Uniformly Starlike and Uniformly Convexity Properties Pertaining to Certain Special Functions

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1 Global Joural of Sciece Frotier Research Volume 11 Issue 7 Versio 1.0 October 011 Tye: Double lid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (US) Olie ISSN : & Prit ISSN: Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios y Rohit Mukheree, mber Srivastava Swami Keshvaad Istitute of Techology, Maagemet ad Gramotha,Jaiur Idia bstract - The aim of this aer is to establish sufficiet coditios for the fuctio z { ψ (z)} to be i the classes of uiformly starlike ad uiformly covex fuctios associated with the arabolic regio Re { ω } > ω 1. Further, covolutio of the fuctios which are aalytic i the oe uit disk with egative coefficiets have bee ivestigated. Fially, similar results usig a itegral oerator have also bee obtaied. Keywords : Uivalet fuctios, Parabolic Starlike fuctios, Uiformly Covex fuctios, Itegral oerator, Geeralized Fox-Wright fuctio. 000 Mathematics Subect Classificatio : 30 C 45 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios Strictly as er the comliace ad regulatios of: 011. Rohit Mukheree, mber Srivastava.This is a research/review aer, distributed uder the terms of the Creative Commos ttributio-nocommercial 3.0 Uorted Licese htt://creativecommos.org/liceses/by-c/3.0/), ermittig all ocommercial use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited.

2 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios Rohit Mukheree α, mber Srivastava Ω 011 bstract - The aim of this aer is to establish sufficiet coditios for the fuctio z{ ψ (z ) } to be i the classes of uiformly starlike ad uiformly covex fuctios associated with the arabolic regio Re { ω } > ω 1. Further, covolutio of the fuctios which are aalytic i the oe uit disk with egative coefficiets have bee ivestigated. Fially, similar results usig a itegral oerator have also bee obtaied. Keywords ad hrases : Uivalet fuctios, Parabolic Starlike fuctios, Uiformly Covex fuctios, Itegral oerator, Geeralized Fox-Wright fuctio. Let deote the class of fuctios of the form I. INTRODUCTION (1.1) that are aalytic i the oe uit disk = {z : z < 1 }. lso let S deote the subclass of cosistig of all fuctios f (z) of the form f (z) = z a z, a 0. (1.) fuctio f is said to be uiformly covex i if f is a uivalet covex fuctio havig the roerty that for every circular arc γ cotaied i with cetre also i,the image curve f(γ) is a covex arc. Deotig the class of all uiformly covex fuctios by UCV, it was show i [10, 13] that To give the geometric iterretatio of (1.3), let Ω = which is the iterior of the arabola v = u 1. The (1.3) (1.4) class closely related to the class UCV is the class of arabolic starlike fuctios deoted by US [15]. This class US ad the class UC P of uiformly covex fuctios have bee studied i [ 1,6,7,11,15,16,17 ]. survey of these fuctios ca the works of [1 ]. Let f S, 0 λ <, ad 0 μ< 1, the [,13] ad f (z) = z + = f UCV < Re + f UC (λ,μ) Re 1+ z f US (λ,μ) Re f (z) a z ( 1 ) { ω : ω = u + i v,re ( ω ) > ω 1 } z f (z) f UCV 1+ Ω f (z) λ, λ 1, (z ). + µ + µ (1.5) (1.6) uthor α : Deartmet of Mathematics, Swami Keshvaad Istitute of Techology, Maagemet ad Gramotha, Jagatura, Jaiur-3005, Idia. rohit@skit.ac.i uthor Ω : Deartmet of Mathematics, Swami Keshvaad Istitute of Techology, Maagemet ad Gramotha, Jagatura, Jaiur-3005 Idia. amber@skit.ac.i October 5 Global Joural of Sciece Frotier Research Volume XI Issue VII v v v v v v v v v v v v v v V ersio I

