Entisar El-Yagubi & Maslina Darus. In order to derive our new generalised derivative operator, we define the analytic function ( ), is defined by

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1 Journal of Quality Measureent and Analysis Jurnal Penguuran Kualiti dan Analisis JQMA 9(1) 2013, SUBCLASSES O ANALYTIC UNCTIONS DEINED BY NEW GENERALISED DERIVATIVE OPERATOR (Suelas ungsi Analisis yang Ditarif oleh Pengoperasi Teritan Teritla) ENTISAR EL-YAGUBI & MASLINA DARUS ABSTRACT A new generalised derivative operator D λλ is introdued. This operator generalised soe well-nown operators studied earlier. New sulasses of analyti funtions in the open unit dis whih are defined using generalised derivative operator are introdued. Inlusion theores are investigated. urtherore, generalised Bernardi-Liera-Livington integral operator is shown to e preserved for these lasses. Keywords: analyti funtions; univalent funtions; starlie funtions; onvex funtions; loseto-onvex funtions; suordination; Hadaard produt; integral operator ABSTRAK Pengoperasi teritan aharu teritla D λλ diperenalan. Pengoperasi ini engitla eerapa pengoperasi terdahulu yang terenal. Suelas aharu fungsi analisis dala aera terua unit diperenalan yang ditarif dengan enggunaan pengoperasi teritan teritla. Teore ranguan diaji. Malah pengoperasi airan Bernardi-Liera-Livington ditunjuan eal untu elas teseut. Kata uni: fungsi analisis; fungsi univalen; fungsi a intang; fungsi eung; fungsi hapir eung; suordinasi; hasil dara Hadaard; pengoperasi airan 1. Introdution Let A denote the lass of funtions of the for f( z) = az, (1.1) = 2 where a is a oplex nuer, whih are analyti in the open unit dis U = {z : z < 1}. Also let S *,C and K denote, respetively, the sulasses of A onsisting of funtions whih are starlie, onvex, and lose to onvex in U. An analyti funtion f is suordinate to an analyti funtion g, written f (z) g(z), (z U ) if there exists an analyti funtion w in U, suh that w (0) = 0, w (z) <1 for z <1 and f(z) = g (w (z)). In partiular, if g is univalent in U, then f (z) g(z) is equivalent to f (0) = g (0) and f (U ) g (U). The onvolution of two analyti funtions φ(z) = z y + az and ψ(z) = = 2 z + z is defined = 2

2 Entisar El-Yagui & Maslina Darus φ(z) ψ(z) = z+ az = ψ(z) φ(z). (1.2) = 2 In order to derive our new generalised derivative operator, we define the analyti funtion 1 + ( λ1+ λ2)( 1) + ( ), λ 1, λ2, z = z + z = λ2 ( 1) + (1.3) where, 0 = {0,1,2, } and λ2 λ1 0. Now, we introdue the new generalised derivative operator D as follows: Definition 1.1. or f Α, the operator D n is defined y D,, : Α Α, D f ( z) = ( z) R f ( z), z U, (1.4) 2 λ1 0, where, 0 = {0}, operator (Rusheweyh 1975), given y n λ and R f ( z ) denotes the Rusheweyh derivative n R f( z) = z+ Cnaz (, ), (n 0, z U ), (1.5) = 2 where Cn (, ) = ( n+ 1) 1 (1) 1. If f is given y (1.1), then we easily find fro equality (1.4) that 1 + ( λ1+ λ2)( 1) + Dλ 1, λ2, f( z) = z+ Cnaz (, ), = λ2 ( 1) + where, 0 = {0}, λ2 λ1 0, ( z U ), (1.6) n+ 1 and Cn (, ) = = ( n+ 1) 1 (1) 1. n Note that, (n) denotes the Pohhaer syol (or the shifted fatorial) defined y (n) = 1 for = 0,n \{0} n(n +1)(n + 2) (n + 1) for,n (1.7) Rear 1.1. Speial ases of the operator D inlude the Rusheweyh derivative operator n,0 0, n in the ase D λ 1, λ 2, (Rusheweyh 1975), the Salagean derivative operator in the ase D1,0,0 S (Salagean 1983), the generalised Salagean derivative operator introdued y Al-Ooudi in the ase D D 0, λ1,0,0 λ1 (Al-Ooudi 2004), the generalised Rusheweyh derivative operator in the n,1 1 ase D,0,0 D λ λ (Al-Shaqsi & Darus 2009), the generalised Al-Shaqsi and Darus derivative n 1 operator in the ase D D (Darus & Al-Shaqsi 2008), the Uralegaddi and Soanatha, 1,0, n 0, derivative operator in the ase D 1,0,1 D (Uralegaddi & Soanatha 1992), the Cho and 48

