Australian Journal of Basic and Applied Sciences. On Certain Classes of Meromorphically P-ValentFunctionsWith Positive Coefficients

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1 ISSN: Austrlin Journl of Bsi Applied Sienes Journl home pge: On Certin Clsses of Meromorphilly P-VlentFuntionsWith Positive Coeffiients Abdul Rhmn S. Jum Husmldin I. Dhye Deprtment Of Mthemtis, Alnbr University, Rmdi-Irq Deprtment Of Mthemtis, Tirit University, Tirit Irq A R T I C L E I N F O Artile history: Reeived 6 April 05 Aepted June 05 Avilble online July 05 Keywords: P Vlent, Poly logrithm funtion, Hdmrd produt, Meromorphi, Positive oeffiients. AMS Subjet Clssifitions : 30C45 A B S T R A C T A purpose of this pper is to introdue the lssn,p α, β, γ of meromorphilly p- vlent funtions by introduing new opertora f(z)ssoited with polylogrithm funtion.we study vrious properties suh s oeffiient inequlity, growth distortion theorems,losure theorems, onvolution properties, rdii of meromorphilly p-vlentstrlieness onvexity, weighted men rithmetimen. 05 AENSI Publisher All rights reserved. To Cite This Artile: Abdul Rhmn S. Jum Husmldin I. Dhye., On Certin Clsses of Meromorphilly P-VlentFuntionsWith Positive Coeffiients. Aust. J. Bsi & Appl. Si., 9(0): -34, 05 INTRODUCTION At lst time,the lssil polylogrithm funtion ws invented in 696,by Leibniz Bernoulli see (Alhindi, K.R. M. Drus, 005), s mentioned in (EL- Ashwh, R.M., 0).For z nturl number with,the polylogrithm funtion (whih is lso nown s Jonquiere's funtion)is defined by the bsolutely onvergent series: z n Li z = n. n= Lter on, mny mthemtiins studied the polylogrithm funtion suh s Euler, Spene, Able,Lobhevsy,Rogers,Rmnujun, mny others (Gorhov, A.B., 994),where they disovered mny funtionl identities by using polylogrithm funtion.however, the wor employing polylogri-thm hs been stopped mny dedes, lter. During the pst four dedes, the wor using polygrithm hs gin been in intensified vividly due to its importne in mny fields of mthemti, suh s omplex nlyti, lgebr, geometry,topology mthemtil physis (quntum field theory)(( Gorhov, A.B., 994; Oi, S., 009; Ponnusmy, S. S. Sbpthy, 996)in 996,Ponnusmy Sbpthy disussed the geometri mpping properties of the generlized polylogrithm. Reently,Al-Shqsi Drus (AlShqsi, K. M. Drus, 008) generlized Rusheweyh slgen opertors,using polylogrithm funtion on lss D of nlyti funtions in the open unit dis = z C: z. By ming use of the generlizedopertor they introdued relted properties. A yer lter, sme uthors gin employed the n th order.polylogrithm funtion to define multiplier trnsformtion on the lss D in (Al- Shqsi, K. M. Drus, 009). Re ll the polylogrithm funtion to be on meromorphi p-vlent type,letσdenote the lss of normlized meromorphi p-vlent funtions of the form f z = z p + n=p n z n, whih re meromorphi p-vlent in the puntured unit disδ = z C 0 < z <. Afuntionf Σ is meromorphistrlie of order ρ, 0 ρ < iff R zf z > ρ z = f z \0. The lss of ll suh funtions is denoted by Σ ρ. A funtion f Σ is meromorpi onvix of order ρ, 0 ρ < iff R+zf"zf z>ρ,z = \0.The lss of ll suh funtions is denoted by Σ ρ. Let Σ p be the lss of funtions f Σof the form f z = + z p n=p n z n, n 0.() The sublss of Σ p onsisting of strlie funtions of order ρis denoted by Σ p ρ, the sublss of Σ p onsisting of onvix funtions of order ρis denoted by Σ p ρ. Corresponding Author: Abdul Rhmn S. Jum, Deprtment Of Mthemtis, Alnbr University, Rmdi-Irq E-mil: dr_ jum@hotmil.om

