On Subordination and Superordination of New Multiplier Transformation

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1 It. J. Ope Probles Copt. Math., Vol., No., March 00 ISSN ; Copyright ICSRS Publicatio, 00 O Subordiatio ad Superordiatio of New Multiplier Trasforatio Aabed Mohaed ad Maslia Darus School of Matheatical Scieces Faculty of Sciece ad Techology, Uiversiti Kebagsaa Malaysia 4600, UKM Bagi, Selagor, Malaysia e-ail: aabedu@yahoo.co e-ail: aslia@u.y (correspodig author) Abstract ( µ ) z Let φ µ ( z,, c) =, z, z <, c /{0,,,...}, µ, = 0! ( + c) be the geeralized Hurwitz Lerch Zeta fuctio. We cosider + c ( µ ) + zf( z,, c) = z ( + c) φµ ( z,, c) = z, = 0 + c! ad defie ( ) z [ zf( z,, c)] [ zf( z,, c)] =, λ >, ( z ) λ + where deotes covolutio (Hadaard product). Let f be the oralized aalytic fuctio i the ope uit dis U. We defie a ew operator ( ) Dµ, c f ( z) = [ zf( z,, c)] f ( z). Moreover, we obtai soe differetial subordiatio ad superordiatio results ivolvig this operator. These results are obtaied by ivestigatig classes of adissible fuctios. Sadwichtype result is also studied. Keywords: Hurwitz Lerch zeta fuctio, Hadaard product, Multiplier Trasforatio, Differetial Subordiatio ad Superordiatio AMS Matheatics Subject Classificatio: 0C45.

2 Aabed Mohaed ad Maslia Darus 0 Itroductio ad Defiitios Let HU ( ) deote the class of holoorphic fuctios i the ope uit dis { : }. For a ad we let + Η [ a, ] = { f HU ( ), f( z) = a+ az + a+ z +..., z U} U = z z z < ad For f j A give by { } A = f H( U), f ( z) = z + a z +..., z U. f ( z) = z + a z, ( j =,), j the Hadaard product (or covolutio) f f of f ad f is defied by ( f f )( z) = z + a, a,z. =, j Let F ad G be aalytic fuctios i the uit dis U. The fuctio F is subordiate to G, writte F G if G is uivalet, F(0) = G(0) ad FU ( ) GU ( ). I geeral, give two fuctios F ad G which are aalytic i U, the fuctio F is said to be subordiate to G, if there exist a fuctio w aalytic i U with such that = w (0) = 0 ad ( z U) : w ( z) < ( z U): F( z) = G( w ( z)). Now let us cosider the geeralized Hurwitz Lerch zeta fuctio ( µ ) z φµ (, zc,) =, (.)! ( + c) = 0 z, z <, c /{0,,,...}, µ,, itroduced by Goyal ad Laddha []. Here ( x ) is Pochhaer sybol (or the shifted factorial, sice () =! ) ad ( λ ) give i ters of the Gaa fuctios ca be writte as Γ ( x + ) ( x) = = x( x + )...( x + ) for =,,,... x ( x) 0 =. Γ( x )

3 0 O Subordiatio ad Superordiatio Note that, the failies ad special cases of the Hurwitz Lerch zeta fuctio are studied by ay authors (aog the) Shy-Der Li & Srivastava [] ad Kaeitsu et.al. [0]. Now we defie the fuctio F( z,, c ) give by ad + c ( µ ) F( z,, c) = ( + c) φµ ( z, s, c) = z = 0 + c! Thus + c ( µ ) + = 0 + c!. zf( z,, c) = z ( + c) φµ ( z,, c) = z + c ( µ ) (,,) z, zf z c = = + c ( )! for z, z <, c /{,,...}, µ,. Now we itroduce the fuctio [ zf( z,, c)] as the followig: z zf z c zf z c z z ( ) for c /{,,...}, µ,. [ (,, )] [ (,, )] =, λ >,, <, z λ + Correspodig to the fuctio [ (,, )] zf z c we defie a ultiplier, trasforatio D λ µ, c o A ad by Hadaard product for fuctio f A, we have Dµ, c = [ zf( z,, c)] f ( z). (.) Sice + c ( λ + ) [ zf( z,, c)] z, = + c ( µ ) =

