Coefficient Inequalities for Classes of Uniformly. Starlike and Convex Functions Defined by Generalized. Ruscheweyh Operator

Transcription

1 International Mathematical Forum, Vol. 6, 2011, no. 65, Coefficient Inequalities for Classes of Uniformly Starlike Convex Functions Defined by Generalized Ruscheweyh Operator Hesam Mahzoon Department of Mathematics Firoozkooh Branch,Islamic Azad university Firoozkooh, Iran mahzoon Abstract. We introduce study the classes V β, b, δ VD α, β, b, δ of analytic functions which are defined by making use of the generalized Ruscheweyh derivative operator. Coefficient inequalities conditions are investigated for these classes. Mathematics Subject Classification: 30C45 Keywords: Uniformly starlike, Coefficient Inequalities Generalized Ruscheweyh derivative

2 3228 H. Mahzoon 1. INTRODUCTION Let A be the class of functions f of the form 1.1 =z + n=2 a n z n which are analytic in the open unit disk U = z C : z < 1}. In the present paper, we use a recent generalization of the Ruscheweyh derivative [1] denoted by D δ, defined as follows: D δ = z 1 z δ D f, f A, z U where sts for the convolution or Hadamard product of two power series. Further, we have, D =1 +zf z,δ > 1, 0, z U. We can easily see that D δ admits a representation of the form D δ =z + [n 1]B n δa n z n where, n=2 δ + 1δ +2...δ + n 1 B n δ =. n 1! We introduce the class V β, b, δ as the subclass of A consisting of functions f obeying the condition 1.2 R 1 2 b + 2 b } D δ >β, z U D δ+1 where b is a non-zero complex number, 0 β<1, 0 δ> 1. This class generalizes the class Vβ, b, δ [4] is of special interest for, it contains many well known as well as new classes of analytic functions. It provides a transition from starlike functions to convex functions more specifically V 0 β,2, 0 is the family of starlike functions of order β V 0 β,1, 1 is the class of convex functions of

3 Coefficient inequalities 3229 order β. Shams, Kulkarni Jahangiri [8] introduced the subclass SDα, β ofa consisting of functions f satisfying } zf z R >α zf 1.3 z 1 + β, z U for some αα 0 β0 β<1. The class KDα, β, another subclass of A, is defined as the set of all functions f obeying } zf z R >α zf 1.4 z f z f z + β, z U for some αα 0 β0 β<1. We introduce the class VD α, β, b, δ as the subclass of A consisting of functions f which satisfy R 1 2 b + 2 D δ+1 } b D δ >α 2 D δ+1 b D δ b + β, z U where b is a non-zero complex number for some α 0, 0 β<1, 0 δ> 1. For the parametric values =0; =0,b=2,δ = 0 =0,b= δ = 1 we obtain the classes VDα, β, b, δ, SDα, β KDα, β respectively. 2. MAIN RESULTS We prove some coefficient inequalities for functions in the class VD α, β, b, δ. Theorem 2.1. If f VD α, β, b, δ with 0 α β, then f V β α 1 α,b,δ. Proof. Since Rω ω for any complex number ω, f VD α, β, b, δ implies that R 1 2 b + 2 D δ+1 } b D δ >α 2 D δ+1 b D δ 2 b + β, z U Equivalently, R 1 2 b + 2 D δ+1 } b D δ > β α,, z U. 1 α If 0 α β, then we have, 0 β α 1 α < 1.

4 3230 H. Mahzoon Corollary 2.2. For the parametric value =0, we get Theorem 2.1 in [3] which reads : If f VDα, β, b, δ with 0 α β, then f V β α,b,δ. 1 α Corollary 2.3. For the parametric values b =2,δ =0 =0, we get Theorem 2.1 in [5] which reads : If f SDα, β with 0 α β, then f S β α. 1 α Corollary 2.4. The parametric values b = δ = 1 = 0yield Corollary 2.2 in [6] stated as: If f KDα, β with 0 α β, then f K β α. 1 α Theorem If f VD α, β, b, δ, then b 1 β n 1[1 + n 1]B n δ bδ + 11 β n 3. Proof. We note that for f VD α, β, b, δ R 1 2 b + 2 D δ+1 } b D δ > β α, z U. 1 α If we define the function p by [ 1 α 1 2 b + 2 D δ+1 ] b D δ pz = β α 1 β z U then p is analytic in U with p0 = 1 Rpz > 0, z U. Let pz =p 1 z + p 2 z , then we have, 1 2 b + 2 b D δ+1 1 βpz+β α D δ = 1 α = 1 β 1 α n=1 p n z n.

