Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR. 1.

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1 Jordan Journal of Mathematics and Statistics JJMS) 42), 2011, pp ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR AISHA AHMED AMER 1) AND 2) MASLINA DARUS Abstract. Motiated by many well-known differential operators, we introduce a new generalied deriatie operator and study its characteriation properties. In addition, we determine conditions under which the partial sums of this operator of bounded turning are also of bounded turning. 1. Introduction The theory of unialent function is a beautiful subject as we can see in recent years, many new articles are written in this area. This field which is often associated with geometry and analysis has raised the interest of many since the beginning of 20th century to recent times. The name unialent functions or schlicht the German word for simple) functions is gien to functions defined on the open unit disc U := { C : < 1} of the complex plane C that are characteried by the fact that such a function proides one-to-one mapping onto its image. Geometrically, the function f is unialent if f 1 ) = f 2 ) implies 1 = 2 in U and is locally unialent at 0 U if it is unialent in some neighborhood of 0. The Koebe function 2000 Mathematics Subject Classification. Primary: 30C45. Key words and phrases. : Unialent Functions, New Deriatie Operator,The Cesáro Sums, Conolution Product, Bounded Turning. Copyright c Deanship of search and Graduate Studies, Yarmouk Uniersity, Irbid, Jordan. ceied: May 15, 2010 Accepted : June 14,

2 92 AISHA AHMED AMER AND MASLINA DARUS k) = /1 ) 2 is a unialent function. In fact, the Koebe function and its rotations e iγ ke iγ ), γ R are the only extremal functions for arious problems. The famous findings of Bieberbach conclude the claims gien. In brief, Bieberbach [1] proed that if f is normalied by f0) = 0 and f 0) = 1 then the second coefficient a 2 2 with equality if and only if f is a rotation of the Koebe function. He also conjectured that a n n, n = 2, 3, ) which is generally alid and this was proed by de Branges in Now operators of normalied analytic functions become ery popular, namely for differential and integral. Many articles discuss on operators and new generaliations of arious authors. To our best knowledge, perhaps Ruscheweyh [4] was the first in 1975 to introduce the differential operator and followed by Salagean [5] in These two operators were quite a while being used to study different properties and problems inoling subclasses of unialent functions. In 2004, Al-Oboudi [7] generalied Salagean operator and followed by Shaqsi and Darus [6],[8]) generalied both differential of Ruscheweyh and Salagean in Many authors started to introduce new operator in their own style based on the Salagean and Ruscheweyh operators. For example see [10], [11]). We use these operators to find another type of differential operator and obtain certain conditions on bounded turning. In addition, we obtain the Cesáro means for the operator defined. This type of problems can be seen in arious work for example see [11],[12],[13]). Let H denote the class of functions of the form 1.1) f) = + a n n, which are analytic in the open unit disk U = { : < 1} on the complex plane C. n=2

3 ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR 93 Let A denote the subclass of H consisting of functions normalied by f0) = 0, f 0) = 1. Let the functions f gien by 1.1) and g ) = + b k k, U). Then the Hadamard product conolution) of f and g, defined by : f g) ) = + a k b k k, U). Now, c) k denotes the Pochhammer symbol or the shifted factorial) defined by c) k 1 for k = 0, = cc + 1)c + 2)..c + k 1) for k N = {1, 2, 3,...}, c C {0}. In order to derie our new generalied deriatie operator, we define the analytic function 1.2) ϕ m λ 1, λ 2, l)) = + λ 1 k 1) + l) m 1 l) m 1 λ 2 k 1)) m k, where m N 0 = {0, 1, 2,...} and λ 2, λ 1, l R such that λ 2 λ 1 0, l 0. Now, we introduce the new generalied deriatie operator I m λ 1, λ 2, l, n)f) as the following: Definition 1.1. For f A the operator I m λ 1, λ 2, l, n) is defined by I m λ 1, λ 2, l, n) : A A 1.3) I m λ 1, λ 2, l, n)f) = ϕ m λ 1, λ 2, l)) R n f) U),