4 Aabed Mohaed ad Maslia Darus 04 Therefore we have + c ( λ + ) µ, c = + = + c ( µ ) D f ( z) z az. (.) I view of (.) we obtai z( D f ( z))' = ( λ+ ) D f ( z) λd f ( z), (.4) λ, λ +, λ, µ, c µ, c µ, c z( D f ( z))' = µ D f ( z) ( µ ) D f ( z) (.5) λ, λ, λ, µ, c ad also z( D f ( z))' = ( c + ) D f ( z) cd f ( z). (.6) + µ, c µ, c µ, c, It is clear that D λ µ, c are ultiplier trasforatios. For, Z, c, µ = adλ = 0 the operator D λ µ, c were studied by Cho ad, Srivastava []. For Z, c =, µ = adλ = 0 the operator D λ µ, c were studied, by Uralegaddi ad Soaatha [], for =, µ = ad λ = 0 the operator D λ µ, c is the itegral operator studied by Owa ad Srivastava [], for ay egative real, uber ad µ =, c =, λ = 0 the operator D λ µ, c is the itegral operator studied by Jug et. al. [6], for ay o-egative iteger uber ad, µ =, c = 0, λ = 0 the operator D λ µ, c is the differetial operator defied by, Salagea [5], for = 0, µ =, λ > the operator D λ µ, c is the differetial, operator defied by Ruscheweyh [4], for Z, λ = 0, µ = the operator D λ µ, c are closely related to the ultiplier trasforatios studied by Flett [7], for, µ = adλ > the operator D λ µ, c is the ultiplier trasforatios defied by, Al-Shaqsi ad Darus [9], for c = 0, µ = ad λ > the operator D λ µ, c is the derivative operator give by Al-Shaqsi ad Darus [8], for, c = 0,, λ N 0 ad µ N the operator D λ µ, c is the liear operator defied by 0,0 0, the authors []. I Particular, we ote that D, c = f ( z) ad D,0 = zf '( z).

5 05 O Subordiatio ad Superordiatio Let p, h Η ( U) ad let ψ ( rst, ; z): U. If p ad ψ (p(z), zp'(z), z p"(z);z) are uivalet ad if p satisfies the (secod-order) differetial superordiatio hz ( ) ψ ( pz ( ), zp'( z), z p''( z); z), z U (.7) the p is called a solutio of the differetial superordiatio of (.7). A aalytic fuctio q is called a subordiat of the differetial superodiatio, if q p for all p satisfyig (.7). A uivalet subordiat q that satisfies q q for all subordiats q of (.7) is said to be the best subordiat. (Note that the best subordiat is uique up to a rotatio of U ). O the other had, a aalytic fuctio q is said to be doiat if p q for all p satisfyig ψ ( p( z), zp'( z), z p''( z); z) h( z), z U (.8) A uivalet doiat q that satisfies q q for all doiats q of (.8) is said to be the best doiat. Recetly Miller ad Mocau [6] obtaied coditios o hq, adψ for which the followig iplicatio holds: hz ( ) ψ ( pz ( ), zp'( z), z p''( z); z) qz ( ) pz ( ) ( z U). Deoted by Q the set of all fuctios q that are aalytic ad ijective o U \ E( q ) where E( q) = { ζ U :li q( z) = } ad are such that q'( z) 0 forζ U \ E( q). Further, let the subclass of Q for which q(0) = a be deoted by Qa ( ) ad Q() = Q. Defitio.. [5, Defitio.a, p. 7]. Let Ω be a set i, q Q ad be a positive iteger. The class of adissible fuctios Ψ [, q] Ω cosists of those fuctios ψ : U that satisfy the adissibility coditio ψ ( rstz,, ; ) Ω wheever r = q( ζ ), s = ζq'( ζ), ad z ζ t ζq"( ζ) R + R +, ( z U, ζ U \ E( q), ). s q'( ζ ) We write Ψ [ Ω, q] as Ψ[ Ω, q].