2 N 0,p 3 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 For funtions f z given by () g z = n=p b n z p + z n we define the Hdmrd produt or onvolution of f gby f g z = + z nb p n=p n z n = g f z. (3) see (Rin, R.K. H.M. Srivstv, 006), whih re nlyti univlent in. Liu Srivstv (Liu, J.L. H.M. Srivstv, 000) defined funtion (α.,.. α q ; β,. β s ; z)by multiplying the well-nown generlized hyprogometri funtion qf s with z p s follows: Φ f z = Ψ z f z p α.,.. α q ; β,. β s ; z = z p qf s α.,.. α q ; β,. β s ; z, where α.,.. α q ; β,. β s re omplex prmeters q s +. p N.Anlogous to Liu Srivstv wor (Liu, J. H.M. Srivstv, 004) orresponding funtion Ψ z given by Ψ z = z Li z = + z p n p+ zn n=p. 4 We onsider liner opertor Φ f z : Σ p Σ p, whih is defined the following Hdmrd produt (or onvolution): = z p + n p + nz n. n=p Now, we define the liner opertor A f z : Σ p Σ p wsstudied byk. R. Alhindi M. Drus [], sfollows: A f z = Φ f z pz p = z p + n p + nz n. 5 n=p+ Now, by ming use of the opertor A f z we define the lssn,p α, β, γ of funtions in Σ p s follows. Definition : A funtion f(z) of the form () is sid to be in the lssn,p α, β, γ if it stisfies the follow- ing inequlity: where 0 α < p, 0 < β, γ <, p N; z. The following re speil lsses of the lssn,p α, β, γ N,p α,, = f Σ p : R z p+ A f z > α, 0 α < p, p N; z, see 8. N 0,p α,, = f Σ p : R z p+ f z > α, 0 α < p, p N; z, see 4. 3 N 0,p α, β, Exmples: Also we note somespeil lsses of the lssn,p α, β, γ s the following: Meromorphilly multivlent funtions hve been extensively studied (for exmple)by mny others suh s Rin Srivstv (Rin, R.K. H.M. Srivstv, 006), Yng, EL- Ashwh, Sif Kilimn, Mostf others.in the present pper, we obtinoeffiient inequlity, growth distortion theorems, losure heorems, Hdmrdprodut, rdii ofmeromorphilly p- vlentstrlieness, onvexity weighted men rithmeti men for these funtionsfor the lssn,p. Coeffiient Inequlity: We now give neessry suffiient onditionfor funtionf tobein the lssn,p Theorem : Let f Σ p given by (). Then f N,p α, β, γ,if only if

3 n= 4 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 where 0 α < p, 0 < β, γ <, p N; z. Proof.Suppose tht (7) is holds. Then z p+ pz For z = r < the left h side of (8) is bounded bove by Thus f N,p Conversely, supposef N,p Then by (6), z p γ z p Sine R z z for ll z, then Now hoosing the vlues of z on the rel xis so tht the funtionz p+ A f z is rel. By lering the denomintor in (9) letting z through positive vlues, we get: Hene the proof is omplete. Corollry : Let the funtion f(z) defined by () be in the lss N,p Then The result is shrp for the funtion: 3. Growth Distortion Theorems: A growth distortion property for the funtionfto be in the lss N,p α, β, γ is given s follows: Theorem : Let the funtion f(z) defined by () be in the lss N,p Then for 0 < z = r <, we hve

4 5 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 with equlity for r p βγ p p + βγ β r p f z r p + βγ p p + βγ β r p, p r p+ Proof.By Theorem, we hve Then for 0 < z = r <, βγ p + βγ β r p f z r p+ + f z = z p + βγ p p + βγ β z p p N 3 p + βγ β n n + βγ β n=p+ n=p+ n=p+ n βγ p p + βγ β f z r p + nr n, n=p+ r p + rp n n=p+ βγ p + βγ β r p, (n p + ) n βγ p. r p + βγ p p + βγ β r p f z r p nr n, n=p+ r p rp n n=p+ r p βγ p p + βγ β r p, whih, together, yield (). Furthermore, it follows from Theorem tht βγ p n n + βγ β. n=p+ Hene f z p z p+ + n n z n, n=p+ f z p z p+ + rp n n p z p+ + f z p z p+ n=p+ βγ p + βγ β r p, n=p+ n n z n, f z p z p+ rp n n n=p+