4 94 AISHA AHMED AMER AND MASLINA DARUS where m N 0 = {0, 1, 2,...} and λ 2 λ 1 0, l 0, and R n f) denotes the Ruscheweyh deriatie operator [4], and gien by R n f) = + cn, k)a k k, n N 0, U), where cn, k) = n+1) k 1 1) k 1. If f is gien by 1.1), then we easily find from the equality 1.3) that I m λ 1, λ 2, l, n)f) = + λ 1 k 1) + l) m 1 l) m 1 λ 2 k 1)) m cn, k)a k k, where n, m N 0 = {0, 1, 2,...}, and λ 2 λ 1 0, l 0, cn, k) = n+1) k 1 1) k 1. Special cases of this operator includes: the Ruscheweyh deriatie operator [4] in the cases: I 1 λ 1, 0, l, n) I 1 λ 1, 0, 0, n) I 1 0, 0, l, n) I 0 0, λ 2, 0, n) I 0 0, 0, 0, n) I m+1 0, 0, l, n) I m+1 0, 0, 0, n) R n, the Salagean deriatie operator [5]: I m+1 1, 0, 0, 0) S n, the generalied Ruscheweyh deriatie operator [6]: I 2 λ 1, 0, 0, n) R n λ, the generalied Salagean deriatie operator introduced by Al-Oboudi [7]: I m+1 λ 1, 0, 0, 0) S n β,

5 ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR 95 the generalied Al-Shaqsi and Darus deriatie operator[8]: I m+1 λ 1, 0, 0, n) D n λ,β, the Al-Abbadi and Darus generalied deriatie operator [9]: I m λ 1, λ 2, 0, n) µ n,m λ 1,λ 2, and finally the Catas driatie operator [10]: I m λ 1, 0, l, n) I m λ, β, l). Using simple computation one obtains the next result. l + 1)I m+1 λ 1, λ 2, l, n)f) = l λ 1 )[I m λ 1, λ 2, l, n) ϕ 1 λ 1, λ 2, l))]f)+ 1.4) λ 1 [I m λ 1, λ 2, l, n) ϕ 1 λ 1, λ 2, l))]. Where U) and ϕ 1 λ 1, λ 2, l)) analytic function and from 1.2) gien by ϕ 1 λ 1, λ 2, l)) = + 1 λ 2 k 1)) k. For 0 β < 1, α > 0, let Ωβ) denote the class of functions f of the form 1.1) so that { f) )α } > β in U. The functions in Ωβ) are called functions of bounded turningcf. [3],ol II). By the Nashiro-Warschowski theorem see, e.g, [3], ol I), the functions in Ωβ) are unialent in U.

6 96 AISHA AHMED AMER AND MASLINA DARUS The th partial sums F ) of the operator 1.2) are gien by 1.5) F ) = + λ 1 k 1) + l) m 1 l) m 1 λ 2 k 1)) m cn, k)a k k, U). We show that if f of the form 1.1) and belongs to the class Ωβ), ) α } i.e. > β then F ) also belong to the class Ωβ), i.e. { f) { ) α } F ) > β, 0 β < 1, α > 0, = 1, 2,. For this purpose,to proe our results we will need the following three lemmas. Lemma 1.1. [2] For U we hae Lemma 1.2. [3] Let P )) > 1/2 in { j k=1 } k 1 k P ) be analytic in U, such that P 0) = 1 and let U. For the function Qanalytic in U, the conolution function P Q takes alues in the conex hull of the image on U under Q. 2. Main sults By making use Lemma 1.1 and Lemma 1.2, we illustrate the conditions under which the th partial sums 1.5) of the functions in also of bounded turning. Ωβ) of bounded turning are Theorem 2.1. Let the function f) = + a belongs to the class Ωβ). If 1 4 < β < 1, α > 0, then F ) Ω 3 1 β)α 3 ).

7 ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR 97 Proof. Let f be of the form 1.1) and belong to Ωβ) for 1 4 < β < 1. Since { f) )α } > β, we hae { ) α } a k k 1 > β > 1 2, { a ) k k 1 α } { > 1 β ) α } a k k 1 > β > 1 2, and then 1+λ 1 k 1)+l) m 1 1+l) m 1 1+λ 2 cn, k)a k 1)) m k k 1 α 1 β > 1 2. Applying the conolution properties of power series to ) α F ) = = 2.1) ) F) α, we may write ) α λ 1 k 1) + l) m 1 l) m 1 λ 2 k 1)) cn, k)a k k 1 m 1 λ 1 k 1) + l) m 1 1 β) l) m 1 λ 2 k 1)) cn, k)a k k 1 m ) 1 β) α k 1 = P ) Q). From Lemma 1.1 for j = 1, we obtain { } k 1 k + 2 3, { } { } 2.2) k 1 k 1 k k=1 k=1 k=1 ) α