6 Aabed Mohaed ad Maslia Darus 06 Defiitio.. [6, Defiitio, p. 87] Let Ω be a set i, q Η[ a, ] with q'( z) 0. The class of adissible fuctios Ψ '[, q] Ω cosists of those fuctios ψ : U that satisfy the adissibility coditio ψ ( rst,, ; ζ ) Ω wheever r = q( z), s = zq'( z), ad t zq"( z) R + R +, s q'( z) z U, ζ U ad. I particular we write Ψ '[ Ω, q ] as Ψ '[ Ω, q ]. Theore.. [5, Theore.b, p. 8]. Let ψ Ψ[ Ω, q] with q(0) = a. If p Η [ a, ] satisfies the p( z) q( z). ψ pz zp z z p z z ( ( ), '( ), ''( ); ) Ω, If the behavior of q is ot ow o the boudary of U, Miller ad Mocau [5] itroduced the followig liitig procedure to prove that p q. Corollary.. [5, Corollary.b., p. 0]. Let Ω ad q be uivalet i U, with q(0) = a. Let ψ Ψ [, ] Ω for soe ρ (0,), where q ( z) = q( ρz). q ρ If p Η [ a, ] ad ψ ( pz ( ), zp'( z), z p''( z); z) Ω, the p( z) q( z). Theore.. [6, Theore, p. 88]. Let ψ Ψ'[ Ω, q] with q(0) = a. If p Qa ( ) ad ψ ( p( z), zp'( z), z p''( z); z) is uivalet i U, the The qz ( ) pz ( ). { ψ ( p( z), zp'( z), z p''( z); z) : z U} Ω I the preset paper, we shall use the ethod of differetial subordiatio ad Superordiatio itroduced by Miller ad Mocau [5, Theore.b, p. 8] ad [6, Theore, p. 88] to derive certai properties of ultiplier trasforatio Dµ, c f ad sadwich-type result is obtaied. Subordiatio Results First, the followig class of adissible fuctios is required i our first result. Defitio.. Let Ω be a set i ad qz ( ) Q H[ q(0),]. The class of adissible fuctios Π [ Ω, q] cosists of those fuctios π : U that satisfy the adissibility coditio ρ

7 07 O Subordiatio ad Superordiatio wheever π ( uvw,, ; z) Ω ζq'( ζ ) + µ q( ζ ) u = q( ζ ), v = µ ( µ )( w u)) ζq"( z) R ( µ ) R +, v u q'( z) ( z U, ζ U \ E( q), ). Now we will derive our first result. Theore.. Let π Π [ Ω, q]. If f A satisfies {,,, (( D λ, cf ( z))',( D λ, c f ( z))'),( D λ µ µ µ, cf ( z))'; z) : z U} π + Ω (.) The ( D f ( z))' q( z). Proof. Defie the aalytic fuctio p i U by p( z) = ( D f ( z))'. (.) µ+, c I view of the relatio (.5) ad fro (.) we get zp '( z ) + µ p( z ) ( Dµ, c f ( z))' =. (.) µ Further, a siple coputatio shows that z p"( z) + µ zp'( z) + µ ( µ ) p( z) ( Dµ, cf ( z))' =. µµ ( ) (.4)