5 6 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 p z p+ βγ p + βγ β r p, whih, together, yield ().It is ler tht the funtion given by (3) is extreml funtion. Hene the proof is omplete. 4. Closure Theorems: We now prove the losure theorems s follws. Theorem 3: Let f p z = z p, f n = z p + βγ p z n, n p +, p N. n + βγ β (n p+) Then f(z) is in the lss N,p α, β, γ if only if it n be expressed in the form where μ n 0 n=p μ n =. f z = μ n f n z, n=p Proof.First suppose tht f z n be expressed of the form Then n=p+ whih shows, thtf N,p f z = μ n f n z, n=p = z p + βγ p μ n n + βγ β n=p+ z n. (n p+) n + βγ β (n p+) βγ p μ n βγ p n + βγ β n=p+ μ n = μ p. (n p+) Conversely, suppose f N,p α, β, γ, then βγ p n, n p +, p N. n + βγ β (n p+) Setting n + βγ β (n p+) μ n = βγ p n we get Hene the proof is omplete. μ p = n=p+ μ n f z = μ n f n z. Theorem 4: Let the funtion f i z = + z p n=p+ n,i z n, i =,,, be in the lss N,p Then the funtion F z = i= μ i f i z, were i= μ i =, is lso in the lss N,p n=p,

6 7 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 Proof.From Theorem, we hve Sine Then i= μ i n=p+ n=p+ n=p+ n + βγ β (n p + ) n βγ p. F z = z p μ i n,i n + βγ β n + βγ β This ompletes the proof of the theorem. Theorem 5: The lss N,p α, β, γ is onvex. n=p+ i= (n p + ) z n, i= μ i n,i (n p + ) n,i βγ p = βγ p. Proof.In order to proof the theorem it is enough to show tht the funtion (z) defined by i= μ i is in the lss N,p α, β, γ, where re in the lss N,p Then z = δf z + δ g z, 0 δ f z = z p + n z n, n 0 n=p+ g z = z p + b n z n, b n 0, n=p+ By using Theorem, we get n=p+ n + βγ β Thus z N,p Hene the proof is omplete. 5. Convolution Properties: For the funtions z = z p + δ n + δ b n z n. (n p + ) n=p+ δ n + δ b n δβγ p + δ βγ p = βγ p. f i z = z p + n,i z n, n,i 0; i =, 4 n=p+, belonging to the lss N,p α, β, γ, wedenote by (f f )(z) the Hdmrd produt (or the onvolution) of the funtionsf (z) f (z), tht is f f z = z p + n, n, z n. 5 n=p+ Theorem 6: Let the funtionsf i z (i=, ) defined by (4) be in the lssn,p α, β, γ,the

7 8 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 f f z N,p ω, β, γ, where βγ p ω = p p + βγ β. The result is shrp for the funtionsfi(z)(i =, ) given by f i z = βγ p + zp p + βγ β z p, i =,, p N. Proof.Sine f i z N,p α, β, γ (i =, ). Then by Theorem we hve: n + βγ β (n p+) βγ p n,i i =,. n=p+ Thus by the Cuhy- Shwrz inequlity, we obtin n + βγ β (n p+) βγ p n=p+ n, n,. 6 To prove the theorem we need to find the lrgest ω suh tht n + βγ β (n p+) βγ p ω n, n,. or we must get: whih is equivlent to From (6), we hve By simplifying it, we get: n=p+ n, n, p ω n, n, n, n, p p ω p βγ p n + βγ β n p + ; p N, n p + ; p N. (n p+) p ω p. βγ p ω p n p + ; p N. n + βγ β (n p+) Now, defining the funtionφ n by βγ p φ n = p n p +. n + βγ β (n p+) This funtion is n inresing funtion of n. Thus, we hve βγ p ω φ p = p p + βγ β. Hene the proof is omplete. Theorem 7: Let the funtionf (z) defined by (4) be in the lss N,p α, β, γ the funtion f (z) defined by (4) be in the lss N,p α, β, γ. Then f f z N,p ψ, β, γ, where ψ = p βγ p p p + βγ β. The result is shrp for the funtions f i (z)(i =, ) given by f z = z p + βγ p p + βγ β z p p N,

8 9 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 f z = z p + βγ p p + βγ β z p p N. Proof.By using the sme tehnique of Theorem 6 we prove the theorem,hene it is omitted. Theorem 8.If f z = z p + n=p+ n, z n N,p α, β, γ f z = + z p, n p +, p N.Then f f z N,p Proof.By using Theorem it is enough to show tht: n + βγ β Sine = n=p+ n=p+ n =p + n =p + Thus f f z N,p α, β, γ. n p+ βγ p n=p+ n, n, n,, n + βγ β n p+ βγ p n, n, n + βγ β n p + βγ p n, n, n + βγ β n p + βγ p n,, z n N,p α, β, γ wit n, Corollry.If f z = + z p n =p + n, z n N,p α, β, γ f z = + z p n =p + n, z n N,p α, β, γ wit 0 n,, n p +, p N.Then f f z N,p α, β, γ. Theorem 9.Let the funtionsf i z (i =, ) defined by (4) be in the lssn,p α, β, γ p + βγ β 4βγ p 0, then the funtion h(z) defined by z = z p + α n, + α n, z n, 7 is lso in the lssn,p α, β, γ. Proof.By Theorem we hve Then Hene n =p + n =p + n =p + n =p + n =p + n + βγ β n p + βγ p n,, n + βγ β n p + βγ p n,. n + βγ β n p + βγ p n + βγ β n p + βγ p α n,, α n,.