3 Sulasses of analyti funtions defined y new generalised derivative operator 0, Srivastave derivative operator in the ase D and Darus derivative operator in the ase 0, D derivative operator in the ase D 1,0, 1,0, 0, λ1,0, λ1, D (Cho & Srivastava 2003), the Eljaal D D (Cǎtas 2008). (Eljaal & Darus 2011), and the Cǎtas To prove our results, we need the following equations throughout the paper: + 1 λ1 λ2 λ1 λ 2 (1 + ) D f ( z) = (1 ( + ) + ) D f ( z) + ( + ) z D f ( z), (1.8) n + 1, = + nd f ( z) z D f ( z) ( n 1) D f ( z). (1.9) Let N e the lass of all analyti and univalent funtions ø in U and for whih ϕ (U) is onvex with ϕ (0) =1 and Re{ϕ(z)}>0, for z U. or ϕ,ψ N, Ma and Minda (1992) studied the sulasses S C ( φ ), and K (, φψ) of the lass A. These lasses are defined using the priniple of suordination as follows: S (φ) := f : f A, z f (z) f (z) φ(z)in U, C(φ) := f : f A, 1+ z f (z) φ(z) in U, (1.10) f (z) K(φ,ψ ) := f : f A, g S (φ) suh that z f (z) ψ (z) in U. g(z) Oviously, we have the following relationships for speial hoies of ϕ and ѱ S 1+ z = S 1 z, 1+ z 1+ z 1+ z C = C, K, = K. 1 z 1 z 1 z (1.11) Using the generalised differential operator D f, λλ new lasses S λ 1, λ 2, ( φ ), C λ 1, λ 2, ( φ ) and K λ, λ, ( φψ, ), are introdued and defined as follows: n { φ } n { φ } n = { φψ } S : = f A : D f ( z) S ( ),, C : = f A : D f ( z) C( ), (1.12), K ( φψ, ): f A: D f ( z) K(, ). It an e shown easily that, f ( z ) C zf ( z ) S. (1.13) Janowsi (1973) introdued the lass S[ AB, ] = S ((1 + Az ) (1 + Bz )), and in partiular for φ (z) = (1 + Az) / (1 + Bz), we set 49

4 Entisar El-Yagui & Maslina Darus 1+ Az Sλ 1, λ2, = S, [ AB, ], (1 A > B 1). (1.14) 1+ Bz In (Oar and Hali 2012), the authors studied the inlusion properties for lasses defined using Dzio-Srivastava operator. This paper investigates siilar properties for analyti funtions in the lasses defined y the generalized differential operator D f. λλ urtherore, appliations of other failies of integral operators are onsidered involving these lasses. 2. Inlusion Properties Involving D λλ,, f To prove our results, we need the following leas: Lea 2.1 (see Eenigenurg et al. 1983). Let φ e onvex univalent in U, with φ(0)=1 and Re{ φ( z) + η} > 0, (,η ). If p is analyti in U with p(0)=1 then p(z) + z p (z) p(z) + η φ(z) p(z) φ(z). (2.1) Lea 2.2 (see Miller and Moanu (1981)). Let φ e onvex univalent in U and w e analyti in U with Re{ w( z)} 0. If p is analyti in U and p(0)= φ(0) then p(z) + w(z)z p (z) φ(z) p(z) φ(z). (2.2) Theore 2.3. or any real nuers, λ 1 and λ, 2 where 0, λ2 λ1 0 and 0. Letφ Ν and Re{ φ( z) + (1 ( λ1+ λ2) + ) ( λ1+ λ2)} > 0, ( n 0). then S S Proof. Let f S λ 1, λ 2, and set p( z) = ( z[ D f ( z)]) ( Dλ 1, λ2, f ( z)), where p is analyti in U, with p (0) = 1. Rearranging (1.8), we have (1 + ) D f ( z) ( + ) z[ D f ( z)] = (1 ( + ) + ) +. (2.3) D f z D f z + 1 λ1 λ 2 λ, 1 λ2 n ( ) ( ) Next, differentiating (2.3) logarithially with respet to z and ultiplying y z, we otain z[ D f ( z)] z[ D f ( z)] (( [,, ( )] ) ( λ, λ, ( )) ) z z D f z D f z + 1 = λ1+ λ2 + λ1+ λ2 D f ( z) D f ( z) ( z[ D f ( z)]) ( D f ( z)) (1 ( ) ) ( ) (2.4) zp ( z ) = pz ( ) +. pz ( ) + (1 ( λ + λ ) + ) ( λ + λ ) 50