3 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios We recall here that the hadamard roduct (or covolutio) of f (z) of the form (1.1 ) ad g (z) = z + b z, is defied as (f g ) z = z a z b, z. (1.7) The geeralized Fox-Wright fuctio [8] aearig i the reset aer is defied by Global Joural of Sciece Frotier Research Volume XI Issue VII Versio I October { Γ ( a + α )} z = 1 =, (1.8) = 0 { (b Γ + β )}! = 1 where α ( = 1,, ) ad β ( = 1,, ) are real ad ositive, 1+ β > α, ad ( = 1,, ) ad ( = 1,, ) ca take o -iteger values. = 1 = 1 For (1.8), we have { a } Γ = 1 or where Now, we defie a liear oerator (a, α ; ) 1, ; ψ (z) = ψ z (b, β ; ) 1, ; z { ψ (z)} = { Γb = 1 } z + { Γ [ a + α ( 1 )]} z = 1 { [ b ( 1 )] } Γ + β ( 1)! = 1 { b } Γ = 1 z Ω = z + { a } Γ = 1 G { [a ( 1)]} z Γ +α = 1 { [b ( 1)] } Γ +β (1)! = 1 b { b } Γ = 1 Ω = Ψ. { a } Γ = 1 : S S as follows: G f (z) = z Ω f (z) (1.9) (1.10)

4 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios { Γb } { Γ [ a + α ( 1 )]} z = 1 = 1 = z - a { a { [ b 1 )] } Γ } Γ + β ( ( 1)! = 1 = 1 or G = z ( a,,, ) a, b z (1.11) 011 (1.1) Corresodig to the oerator G defied i (1.11), we let S (λ) for -1 γ < 1, deotes the subclass of fuctios f S satisfyig the ieuality I the reset aer we shall use the followig Lemmas [, 18] to establish our mai results: Lemma 1.1. fuctio f (z) of the form (1.1) is i the class US (λ,μ) if where M 1 > 0 is a suitable costat. Lemma 1.. fuctio f(z) of the form (1.1) is i the class UC where M > 0 is a suitable costat. Theorem.1. If Where (a, b,,, ) { Γb } = 1 = { Γa } = 1 z(g f (z)) z(g f (z)) Re γ 1 G f (z) G f (z) [(1 +λ) ( λ+µ )] a (1 µ ) M 1, II. (λ,μ) if [(1 +λ) ( λ+µ )] a (1 µ ) M, MIN RESULTS b > a + 1, ad 1+ β> α, = 1 = 1 = 1 = 1 { Γ[a +α (1)]} = 1 { [b 1)] } Γ +β ( (1)! = 1 the a sufficiet coditio for the fuctio z{ ψ ( z ) } to be i the class US (λ, μ), 0 λ <, ad 0 μ< 1,, is { a} 1 (a +α, α; ) 1, ; (a, ; ) ; Γ +λ α 1, = 1 ψ M. 1 µ (b +β, β; ) 1, ; ψ 1+ (b, β; ) 1, ; { b} Γ = 1 Proof. Sice (1.13) (1.14) (1.15) (.1) October 7 Global Joural of Sciece Frotier Research Volume XI Issue VII v v v v v v v v v v v v v v V ersio I

5 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios z{ ψ (z)} = { Γa = 1 { Γb = 1 } } z + Global Joural of Sciece Frotier Research Volume XI Issue VII Versio I October { Γ [ a + α ( 1 )]} z = 1 { [ b ( 1 )] } Γ + β ( 1)! = 1 so by virtue of Lemma 1.1, we eed oly to show that { [a ( 1)]} Γ +α = 1 [(1 +λ) ( λ+µ )] { [b ( 1)]} Γ +β (1)! = 1 (1 µ ) M 1 Now, we have { Γ [a +α ( 1)]} = 1 [(1 +λ) ( λ+µ )] { Γ [b +β (1)]} (1)! = 1 = [(1 +λ )( + )-( λ+µ )] = 0 { Γ [(a ) ]} +α + α = 1 = (1 +λ) = 0 { Γ [(b ) ]} +β + β! = 1 { Γ [a +α ( + 1)]} = 1 { Γ [b +β (+ 1)]} (+ 1)! = 1 { (a )} Γ +α = (1 µ ) 0! = { Γ (b +β)} = 1 = 1 = 1 (.) { Γa} { Γb} ( a +α, α ;, ) 1 ; = ( 1 + λ ) ψ 1 ( b + β β ; ;,, ) 1