8 Aabed Mohaed ad Maslia Darus 08 Defie the trasforatios fro to by s + µ r u( r, s, t) = r; v( r, s, t) =, µ t + µ s + µ ( µ ) r w ( r, s, t) =. µµ ( ) (.5) Let ψ ( rstz,, ; ) = π ( uvw,, ; z) s + µ r t + µ s + µ ( µ ) r = π ( r,, ; z) µ µ ( µ ) (.6) By aig use of Theore., ad usig equatios (.), (.) ad (.4), also fro (.6), we obtai ψ ( p( z), zp'( z), z p''( z); z) = π (( D + f ( z)) ', ( D f ( z)) ', ( D f ( z)) '; z) (.7) µ, c µ, c µ, c Hece (.) becoes π (( D f ( z))',( D f ( z))',( D f ( z))'; z) µ, c µ, c = ψ Ω ( pz ( ), zp'( z), z p''( z); z). (.8) It reais to show that the adissibility coditio for π Π [, q] Ω is equivalet to the adissibility coditio for ψ as give i Defiitio.. Note that t ( µ )( w u)) + = (µ ), s v u ad hece ψ Ψ [ Ω, q]. By Theore.,, p( z) q( z),or( D λ f ( z))' q( z). µ+, c We ext cosider the special situatio whe Ω is a siply coected doai. I this case Ω= hu ( ), where h is a coforal appig of U oto Ω. I this case the class Π [ hu ( ), q] is writte as Π [ hq, ]. The followig result is a iediate cosequece of Theore..

9 09 O Subordiatio ad Superordiatio Theore.. Let π Π [ hq, ] with q (0) =. If f ( z) A satisfies π (( D + f ( z))',( D f ( z))'),( D f ( z))'; z) h( z), (.9) λ, λ, λ, µ, c µ, c µ, c The ( D f ( z))' q( z). By aig use Corollary., we give a extesio of Theore. i the case where the behavior of qo U is ot ow. Corollary.. Let Ω ad let q be uivalet i U, q (0) =. Let π Π [, q ] Ω ρ for soe ρ (0,) where qρ ( z) = q( ρz). If f A ad π (( D + f ( z))',( D f ( z))',( D f ( z))'; z) Ω, λ, λ, λ, µ, c µ, c µ, c the ( D f ( z))' q( z). Proof. Theore. yields qρ ( z) q( z). ( D + f ( z))' q ( z). The result is ow deduced fro µ, c ρ Theore.. Let h ad q be uivalet fuctio i U, with q (0) = ad set qρ ( z) = q( ρz). ad hρ ( z) = h( ρz). Let π : U, satisfy oe of the followig coditios: (i) π Π [ hq, ρ ] for soe ρ (0,), or (ii) there exists ρ0 (0,), such that π Π [ h, q ρ ρ ] for all ρ ( ρ0,). If f A satisfies (.9), the ( D f ( z))' q( z). Proof. Followig the sae arguet i [5, Theore.d, p. 0], we have (i) By applyig Theore. we obtai ( D f ( z)) ' qρ( z). Sice qρ ( z) q( z). we deduce ( D f ( z))' q( z). (ii) If we let ( D f ( z))' = ( D f ( z))', the ρ ρ