5 Sulasses of analyti funtions defined y new generalised derivative operator Sine (z[d λ1,λ 2, f (z)]) (D λ1,λ 2, f (z)) φ(z) and applying Lea 2.1, it follows that p φ. Thus f S λ, λ,. Theore 2.4. Let, λ1, λ2, where 0, λ 2 λ 1 0 and n 0. Then S S ( 0, φ Ν). n+ 1, Proof. Let f S λ, λ, and fro (1.9), we otain that n+ 1, nd f z z D f z = + ( n 1). (2.5) D f z D f z n+ 1, ( ) [ ( )] ( ) ( ) Maing use of the differentiating (2.5) logarithially with ultiplying y z and setting p( z) = ( z[ D f ( z)]) ( Dλ 1, λ2, f ( z)), we get the following: z[d n+1, λ1 f (z)],λ 2, D n+1, λ1,λ 2, f (z) = p(z) + z p (z) p(z) + (n 1) φ(z). (2.6) Sine n 0. and Re{ φ ( z) + ( n 1)} > 0, using Lea 2.1, we onlude that f S λ, λ, Corollary 2.5. Let λ2 λ1 0, n 0, and 1 A > B 1. Then S + [ nab ;, ] S [ nab ;, ] and 1,, Theore 2.6. Let λ2 λ1 0, and n 0. Then C C C. n+ 1, Proof. Using (1.12) and Theore 2.3, we oserve that, 1 f ( z) C ( ) n + zf z S + 1,, zf ( z ) S,, S [ n+ 1; AB, ] S [ n; AB, ].,, C and + 1,, D,, zf ( z ) S zd [ f( z)] S λ, λ, Dλ, λ, f ( z) C f ( z) C. (2.7),, 51

6 Entisar El-Yagui & Maslina Darus To prove the seond part of the theore, we use siilar steps and apply Theore 2.4, the result is otained. Theore 2.7. Let λ2 λ1 0, 0 and Re{ φ( z) + (1 ( λ1+ λ2) + ) ( λ1+ λ2)} > 0. Then K ( φψ, ) K ( φψ, ) and K + 1 ( φψ, ) K ( φψ, ), (, φψ) N. n+ 1, Proof. Let funtion + 1 g S λ, λ, f K λ, λ, ( φψ, ). In view of the definition of the lass + 1 suh that K,, ( φψ, ), + 1 there is a z[ D f ( z)] p ψ ( z ) (2.8) D g( z) Applying Theore 2.3, then q(z) = (z[d λ1 g(z) ]) (D,λ 2, λ1 g(z)) φ(z).,λ 2, Let the analyti funtion p with p(0)=1, as follows: g S λ, λ, and let zd [ f( z)] pz ( ) =. (2.9) D gz ( ) λ 1, λ 2, λ 1, λ 2, Thus, rearranging and differentiating (2.9), we have [ D zf ( z )] p( z )[ D g( z )] = + p ( z). (2.10) D g( z) D g( z) Maing use of (1.8), (2.9), (2.10), and q (z), we otain that + 1 z[ D f ( z)] [ D zf ( z )] = + 1 D g( z) D g( z) (1 ( λ + λ ) + ) D zf ( z ) + ( λ + λ ) z[ D zf ( z )] = (1 ( + ) + ) D g( z) + ( + ) z[ D g( z)] λ1 λ2 λ1 λ2 (2.11) = ((1 ( λ1+ λ2 ) + ) D ) ( ) ( ) ( ) zf ( z ) Dλ 1, λ2, g( z ) + ( λ1+ λ2 ) z[ D zf ( z )] Dλ 1, λ2, g( z ) (1 ( λ1+ λ2) + ) + (( λ1+ λ2 ) zd [ λ, λ, gz ( )]) Dλ, λ, ( gz ( )) (1 ( λ1 + λ2) + pz ) ( ) + ( λ1 + λ2)[ pzqz ( ) ( ) + p ( z)] = (1 ( λ + λ ) + ) + ( λ + λ ) qz ( ) 52