8 98 AISHA AHMED AMER AND MASLINA DARUS Applying a simple algebra to inequality 2.2) and Q) in 2.1) yields {Q)} = { } 1 β) α k 1 > 1 β) α ) 1 3 ) 3 1 β)α =. 3 On the other hand, the power series P ) in 2.1) yield P )) > 1/2. Therefore, by Lemma 1.2, {P )} = for U). Thus { ) α } 1 λ 1 k 1) + l) m 1 1 β) α l) m 1 λ 2 k 1)) cn, k)a k k 1 > 1 m 2, { ) α } F ) > 3 1 β)α. 3 This concludes the Main Theorem. Next we determine the bounded turning for the Cesáro sums of order. From the partial sum s ) = + a k k, U), with s 1 ) = we construct the Cesáro means σ ) of f A by

9 ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR 99 σ, f) = 1 s k ) k=1 = 1 [s 1) s )] = 1 [ + + a2 2 ) a ) ] = 1 [ + 1)a a ) ] ) k + 1 = + a k k [ ) ] k + 1 = f) + k = f) g ), where g ) = + ) k + 1 k. Now, we hae the following result: Theorem 2.2. Let the function f) = + a belongs to the class 1 < β < 1, then σ 4, I m λ 1, λ 2, l, n)f)) Ω 3α 1 β) α ). 3 α Ωβ), If Proof. σ,i m λ 1,λ 2,l,n)f)) = = ) α ) ) α λ 1 k 1) + l) m 1 k + 1 cn, k) )a l) m 1 λ 2 k 1)) m k k 1 1 λ 1 k 1) + l) m 1 1 β) l) m 1 λ 2 k 1)) cn, k)a k k 1 m ) α

10 100 AISHA AHMED AMER AND MASLINA DARUS 1 β) α k + 1 = P ) Q). ) α k 1 ) Thus as k, a small computation gies { 1 β) α {Q)} = α k 1 } > 3α 1 β) α 3 α. This ends the proof. Acknowledgement The work presented here was supported by UKM-ST-06-FRGS ferences [1] L. Bieberbach, Über einige extremal probleme im gebiete der konformen abbdildung. Math. Ann. 77, 1916), [2] J. M. Jahangiri, K. Farahmad, Partial sums of functions of bounded turning, J. Ineq. Pure and Appl. Math., 44) Art. 79,2003), 1-3. [3] A.W. Goodman, Unialent Functions, Vols. I and II, Mariner Pub. Co. Tampa, FL., [4] St. Ruscheweyh, New criteria for unialent functions, Proc. Amer. Math. Soc. Vol. 49,1975), [5] G. S. Salagean, Subclasses of unialent functions, Lecture Notes in Math. Springer-Verlag), 1013, 1983), [6] K. Al-Shaqsi and M. Darus, An operator defined by conolution inoling polylogarithms functions, J. Math. Stat, 41), 2008), [7] F.M. AL-Oboudi, On unialent functions defined by a generalised Salagean Operator, Int. J. Math. Math. Sci. 27, 2004), [8] K. Al-Shaqsi and M. Darus, Differential Subordination with generalised deriatie operator, Int. J. Comp. Math. Sci, 22),2008),

11 ON SOME PROPERTIES FOR NEW GENERALIZED DERIVATIVE OPERATOR 101 [9] M. H. Al-Abbadi and M. Darus,Differential Subordination for new generalised deriatie operator, Acta Uniersitatis Apulensis, 20, 2009), [10] A. Catas,On a Certain Differential Sandwich Theorem Associated with a New Generalied Deriatie Operator, General Mathematics ), [11] M. Darus and R. W. Ibrahim, On Cesaro means for Fox-Wright functions, J. Math. and Stat. 43)2008), [12] M. Darus and R. W. Ibrahim, On some properties of differential operator, Acta Didactica Napocensia. 22)2009), [13] M. Darus and R. W. Ibrahim, Generalied Cesaro integral operator, International J. Pure and Applied Mathematics, 694)2011), School of Mathematical Sciences, Faculty of Science and Technology, Uniersiti Kebangsaan Malaysia 43600,Bangi, Selangor, Malaysia. address: 1) eamer 80@ukm.my address: 2) maslina@ukm.my