CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED BY A GENERALISED DERIVATIVE OPERATOR

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1 Journal of Quality Measureent and Analysis Jurnal Penguuran Kualiti dan Analisis JQMA 8(2) 202, CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED BY A GENERALISED DERIVATIVE OPERATOR (Sifat Tertentu bagi Fungsi Analisis yang Ditarif oleh Pengoperasi Terbitan Teritla) AISHA AHMED AMER & MASLINA DARUS ABSTRACT In this paper, soe iportant properties of analytic functions with negative coefficients defined by a generalised derivative operator are investigated The properties include the necessary and sufficient conditions, radius of starlieness, convexity and close-to-convexity Keywords: derivative operator; radius of starlieness; convexity; close-to-convexity ABSTRAK Dala aalah ini diaji beberapa sifat penting bagi fungsi analisis berpeali negatif yang ditarif oleh pengoperasi terbitan teritla Sifat tersebut terasulah syarat perlu dan cuup, jejari ebabintangan, ecebungan dan deat-dengan-ecebungan Kata unci: pengoperasi terbitan; jejari ebabintangan; ecebungan; deat-denganecebungan Introduction Let A denote the class of functions f in the open unit disc U = { z C : z < }, and let T denote the subclass of A consisting of analytic functions of the for f( z) = z, ( z U ), =2 which are analytic in the unit disc U Definition Let f A Then f is said to be convex of order µ (0 µ < ) if and only if zf ( z ) R + > µ, z U f ( z) Definition 2 Let f A Then f is said to be starlie of order µ (0 µ < ) if and only if zf ( z ) R > µ, z U f ( z) Aer and Darus (20; 202) have recently introduced a new generalised derivative operator I ( λ, λ, lnf, ) ( z) as follows: 2 37

2 Aisha Ahed Aer & Maslina Darus Definition 3 Let f A, then the generalised derivative operator is given by where I ( λ, λ, lnf, ) ( z)= z, () 2 =2 ( + λ ( ) + l) = cn (, ), ( + l) ( + λ2 ( )) ( n + ) λ2 λ 0, l 0, and cn (, )= () n, N = {0,,2,}, Definition 4 Let a function f be in T Then f is said to be in the class of ταβλ (,,, λ, l, n) if, 2 R I (λ,λ 2,l,n) f (z) z[i (λ,λ 2,l,n) f (z)] > α I (λ,λ 2,l,n) f (z) z[i (λ,λ 2,l,n) f (z)] where n, N 0 = {0,,2,}, l 0, 0 α <, and 0 β < 0 + β, (2) The faily ταβλ (,,, λ 2, l, n) is a special interest as it contains any well-nown classes of analytic univalent functions This faily is studied by (Najafzadeh & Ebadian 2009), and also (Tehranchi & Kularni 2006a; 2006b) 2 Necessary and Sufficient Conditions Theore 2 Let f T Then f ταβλ (,,, λ2, l, n) if and only if, [( + α) ( α + β)] a < (3) β =2 Proof: Let us assue that f ταβλ (,,, λ2, l, n) So by using the fact that R( ω) > α ω + β if, and only if [ ( i i R ω + αe θ ) α θ ] > β I ( λ, λ2, lnf, ) ( z) and letting ω = [ zi ( λ, λ, lnf, ) ( z )] So 2 in (2), we obtain [ ( i i R ω + αe θ ) α θ ] > β 38

3 Certain properties for analytic functions defined by a generalised derivative operator then z =2 iθ iθ R ( + αe ) αe β > 0, z =2 β ( β ) α ( ) R iθ e =2 =2 =2 > 0 i The above inequality ust hold for all z in U Letting z = re θ where 0 r <, we obtain iθ β [( β) + αe ( )] ar =2 R ar =2 > 0 i By letting r through half line z = re θ and by ean value theore, we have R β β + α [( ) ( )] ar > 0, =2 then we get [( + α) ( α + β)] a β =2 < Conversely, let (3) holds We will show that (2) is satisfied and so f ταβλ (,,, λ2, l, n) By using the fact that R ( ω)> β if and only if ω ( + β) < ω+ ( β), it is enough to show that ω ( + α ω + β) < ω+ ( α ω + β), if R = ω+ ( α ω + β) = 2 z βz [ + ( β) + α α] zi [ ( λ, λ, lnf, ) ( z)] 2 =2 This iplies that z R> 2 β [ + ( + α) ( α + β) ] zi [ ( λ, λ, lnf, ) ( z)] 2 =2 39