7 Sulasses of analyti funtions defined y new generalised derivative operator = p(z) + z p (z) ψ (z). q(z) + (1 (λ 1 + λ 2 ) + ) (λ 1 + λ 2 ) Sine q(z) φ(z) and Re{(1 ( λ1+ λ2) + ) ( λ1+ λ2)} > 0, then Re{ qz ( ) + (1 ( λ + λ ) + ) ( λ + λ )} > 0. Using Lea 2.2, we onlude that pz ( ) ψ ( z ) and thus f K λ, λ, ( φψ, ). By using siilar anner and (1.9), we otain the seond result. In suary, y using suordination tehnique, inlusion properties have een estalished for ertain analyti funtions defined via the generalised differential operator. 3. Inlusion Properties Involving f In this setion, we deterine properties of generalised Bernardi-Liera-Livington integral operator defined y (Bernardi 1969; Jung et al. 1993; Liera 1965; Livington 1966). z [ ( )] ( ) z 0 f z = t f t dt ( f Α, > 1). + 1 = z+ az n n, (3.1) n + n = 2 and satisfies the following: Dλ 1, λ2, [ f ( z)] + z D [ f ( z)] = ( + 1) Dλ 1, λ2, f ( z). (3.2) Theore 3.1. If f S λ, λ, then f S λ, λ,. ( ) ( D λ1,λ 2, ) φ(z).taing the Proof. Let f S λ 1, λ 2, then z[d λ1 f (z)] f (z),λ 2, differentiation on oth sides of (3.2) and ultiplying y z, we otain z D λ1 f (z) z D,λ, 2 λ1,λ, = 2 [ f (z)] z z D λ1,λ, 2 [ f (z)] ( D λ1,λ, 2 [ f (z)]) + D λ1 f (z) D,λ, λ1,λ, 2 2 [ f (z)] z D λ1,λ, 2 [ f (z)] D λ1,λ, 2 [ f (z)] ( ) + (3.3) 53

8 Entisar El-Yagui & Maslina Darus Setting we have n ( ), p( z) = z D [ f( z)] D [ f( z)], z[ D f ( z)] zp ( z ) = pz ( ) +. D f( z) pz ( ) + (3.4) Lea 2.1 iplies z D λ1,λ 2, ( ) φ(z). Hene [ f (z)] D λ1,λ 2, [ f (z)] f S λ, λ,. Theore 3.2. Let f C λ 1, λ 2, then f C λ 1, λ 2,. Proof. By using (1.12) and Theore 3.1, it follows that f C λ, λ, zf ( z ) S λ, λ, [ zf ( z )] S z[ [ f ( z)]] S [ f ( z)] C. (3.5) Theore 3.3. Let φψ, N and Proof. Let f K φψ z[d ( λ1 f (z)],λ 2, ) D λ1 g(z),λ 2, f K λ, λ, ( φψ, ), then f K λ, λ, ( φψ, ). (, ), then there exists a funtion ( ) ψ (z). Sine [ g( z)] S. Then let Set g S λ, λ, suh that g S λ, λ, therefore fro Theore 3.1, q(z) = z D λ 1,λ 2, [g(z)] φ(z). (3.6) [g(z)] D λ1,λ 2, z D [ f ( z)] pz ( ) =. (3.7) D [ g( z)] By rearranging and differentiating (3.7), we otain that [ D [ zf ( z)]] p( z)[ D [ g( z)]] p ( z)[ D [ g( z)]] = +. (3.8) D [ g( z)] D [ g( z)] D [ g( z)] Maing use of (3.2), (3.7), and (3.6), it an e derived that 54

