APPLYING RUSCHEWEYH DERIVATIVE ON TWO SUB-CLASSES OF BI-UNIVALENT FUNCTIONS
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1 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 68 APPLYING RUSCHEWEYH DERIVATIVE ON TWO SUB-CLASSES OF BI-UNIVALENT FUNCTIONS ABDUL RAHMAN S. JUMA 1, FATEH S. AZIZ DEPARTMENT OF MATHEMATICS, ALANBAR UNIVERSITY, RAMADI-IRAQ 1, DEPARTMENT OF MATHEMATICS, SALAHADDIN UNIVERSITY,ERBIL-IRAQ, Abstract. The Ruscheweyh derivative has been applied in this paper to investigate two subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. We find estimates on the coefficients a a 3 for functions in these subclasses. Keywords : Analytic univalent functions; Bi-univalent functions; λ - convex functions; Ruscheweyh derivative; Coefficient bounds. AMS Subject Classifications : 30C45 1. Introduction definitions Let Ω denote the class of all functions of the form: (1.1) f(z) = z + a n z n, which are analytic in the open unit disc U = {z : z < 1}. Let M(λ) denote the class of λ-convex functions in U defined as follows see [5]: M(λ) = {f Ω : Re[(1 λ) zf (z) f(z) + λ(1 + zf (z) f (z) )] > 0, λ 0}. Further, by S we shall denote the class of all functions in Ω which are univalent in U ( for details, see [3],[4],[10]). It is well known that every function f S has an inverse f 1, defined by f 1 (f(z)) = z (z U), f(f 1 (w)) = w ( w < r o (f) : r o (f) 1 4 ), where n= f 1 (w) = w a w + (a a 3 )w 3 (5a 3 5a a 3 + a 4 )w A function f(z) Ω is said to be bi-univalent in U if both f(z) f 1 (z) are univalent in U see [10]. 1 dr juma@hotmail.com fatehsaber@gmail.com December 01 IJENS I J E N S
2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 69 Let Σ denote the class of bi-univalent functions in U given by (1.1). Brannan Taha [] (see also [6]) introduced certain subclasses of the bi-univalent function class Σ similar to the familiar subclasses δ (α) K(α) of starlike convex functions of order α(0 < α 1), respectively (see [7]). Thus, following Brannan Taha [] (see also [6]), a function f(z) Ω is in the class δσ (α) of strongly bi-starlike functions of order α(o < α 1) if each of the following conditions is satisfied: f Σ, arg{ zf (z)) f(z) < απ (0 < α 1, z U), arg{ wg (w)) g(w) < απ (0 < α 1, w U) where g is the extension of f 1 to U. The classes δσ (α) K Σ(α), of bi-starlike functions of order α bi-convex functions of order α, corresponding (respectively) to the function classes δ (α) K(α), were also introduced analogously. For each of the function classes δσ (α) K Σ (α), they found non-sharp estimates on the first two Taylor-Maclaurin coefficients a a 3. The object of the present paper is to introduce two subclasses of the function class Σ applying the Ruscheweyh derivative, where Ruscheweyh [9] observed that (1.) = z(zn 1 f(z)) (n), n! for n N o = {0, 1,,...}. This symbol, n N o is called by Al- Amiri [1], the n th order Ruscheweyh derivative of f(z). We note that D o f(z) = f(z), D 1 f(z) = zf (z) (1.3) = z + where σ(n, k)a k z k, k= ( ) n + k 1 (1.4) σ(n, k) =, n find estimates on the coefficients a a 3 for functions in these subclasses of the function class Σ employing the techniques used by Xiao-FeiLi et al.[11]. For deriving our main results, the following lemma needed to be mentioned [8]. Lemma 1.1. If h P then c k for each k, where P is the family of all functions h analytic in U for which Re(h(z)) > 0, h(z) = 1 + c 1 z + c z + c 3 z for z U December 01 IJENS I J E N S