5 That is, D δ+1 D δ+1 = bd δ+1 Coefficient inequalities 3231 [ 1 β 1 α n=1 ] p n z n. Therefore, 2.3 implies that 2[n 1] B n δn 1a n δ +1 b1 β = 1 α [p n 1 +B 2 δa 2 p +2B 3 δa 3 p n []B n 1 δa n 1 p 1 ] Applying the coefficient estimates such that p n 2 functions [2], we obtain that n 1 for Caratheodory a n n 1[1 + n 1]B n δ [B 2δ a B 3 δ a [n 2]B n 1 δ a n 1 ] For n = 2, we have b 1 β 1 + which proves 2.1. For n =3, a 3 [ B 3 δ ]

6 3232 H. Mahzoon Therefore 2.2 holds for n = 3. Suppose that 2.2 is true for n = k. Consider, a k+1 k1 + kb k+1 δ + 2 = k 1 k1 + kb k+1 δ k 2 k 1 }. Therefore, the result is true for n = k + 1. Using mathematical induction, 2.2 holds true for all n 3. Corollary 2.6. The parametric value =0yields Theorem 2.4 in [3] which states that: If f VDα, β, δ, then n 1[1 + n 1]B n δ b 1 β Since VD 0,β,b,δ V β, b, δ, we have the following Corollary. Corollary If f V β, b, δ, then n 1[1 + n 1]B n δ b 1 β 1 +, n 3. b δ + 11 β, n 3. j

7 Coefficient inequalities 3233 Corollary 2.8. For the parametric values b =2,δ=0 =0, we get Theorem 2.3 in [5] which states that: If f SDα, β, then Corollary a result by Robertson [6]. 21 β n 1 21 β Putting α =0in Corollary 2.8, we get n j=2 21 β, n 3. j 2β, n 2. n 1! Corollary For the parametric values b = δ =1 =0, we obtain Corollary 2.5 in [5] given by : If f KDα, β,then β nn 1 1 β 21 β, m 3. Putting α =0in Corollary 2.10, we get the inequality by Robert- Corollary son [6] given by : 2.13 n j=2 j 2β, n 2. n! References [1] Al-Shaqsi. K, Darus. M, On certain subclass of analytic univalent of functions with negative cofficients, Appl. Math. Sci., Vol. 1, 2007, No. 3, [2] Caratheodory. C, Über den variabilitätsbereich der Fourier schen konstanten von possitiven harmonischen funktionen, Rend.Circ. Palermo., , [3] Latha. S, Coefficient inequalities for classes of Ruscheweyh type analytic functions, J. Ineq. in Pure Applied. Math., Volume 9, Issue 2, Article

8 3234 H. Mahzoon [4] Latha. S Nanjunda Rao. S, Convex combinations of n analytic functions in generalized Ruscheweyh class, Int. J. Math. Sci. Technology., , [5] Owa. S, Polatoglu. Y Yavuz. E, Cofficient inequalities for classes of uniformly starlike convex functions, J. Ineq. in Pure Applied. Math., Volume 7 Isuue 5 Article 160 Article [6] Robertson. M. S, On the theory of univalent functions, Ann. Math., , [7] Ruscheweyh. S, A new criteria for univalent function, Proc. Amer. Math. Soc., , [8] Shams. S, Kulkarni. S. R Jahangiri. J. M, Classes of uniformly starlike convex functions, Internat. J. Math. Math. Sci., 55, 2004, Received: May, 2011