4 Aisha Ahed Aer & Maslina Darus Siilarly, if L = ω ( α ω + β), we get z L< β + [ + ( + α) ( α + β) ] zi [ ( λ, λ, lnf, ) ( z)] 2 =2 It is easy to verify that R L >0 and so the proof is coplete Corollary 2 Let f ταβλ (,,, λ2, l, n), then a β < [( + α) ( α + β)] Proof: For 0 µ <, we need to show that Now, let us show that zf ( z ) < µ f ( z) zf ( z ) f ( z ) = f ( z) =2 ( ) =2 =2 ( ) a z a z =2 <µ µ a z =2 µ < By Theore 2, it is enough to consider ( µ )[( + α) ( α + β)] z < ( µ )( β) Theore 22 Let f ταβλ (,,, λ2, l, n) Then f is convex of order µ (0 µ < ) in z < r = r ( αβλ,,, λ, l, n, µ ) where 2 2 ( µ )[( + α) ( α + β)] r2( αβλ,,, λ ( µ )( β) Proof: For 0 µ <, we need to show that Now again let us show that zf ( z ) < µ f ( z) 40

5 Certain properties for analytic functions defined by a generalised derivative operator =2 ( ) =2 a z =2 µ a z =2 µ ( ) a z a z =2 < <µ By Theore 2, it is again enough to consider ( µ )[( + α) ( α + β)] z < ( µ )( β) Theore 23 Let f ( z) ταβλ (,,, λ2, l, n) Then f ( z ) is close-to-convex of order µ (0 µ < ) in z < r = r3( αβλ,,, λ 2, l, n, µ ) where ( µ )[( + α) ( α + β)] r3( αβλ,,, λ ( β) Proof: For 0 µ <, we ust show that f ( z ) < µ Siilarly we show that ( ) = =2 =2 f z a z a z µ =2 a µ < z By Theore 2, the above inequality holds true if, ( µ )[( + α) ( α + β)] z < ( β) 3 Radius of Starlieness, Convexity and Close-to-convexity In this section, we will calculate Radius of Starlieness, Convexity and Close-to-convexity for the class (,,, 2, l, n) ταβλ λ 4

6 Aisha Ahed Aer & Maslina Darus Theore 3 Let f ταβλ (,,, λ2, l, n) Then f is starlie of order µ (0 µ < ) in z < r = r( αβλ,,, λ, l, n, µ ), where 2 ( µ )[( + α) ( α + β)] r( αβλ,,, λ ( µ )( β) Proof: For 0 µ <, we need to show that Now, we have to show that zf ( z ) < µ f ( z) zf ( z ) f ( z ) = f ( z) =2 ( ) =2 =2 ( ) a z a z =2 <µ µ a z =2 µ < By Theore 2, it is enough to consider ( µ )[( + α) ( α + β)] z < ( µ )( β) Theore 32 Let f ταβλ (,,, λ2, l, n) Then f is convex of order µ (0 µ < ) in z < r = r ( αβλ,,, λ, l, n, µ ), where 2 2 ( µ )[( + α) ( α + β)] r2( αβλ,,, λ ( µ )( β) Proof: For 0 µ <, we need to show that We have to show that zf ( z ) < µ f ( z) 42

7 Certain properties for analytic functions defined by a generalised derivative operator =2 ( ) =2 a z =2 ( ) a z a z =2 <µ µ a z =2 µ < By Theore 2, it is enough to consider ( µ )[( + α) ( α + β)] z < ( µ )( β) Theore 33 Let f ταβλ (,,, λ2, l, n) Then f is close-to-convex of order µ (0 µ < ) z < r = r ( αβλ,,, λ, l, n, µ ), where in 3 2 ( µ )[( + α) ( α + β)] r3( αβλ,,, λ ( β) Proof: For 0 µ <, we ust show that f ( z ) < µ We have to show that ( ) = =2 =2 f z a z a z µ =2 a µ < z By Theore 2, the above inequality holds true if ( µ )[( + α) ( α + β)] z < ( β) Acnowledgeent The wor presented here was partially supported by UKM-ST-06-FRGS

8 Aisha Ahed Aer & Maslina Darus References Aer A A & Darus M 20 On soe properties for new generalized derivative operator Jordan Journal of Matheatics and Statistics (JJMS) 4(2): 9-0 Aer A A & Darus M 202 On preserving the univalence integral operator Applied Sciences 4: 5-25 Najafzadeh Sh & Ebadian A 2009 Neighborhood and partial su property for univalent holoorphic functions in ters of Koatu operator Acta Universitatis Apulensis 9: 8-89 Tehranchi A & Kularni SR 2006a Soe integral operators defined on p-valent functions by using hypergeoetric functions Studia Univ Babes, Bolyai Matheatica (Cluj) : Tehranchi A & Kularni SR 2006b Study of the class of univalent functions with negative coefficients defined by Ruscheweyh derivative J Rajasthan acadey of Physical Science 5(): School of Matheatical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia UKM Bangi Selangor DE, MALAYSIA E-ail: eaer_80@yahooco, aslina@uy* * Corresponding author 44