9 30 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 n =p + n + βγ β n p + βγ p To proof the theorem it is suffiient to show tht n + βγ β n p + βγ p n =p + Thus the inequlity (8) will be stisfied if, for n p or if n + βγ β α n, + α n, α n, + α n,.. 8 n + βγ β n p + n + βγ β n p + βγ p βγ p 4βγ p 0, n = p, p +, p +, n p +, The left h side of the bove inequlity is n inresing funtion of n, so itstisfied for ll n if p + βγ β 4βγ p 0, whih is given by our hypothesis. This ompletes the proof of the theorem. Theorem 0: Let the funtionsf i z (i =, ) defined by (4) be in the lssn,p α, β, γ. Then the funtion (z )defined by (7) belongs to the lss N,p Ω, β, γ, where 4βγ p Ω = p p + βγ β. The result is shrp for the funtions f i z (i =, ) given by f i z = βγ p p + z p + βγ β z p, i =,, p N. Proof.From Theorem, we hve Now n =p + n =p + n + βγ β n p + βγ p n + βγ β n p + βγ p n,i i =,. α n,i for f i z N,p α, β, γ i =,, we hve n =p + n + βγ β n p + βγ p n =p + n + βγ β n p + βγ p α n, + α n, Hene, we hve to find the lrgest Ω suh tht n + βγ β p Ω n p + 4βγ p n p +, or n,i. 9 4βγ p Ω = p n p +. n + βγ β n p + We observe tht the right h side of the bove inequlityis n inresing funtion of n,we get i =,,

10 3 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 4βγ p Ω = p p + βγ β. Whih ompletes the proof of Theorem Rdii of Meromorphilly p-vlentstrliness Convexity: Theorem : Let the funtionf(z)defined by () be in the lss N,p α, β, γ. Then f(z) is meromorphilly p- vlentstrlie of order δ 0 δ < p in the dis z < r, where r = The result is shrp. Proof.From Theorem, we hve: n =p + inf n p + n + βγ β n + βγ β n p + βγ p n δ p δ p +n n p + n βγ p, f (z ) is sid to be meromorphilly p-vlentstrlie of order δ 0 δ < p, if R z f z > δ, f z or z f z + p f z p δ 0 δ < p. f z Now z f z + pf z n =p + p + n n z n = f z z p + z n n =p + n =p + n p + n n z + n =p + n z p +n. To prove the theorem the bove inequlity must be less thn or equl to p δ, so n δ p δ n z p +n p δ. 0 n =p + Then by Corollry the inequlity (0) will be true if tht is, p +n n + βγ β z p +n n p + p δ βγ p n δ z n + βγ β n p + βγ p n δ p δ The infimum of the bove quntity is the rdii of strlieness of the funtion f (z ) in the lss N,p α, β, γ The shrpness follows by hoosing the sme extreml funtion (0). Whih ompletes the proof of theorem. Theorem : Let the funtion f(z)defined by () be in the lss N,p α, β, γ. Then f(z) is meromorphilly p-vlent onvex of order ζ 0 ζ < p in the dis z < r, where r = The result is shrp. Proof.It is enough to show tht inf n p + p p ζ + βγ β n p + βγ p p + n ζ, p +n, p +n..

11 3 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 or zf "z R + f z z f z + p f z f z > ζ 0 ζ < p, z < r, p N, = n n =p + n p + n n z p + z p + n =p + n n n z p +n n =p + n p + n n z p n =p + n n z p +n. To prove the theorem the bove inequlity must be less thn or equl top ζ, or n p + n ζ p p ζ n z p +n. From Theorem,we get Thus z n =p + p p ζ + βγ β z p +n n p +. βγ p p + n ζ p p ζ + βγ β n p + βγ p p + n ζ p +n n p +, p N. By hoosing r to be the infimum of the bove quntity we get the result.the shrpness follows by hoosing the sme extreml funtion (0).This ompletes the proof of the theorem. 7. Weighted Men Arithmeti Men: Definition : If the funtions f (z ) g (z ) defined by () re in the lssn,p α, β, γ, then the weighted men i (z ) of the two funtions is defined sfollows i z = i f z + + i g z. Theorem 3: Let the funtions f z g z defined by () re in the lss N,p α, β, γ. Then their weighted men is lso in the lssn,p α, β, γ. Proof.the weighted men of f (z ) g (z ) is: i z = i f z + + i g z, = i f z = z p + n z n + + i g z = z p + b n n =p + n =p + z n = z p + n =p + i n + + i b n z n. By using Theorem, it is suffiient to show tht n + βγ β n p + n =p + i n + + i b n z n = i n + βγ β n p + n n =p i n + βγ β n p + b n n =p +