10 Aabed Mohaed ad Maslia Darus 0 π(( D f ( z))',( D f ( z))',( D f ( z))'; ρz) ρ µ, c ρ µ, c ρ = π(( D f ( ρz))',( D f ( ρz))',( D f ( ρz))'; ρz) h ( U). µ, c µ, c ρ By usig Theore. ad the coet associated with (.8) with w ( z) = ρz which appig U it ou, we obtai ( D f ρ( z))' qρ( z),for ρ ( ρ0,). By lettig ρ, we obtai ( D f ( z))' q( z). The ext Theore yields best doiat of the differetial subordiatio (.9) Theore.4. Let h be uivalet i U, ad π : U. suppose the differetial equatio zq '( z ) + µ q( z ) z q "( z ) + µ zq '( z ) + µ ( µ ) q( z ) π qz ( ),, ; z = hz ( ) µ µ ( µ ) (.0) has a solutio q with q (0) = ad oe of the followig coditios is satisfied: (i) qz ( ) Q ad π Π [ hq, ], (ii) qz ( ) is uivalet i U ad π Π [ hq, ρ ], for soe ρ (0,) or (iii) qz ( ) is uivalet iu ad there exists ρ0 (0,) such that π Π [ h, q ρ ρ], for all ρ ( ρ0,). If f ( z) A satisfies (.9), the ( D f ( z))' q( z) ad qz ( ) is the best doiat. Proof. By usig sae ethod give by [5, theore.e, p. ], we deduce that fro Theores. ad. above, q is a doiat of (.9). Sice q satisfies (.0), it is a solutio of (.9) ad therefore q will be doiated by all doiats of (.9). Hece q will be the best doiat of (.9). Superordiatio ad Sadwich Results I this sectio the correspodig differetial superordiatio proble is ivestigated ad sadwich-type result is give. Defitio.. Let Ω be a set i, qz ( ) Hq [ (0),] with zq '( z ) 0. The class of adissible fuctios Π '[, q ] Ω cosists of those fuctios π : U that satisfy the adissibility coditio π( uvw,, ; ζ) Ω wheever

11 O Subordiatio ad Superordiatio zq '( z ) + ( µ + ) q( z ) u = q( z), v =, µ ( µ )( w u)) ζq"( z) R ( µ ) R +, v u q'( z) ( z U, ζ U, ). Theore.. Let π Π'[ Ω, q]. If f A,( D f ( z))' Q ad π µ, c µ, c (( D f ( z))',( D f ( z))'),( D f ( z))'; z) is uivalet i U, the λ {, λ, λ ((, π Df ( z))',( Dµ, c f ( z))'),( Dµ, cf ( z))'; z): z U} Ω (.) iplies qz ( ) ( D f( z))'. Proof. Let p be defied by (.) ad π by (.6). Sice π Π'[ Ω, q ],(.7) ad (.) yield { π p z zp z z p z z z U} Ω ( ( ), '( ), "( ); ) :. Fro (.5), the adissibility coditio for π Π'[ Ω, q ], is uivalet to the adissibility coditio for ψ as give i Defiitio.. Hece ψ Ψ '[ Ω, q ], ad by Theore., qz ( ) pz ( ) or qz ( ) ( D f( z))'. Siilarly as i the previous sectio, we ext cosider the special situatio whe Ω is a siply coected doai. I this case Ω = hu ( ), where h is a coforal appig of U oto Ω. I this case the class Π '[ hu ( ), q ] is writte as Π '[ hq, ]. The followig result is a iediate cosequece of Theore.5. Theore.. Let qz ( ) Hq [ (0),], hz ( ) be aalytic i U ad π Π '[ hq, ]. If f A,( D f ( z))' Q ad µ+, c π µ, c µ, c (( D f ( z))',( D f ( z))'),( D f ( z))'; z)

12 Aabed Mohaed ad Maslia Darus is uivalet i U, the hz ( ) π (( D + f( z))',( D f( z))'),( D f( z))'; z): z U λ, λ, λ, µ, c µ, c µ, c iplies qz ( ) ( D f( z))'. Theore.. Let h be aalytic i U, ad π : U. suppose the differetial equatio zq '( z ) + µ q( z ) z q "( z ) + µ zq '( z ) + µ ( µ ) q( z ) π qz ( ),, ; z = hz ( ) µ µ ( µ ) has a solutio q Q. If Π'[ hq, ], f A,( D f( z))' Q ad is uivalet i U, the π π µ, c µ, c (( D f ( z))',( D f ( z))'),( D f ( z))'; z) hz ( ) π (( D + f( z))',( D f( z))'),( D f( z))'; z): z U λ, λ, λ, µ, c µ, c µ, c iplies qz ( ) ( D f( z))', ad q( z ) is the best subordiat. Proof. The proof is siilar to the proof of Theore.4 ad is oitted. Cobiig Theores. ad., we obtai the followig sadwich-type theore. Corollary.. Let h ( z ) ad q ( z ) be aalytic fuctios i U, h( z ) be uivalet i U, q Q, with q(0) = q(0) =, ad π Π[ h, q] Π' [ h, q]. If f A,( D f ( z))' H[ q(0),] Q ad is uivalet i U, the π µ, c µ, c (( D f ( z))',( D f ( z))'),( D f ( z))'; z) h ( z) (( D f ( z))',( D f ( z))'),( D f ( z))'; z) h ( z) π µ, c µ, c iplies q ( z) ( D f ( z))' q ( z). 4 Ope Proble The defiitios, theores ad corollaries we established ca be exteded by usig the cocept of the strog differetial subordiatio itroduced i [7] by Atoio ad Roaguera ad studied i [4] by Oros ad Oros.