9 Sulasses of analyti funtions defined y new generalised derivative operator z[ D f ( z)] zp ( z ) = pz ( ) +. D gz ( ) qz ( ) + (3.9) Hene, applying Lea 2.2, we onlude that p(z) ψ (z), and it follows that [ f( z)] K λ 1, λ 2, ( φψ, ). or analyti funtions in the lasses defined y generalised differential operator, the generalised Bernardi-Liera-Livington integral operator has een shown to e preserved in these lasses. 4. Conlusion Results involving funtions defined using the generalised differential operator, naely, inlusion properties and the Bernardi-Liera-Livington integral operator were otained using suordination priniples. In Oar and Hali (2012), siilar results were disussed for funtions defined using the Dzio-Srivastava operator. Anowledgent The wor presented here was partially supported y GUP Referenes Al-Ooudi.M On univalent funtions defined y derivative operator. International Journal of Matheatis and Matheatial Sienes 27: Al-Shaqsi K. & Darus M On univalent funtions with respet to -syetri points defined y a generalized Rusheweyh derivative operators. Journal of Analysis and Appliations 7(1): Bernardi S.D Convex and starlie univalent funtions. Transations of the Aerian Matheatial Soiety 135: Cǎtas A On ertain lasses of p-valent funtions defined y new ultiplier transforations, TC Istanul Kultur University Puliations, Proeedings of the International Syposiu on Geoetri untion Theory and Appliations (GTA 07), Istanul, Turey, August 2007, vol. 91, pp Cho N.E. & Srivastava H. M Arguent estiates of ertain analyti funtions defined y a lass of ultiplier transforations. Matheatial and Coputer Modelling 37(1-2): Darus M. & Al-Shaqsi K Differential sandwih theores with generalised derivative operator. International Journal of Coputing and Matheatial Sienes 22: Eenigenurg P., Miller S.S., Moanu P.T. & Reade M. O On a Briot-Bouquet differential suordination. General Inequalities 3: Eljaal E.A. & Darus M Suordination results defined y a new differential operator. Ata Universitatis Apulensis 27: Janowsi W Soe extreal proles for ertain failies of analyti funtions I. Annales Polonii Matheatii 28: Jung I.B., Ki Y. C. & Srivastava H. M The Hardy spae of analyti funtions assoiated with ertain oneparaeter failies of integral operators. Journal of Matheatial Analysis and Appliations 176(1): Liera R.J Soe lasses of regular univalent funtions. Proeedings of the Aerian Matheatial Soiety 16: Livington A.E On the radius of univalene of ertain analyti funtions. Proeedings of the Aerian Matheatial Soiety 17: Ma W. & Minda D A unified treatent of soe speial lasses of univalent funtions. In Z. Li,. Ren, L. Yang & S. Zhang (Eds.). Proeedings of the Conferene on Coplex Analysis, pp , International Press, Caridge, Mass, USA. 55

10 Entisar El-Yagui & Maslina Darus Miller S.S. & Moanu P.T Differential suordination and univalent funtions. The Mihigan Matheatial Journal 28: Oar R. & Hali S.A Classes of funtions defined y Dzio-Srivastava operator. ar East Journal of Matheatial Sienes 66(1): Rusheweyh S New riteria for univalant funtion. Proeedings of the Aerian Matheatial Soiety 49: Salagean G.S Sulasses of univalent funtions. In Proeedings of the Coplex Analysis 5th Roanian- innish Seinar Part 1, 1013: Uralegaddi B.A. & Soanatha C Certain lasses of univalent funtion. In Current Topis in Analyti untion Theory. River Edge, NJ: World Sientifi, pp Shool of Matheatial Sienes aulty of Siene and Tehnology University Keangsaan Malaysia UKM Bangi Selangor DE, MALAYSIA E-ail: entisar_e1980@yahoo.o, aslina@u.y* *Corresponding author 56