12 33 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 i βγ p + + i βγ p Hene i z N,p α, β, γ Whih ompletes the proof of theorem. = βγ p. Theorem 4: If the funtions f i (z )(i =,..., d ) defined by f i z = z p + n,i z n, n,i 0, n p +, i =,,, d, n =p + belongs to the lss N,p α, β, γ, then their rithmeti men defined by is lso in the lss N,p α, β, γ. Proof.Sine Then by using Theorem, we must show tht n =p + n + βγ β z = d i = z = z p + d n p + d d d i = Hene z N,p α, β, γ. This ompletes the proof of the theorem. d i = n =p + n,i = d f i z d i = d n,i i = n =p + z n. βγ p = βγ p. n + βγ β n p + n,i REFERENCES Alhindi, K.R. M. Drus, 005. A new lss of meromorphi funtions involving the poly logrithm funtion, Hindwi Publishing Corportion, Journl of Complex Anlysis, Artile ID864805, 5. AlShqsi, K. M. Drus, 008. An opertor defined by onvolution involving the polylogrithms funtions, Journl of Mthemtis sttistis, 4(): Al-Shqsi, K. M. Drus, 009.''A multiplier trnsformtion defined by onvolution involving nth order polylogrithm funtions,'' Interntionl Mthemtil Forum, 4(37-40): Aouf, M.K., 008. A lss of meromorphi multivlent funtions with positive oeffiients.tiwnese journl of mthemtis, 9: EL- Ashwh, R.M., 0. Properties of ertin lss of p-vlentmeromorphi funtions ssoited with new integrl opertor, At Universittis Apulensis, 9: Gerhrdt, C.I. G.W. Leibniz, 97. Mthe mtishe Shriften III / I, Georg Olms, NewYor, NY, USA. Gorhov, A.B., 994. Polylogrithms in rithmeti geometry, in Proeedings of the Interntionl Congress of Mthemtiins, Zu rih, Switzerl, Jum, A.R.S. HzhZirr, 03. On ertin lsses of meromorphilly p-vlent funtions with positive oeffiients defined by linier opertor Interntionl Journl of Bsi & Applied Sienes, 05: Lewin, L., 98. Poly logrithms Assoited Funtions, North-Holl, Oxford, UK. Liu, J.L. H.M. Srivstv, 000. A Liner opertor ssoited fmilies of meromorphilly multivlent funtions, J. Mth. Anl. Appl., 59: Liu, J. H.M. Srivstv, 004. lsses of meromorphilly multivlent funtions ssoited with thegenerlized hypergeometri funtion, Mthemtil omputer Modelling, 39(): -34. Mostf, A.O., 0. Inlusion results for ertin sublsses of p-vlentmeromorphi funtions ssoited with new opertor, Journl of inequlities Applition, 69. Oi, S., 009. Guss hypergeometri funtions, multiple polylogrithms, multiple zet vlues, Publitions of the Reserh Institte for Mthemtil Sienes, 54(4):

13 34 Abdul Rhmn S. Jum Husmldin I. Dhye, 05 Ponnusmy, S. S. Sbpthy, 996. Polylogrithms in the theory of univlent funtions, Results inmthemtis, 30(-): Rin, R.K. H.M. Srivstv, 006. A new lss of meromorphilly multivlent funtions withpplitions to generlized hypergeometri funtions, Mth. Comput. Modelling, 43: Sif, A. A. Kilimn, 0. On ertin sublsses of meromorphilly p-vlent funtions ssoited by the liner opertor D λ n, Journl of Inequlity Applition, Artile ID 4093, 6. Shrm, R.K. D. Singh, 0. An inequlity of sublsses of univlent funtions relted to omplex order by onvolution method, Gen.Mth. Notes, 3(): Yng, D., 996. On lss of meromorphistrlie multivlent funtions, Bull. Inst. Mth. Ad.Sini, 4: 5-57.