3 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No: Coefficient bounds for the function class F Σ (α, λ) Definition.1. A function f(z) given by (1.1) is said to be in the class F Σ (α, λ) if the following conditions are satisfied: (.1) f Σ, arg{(1 λ) z(dn f(z)) () ]} < απ (0 < α 1, λ 0, z U), (.) arg{(1 λ) w(dn g(w)) D n g(w) + λ[1 + w(dn g(w)) (D n g(w)) ]} < απ (0 < α 1, λ 0, w U), where the function g is the extension of f 1 given by (.3) g(w) = w a w + (a a 3 )w 3 (5a 3 5a a 3 + a 4 )w We note that for n = 0 the class F Σ (α, λ) reduces to the class B Σ (α, λ) introduced studied by Xiao-FeiLi et al.[11]. Theorem.. Let f(z) given by (1.1) be in the class F Σ (α, λ), 0 < α 1 λ 0. Then (.4) a (.5) a 3 Proof: α α(1 4nλ λ ) αn (λ + 4λ + 1) + (n + 1) (λ + 1), α (n + 1)(n + )(1 + λ) + 4α (n + 1) (λ + 1). We can write the argument inequalities in (.1) (.) equivalently as follows: (.6) (1 λ) z(dn f(z)) (.7) (1 λ) w(dn g(w)) D n g(w) () ] = [p(z)] α, + λ[1 + w(dn g(w)) (D n g(w)) ] = [q(w)] α, where p(z) q(w) satisfy the following inequalities Re(p(z)) > 0 (z U) Re(q(w)) > 0 (w U). Furthermore, the functions p(z) q(w) have the forms (.8) p(z) = 1 + p 1 z + p z + p 3 z , (.9) q(w) = 1 + q 1 w + q w + q 3 w December 01 IJENS I J E N S
4 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 71 And g(w) is given as in (.3). Now equating the coefficients in equations (.6) (.7), we get (.10) (n + 1)(1 + λ)a = p 1 α, (.11) (n + 1)(n + )(1 + λ)a 3 = p α + (.1) (n + 1)(1 + λ)a = q 1 α, (.13) (n + 1)(n + )(1 + λ)(a a 3 ) = q α + From equations (.10) (.1), we get (.14) p 1 = q 1, also we get (.15) (n + 1) (λ + 1) a = α (p 1 + q 1). From (.11),(.13) (.15) we obtain (.16) a = α(α 1) p λ (1 + λ) p 1α. α(α 1) q λ (1 + λ) q 1α. α (p + q ) α(1 4nλ λ ) αn (λ + 4λ + 1) + (n + 1) (λ + 1). Applying lemma (1.1) for the coefficients p q, we get (.17) a α α(1 4nλ λ ) αn (λ + 4λ + 1) + (n + 1) (λ + 1). Next, in order to find the bound on a 3, by subtracting (.13) from (.11), we get (.18) (n + 1)(n + )(1 + λ)a 3 (n + 1)(n + )(1 + λ)a = α(p q ) + α(α 1) (p 1 q 1) λ (1 + λ) α (p 1 q1). Upon substituting the value of a from (.15) observing that p 1 = q 1 it follows that (.19) a 3 = α(p q ) (n + 1)(n + )(1 + λ) + α (p 1 + q1) (n + 1) (λ + 1), Applying lemma (1.1) once again for the coefficients p 1, p, q 1 q, we readily get (.0) a 3 This completes the proof of Theorem.. α (n + 1)(n + )(1 + λ) + 4α (n + 1) (λ + 1), Putting n = 0 in Theorem. we have December 01 IJENS I J E N S
5 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 7 Corollary.3. Let f(z) given by (1.1) be in the class M Σ (α, λ), 0 < α 1, λ 0, z U. Then (.1) a α α(1 λ ) + (1 + λ), (.) a 3 α 1 + λ + 4α (1 + λ).. 3. Coefficient bounds for the function class F Σ (β, λ) Definition 3.1. A function f(z) given by (1.1) is said to be in the class F Σ (β, λ) if the following conditions are satisfied: (3.1) (3.) f Σ, Re{(1 λ) z(dn f(z)) Re{(1 λ) w(dn g(w)) D n g(w) where the function g(w) is given as in (.3). () ]} > β (0 β < 1, λ 0, z U), + λ[1 + w(dn g(w)) (D n g(w)) ]} > β (0 β < 1, λ 0, w U), We note that for n = 0 the class F Σ (β, λ) reduces to the class B Σ (β, λ) introduced studied by Xiao-FeiLi et al.[11]. Theorem 3.. Let f(z) given by (1.1) be in the class F Σ (β, λ), 0 β < 1 λ 0. Then (1 β) (3.3) a (n + 1)((1 n)λ + 1), (3.4) a 3 The argument inequalities in (3.1) (3.) equivalently can be written as follows: Proof: (1 β) (n + 1)(n + )(1 + λ) + 4(1 β) (1 + n) (1 + λ). (3.5) (1 λ) z(dn f(z)) () ] = β + (1 β)p(z), (3.6) (1 λ) w(dn g(w)) D n + λ[1 + w(dn g(w)) g(w) (D n g(w)) ] = β + (1 β)q(w), December 01 IJENS I J E N S