13 O Subordiatio ad Superordiatio ACKNOWLEDGMENT: The wor here is fully supported by UKM-GUP- TMK , Malaysia. Refereces [] A. Mohaed ad M. Darus, A operator defied by covolutio ivolvig the geeralized Hurwitz Lerch zeta fuctio, (009) (subitted). [] B. A. Uralegaddi ad C. Soaatha, Certai classes of uivalet fuctios, I Curret Topics i Aalytic Fuctio Theory, (Edited by H.M.Srivastava ad S. Owa), pp. 7-74, World Scietific, Sigapore, (99). [] N.E. Cho ad H.M. Srivastava, Arguet estiates of certai aalytic fuctios defied by a class of ultiplier trasforatios. Matheatical ad Coputer Modellig, 7(00), [4] G. I. Oros ad G. Oros, Strog differetial subordiatio, Tur, J. Math., (009), [5] G.S. Salagea. Subclasses of uivalet fuctios, Lecture otes i Math,Spriger Verlag 0(98), 6-7. [6] I. B. Jug, Y. C. Ki ad H. M. Srivastava, The Hardy space of aalytic fuctios associated with certai oe paraeter failies of itegral operators, J. Math. Aal. Appl. 76, (99), [7] J. A. Atoio ad S. Roaguera, Strog differetial subordiatio to Briot- Bouquet differetial equatios, Joural of Differetial Equatios 4(994), [8] K. Al-Shaqsi ad M. Darus, A operator defied by covolutio ivolvig the polylogariths fuctios, Joural of Matheatics ad Statistics, 4(008), [9] K. Al-Shaqsi ad M. Darus, A ultiplier trasforatio defied by covolutio ivolvig th order polylogariths fuctios, Iteratioal Matheatical Foru, 4(009) [0] S. Kaeitsu, M. Katsurada ad M. Yoshioto, O the Hurwitz-Lerch zetafuctio, Aequatioes Math., 59(000), -9 [] S. Li ad H. M. Srivastava, Soe failies of the Hurwitz- Lerch zeta fuctios ad associated fractioal derivative ad other itegr represetatios, Applied Matheatics ad Coputatio. 54(004), 75-7.

14 Aabed Mohaed ad Maslia Darus 4 [] S. Owa ad H. M. Srivastava, Soe applicatios of the geeralized Libera itegral operator, Proc. Japa Acad. Set. A Math. Sei. 6, (986), 5-8. [] S.P. Goyal ad R.K. Laddha, O the geeralized Riea zeta fuctios ad the geeralized Labert trasfor, Gaita Sadesh (997), [4] St. Ruscheweyh, New criteria for uivalet fuctios, Proc. Aer. Math. Soc., 49(975), [5] S. S. Miller ad P. T. Mocau, Differetial subordiatios: Theory ad Applicatios, Marcel Deer Ic., New Yor, 000. [6] S. S. Miller ad P. T. Mocau, Subordiats of differetial superordiatios, coplex Variables Theory Appl. 48(0), (00), [7] T. M. Flett, The dual of a iequality of Hardy ad Littlewood ad soe related iequalities, J. Math. Aal. Appl. 8, (97),