ+1 fz. we obtain zp zd β fz + zpz D β fz = z fz. This implies that zpz D β fz+ 1pzDβ fz+pzdβ+1 fz = 1 fz+d β+2 fz. Therefore, we have that zpz D β fz fz β D + 1pz that is, that D β+2 fz fz = 1 pz Therefore, we obtain β+2 fz D fz + pz = 1 + fz fz, pz 2 + zp z. F z D β F z = pz2 + zp z + a + 1 1 pz pz + a + 1 1 zp z = pz + pz + a + 1 1,
8 Mugur Acu et al. where a C, Re a 0, β 0 and 1. If we denote Dβ+1 F z D β F z = hz, with h0 = 1, we have from F z ML β,α q see the proof of the above Theorem: or J β, α, F ; z = hz + α zh z hz qz Using the hypothesis, from Theorem 2, we obtain hz qz zp z pz + pz + a + 1 1 qz, where a C, Re a 0 and 1. By using the Theorem 2 and the hypothesis we have or pz qz fz D β fz qz, where β 0 and 1. This means fz = L a F z SL β q. Remark 3.6 It is easy to observe that if we choose different values for the parameters β and, and different functions qz, in the present results, we obtain variously previously results for example see [2], [3]. 4 Open problem In [1] the author define a generic class of analytical functions from which it is easy to obtain a great number of subclasses of starlike, convex and close to convex functions, and so to study properties for this generic class and simply translate them to the corresponding subclasses. We recall here the main ideas of this paper: We have already see in the Preliminary results section the definitions for the classes SL β q and SLc β q. More, in [6] the authors consider the following general class of close to convex functions:
On Some α-convex Functions 9 Definition 4.1 Let qz H u U, with q0 = 1 and qu = D, where D is a convex domain contained in the right half plane, β, R, β 0 and 0. We say that a function fz A is in the class CCL β q, with respect to the function gz SL Dβ+1 fz β q, if D β gz qz, z U. We observe that the conditions from the definitions of the classes SL β q, SL c β q, CCL βq can be written where pz = Dβ+1 fz D β fz pz qz or Dβ+2 fz fz or D β+1 fz D β gz. 3 From the above notations we have p0 = 1. This mean that instead of all three classes, we may define a generic class of analytic functions by CLASSp, q, β, = {f A : p HU having the form 3 satisfy pz qz}. Remark 4.1 In the case of the third form of pz from 3 it is sufficiently to us to consider that g is also in the class CLASSp, q, β,. The open problem is to define a generic class of analytical functions such that the class ML β,α q is contained inside and is possible to obtain, in a similarly way with the results from [1], where results for all the four general classes are included. References [1] M. Acu, On a generic class of analytic functions, Proceedings of Geometric Function Theory and its Applications - Istanbul 2007. [2] M. Acu, On a subclass of α-uniform convex functions, Archivum Mathematicum, no. 2, Tomus 41/2005, No. 2, 175-180. [3] M. Acu, Some subclasses of α-uniformly convex functions, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 212005, 49-54. [4] M. Acu and S. Owa, Note on a class of starlike functions, Proceedings of The International Short Joint Research Work on Study on Calculus Operators in Univalent Function Theory - Kyoto 2006, 1-10.
10 Mugur Acu et al. [5] M. Acu and S. Owa, Note on a class of convex functions, Proceedings of The International Short Joint Research Work on Study on Calculus Operators in Univalent Function Theory - Kyoto 2006, 11-20. [6] M. Acu, S. Owa, Note on a class of close to convex functions, International Journal of Computing and Mathematical Applications, Vol. 1, no. 1, June 2007, 1-10. [7] F. M. Al-Oboudi, On univalent funtions defined by a generalized S al agean operator, Ind. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436. [8] P. L. Duren, Univalent functions, Springer Verlag, Berlin Heildelberg, 1984. [9] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 161965, 755-758. [10] S. S. Miller and P. T. Mocanu, Differential subordination and univalent functions, Mich. Math. 281981, 157-171. [11] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet differential equation, J. Differential Equations 561985, 297-308. [12] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordination, Mich. Math. 321985, 185-195. [13] N. N. Pascu, Alpha-close-to-convex functions, Romanian-Finish Seminar on Complex Analysis, Bucharest 1976, Proc. Lect. Notes Math. 1976, 743, Spriger-Varlag, 331-335. [14] G. S. Sălăgean, On some classes of univalent functions, Seminar of geometric function theory, Cluj-Napoca, 1983.