6 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 73 where g(w), p(z), q(w) have the forms (.3),(.8) (.9) respectively. Equating coefficients in equations (3.5) (3.6) yields (3.7) (n + 1)(1 + λ)a = p 1 (1 β), (3.8) (n + 1)(n + )(1 + λ)a 3 = p (1 β) λ (1 + λ) p 1(1 β). (3.9) (n + 1)(1 + λ)a = q 1 (1 β), (3.10) (n + 1)(n + )(1 + λ)(a a 3 ) = q (1 β) λ (1 + λ) q 1(1 β). From equations (3.7) (3.9), we get (3.11) p 1 = q 1, also we get (3.1) (n + 1) (λ + 1) a = (1 β) (p 1 + q 1). Now adding (3.8) to (3.10) gives (3.13) (n + 1)(n + )(1 + λ)a = (1 β)(p + q ) λ (1 + λ) (p 1 + q 1)(1 β), substituting value of p 1 + q 1 from (3.1) in (3.13) we get (3.14) a = (1 β)(p + q ) (n + 1)(1 + λ(1 n)). Applying lemma (1.1) for the coefficients p q we have (1 β) (3.15) a (n + 1)((1 n)λ + 1), Next, in order to find the bound on a 3, by subtracting (3.10) from (3.8) we get (3.16) (n + 1)(n + )(1 + λ)a 3 = (1 β)(p q ) + (n + 1)(n + )(1 + λ)a, putting value of a from (3.1) in (3.16) we get (3.17) a 3 = (1 β)(p q ) (n + 1)(n + )(1 + λ) + (1 β) (p 1 + q1) (n + 1) (λ + 1), Applying lemma (1.1) once again for the coefficients p 1, p, q 1 q, we readily get (3.18) a 3 This completes the proof of Theorem 3.. (1 β) (n + 1)(n + )(1 + λ) + 4(1 β) (1 + n) (1 + λ), Putting n = 0 in Theorem 3. we have December 01 IJENS I J E N S
7 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 74 Corollary 3.3. Let f(z) given by (1.1) be in the class M Σ (β, λ), 0 β < 1, λ 0, z U. Then (3.19) a (1 β) 1 + λ, (3.0) a 3 1 β 1 + λ + 4(1 β) ) (1 + λ).. References [1] H.S. Al-Amiri, On Ruscheweh derivatives, Ann. Poln. Math., 38 (1980) pp [] D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis its Applications, Kuwait; February 18-1, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, (1988) pp see also Studia Univ. Babes-Bolyai Math., 31() (1986) pp [3] D. Breaz, N. Breaz, H.M. Srivastava, An extention of the univalent condition for a family of integral operators, Appl. Math. Lett. (009) pp [4] P.L. Duren, Univalent functions, in: Grunddlehren der mathematischen Wissenschaften, B 59, Springer -Verlag, New York, Berlin, Hidelberg Tokyo, [5] Fekete M., Szegö G.,Eine Bermerkung uber ungeraade schlichte funktionen [J].J., London Math. Soc., 8 (1933) pp [6] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18(1967) pp [7] E. Netanyahu, The minimal distance of the image boundary from the origin the second coefficient of a univalent function in z < 1, Arch. Rational Mech. Anal., 3 (1969) pp [8] Ch. Pommerenke, Univalent functions, Venhoeck Ruperchi, Göttingen (1975). [9] S. Ruscheweyh, New criteria for univalent functions, Proc.Amer.Math. Soc., 49 (1975) pp [10] H.M. Srivastava, A.K. Mishra, P. Gochhayat, Certain subclasses of analytic bi-univalent functions, Appl. Math. Lett., 3 (010) pp [11] Xiao-Fei-li An-Ping Wang, Two new subclasses of bi-univalent funcions, International Mathematical Forum, Vol. 7, no. 30 (01) pp December 01 IJENS I J E N S
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