6 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios { Γa} (a, ; ) ; α 1, = 1 + (1 µ ) ψ 1 (1 µ ) (b, ; ) ; β 1, { Γb} = 1 (1 µ ) M 1 which i view of Lemma 1.1 gives the desired result. Theorem.. Let f (z) be give by (1.1) ad -1 γ < 1, the f(z) S ( γ) if (1 γ) (a,b,,,) a 1γ = (.3) 011 October where ( a, b,,, ) is give by (1.1). Proof. y virtue of (1.13), it is sufficiet to show that z(g f (z)) z(g f (z)) Re γ 1 G f (z) G f (z) z(g f (z)) or 1 1 γ G f (z) z 1 (a,b,,,) a z 1 or = 1 1γ z (a,b,,,) a z = or (1)(a,b,,,) a = (1 γ) 1 (a,b,,,) a = or (1 γ) (a,b,,,) a 1γ = Hece the theorem. III. N INTEGRL OPERTOR 9 Global Joural of Sciece Frotier Research Volume XI Issue VII v v v v v v v v v v v v v v V ersio I (a, α ; ) 1, ; z I this sectio we obtai sufficiet coditios for the fuctio ϕ z = ψ (x) dx to be i the (b,β ; ) 1, ; 0 classes US (λ,μ) ad UC (λ,μ). Theorem 3.1. If b > a + 1, ad 1+ β> α, the a sufficiet coditio for the fuctio = 1 = 1 = 1 = 1 z ϕ (z) = ψ(x) dx to be i the class US (λ,μ), 0 λ < ad 0 μ< 1, is 0

7 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios (1 +λ) ( λ+µ ) (a α, α; ) 1, ; ψ(1) ψ 1 (1 µ ) (1 µ ) (b β, β; ) 1, ; Global Joural of Sciece Frotier Research Volume XI Issue VII Versio I October Proof. Sice Now, we have { Γ(a )} {a} α Γ ( λ+µ ) = 1 = 1 + M 1 +. (1 µ ) { Γ(b )} { b} β Γ = 1 = 1 z ϕ (z) = ψ(x) dx 0 {a} Γ { [(a ) ]} Γ α +α z = 1 = 1 = z +! { b} = Γ { Γ[(b β ) +β]} = 1 = 1 { Γ[(a ) ]} α +α = 1 [(1 +λ) ( λ+µ )] { Γ[(b ) ]} β +β! = 1 { (a )} Γ +α = 1 = (1 +λ) = 0 { Γ (b )} +β! = 1 { (a )} Γ = 1 { Γ(b )}! = 1 [(a ) ]} Γ α +α = 1 ( λ+µ ) = 0 Γ[(b ) ]} β +β! = 1 (3.1) { (a )} {a} Γ α Γ = 1 = 1 { Γ(b )} { b} β Γ = 1 = 1

8 Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios (a α, α; ) 1, ; = (1 +λ) ψ(1) ( λ+µ ) ψ 1 (b β, β; ) 1, ; { Γ(a α)} {a} Γ = 1 = 1 + ( λ+µ ) (1 µ ) { Γ(b )} { b} β Γ = 1 = 1 which i view of Lemma 1.1, leads to the result (3.1). 011 Theorem 3.. If b > a + 1, ad 1+ β> α, = 1 = 1 = 1 = 1 z the a sufficiet coditio for the fuctio ϕ (z) = ψ(x) dx to be i the class UC (λ,μ), 0 λ < ad 0 0 μ< 1, is Proof. The result follows as direct coseuece of the Theorem 3.1, keeig Lemma 1. i view. IV. PRTICULR CSES 4.1 For λ= ad μ=0, Theorem.1 ad Theorem 3.1 corresods to the results recetly obtaied by Chaurasia ad Kumawat [3] for α = For = 1( = 0,1,,) ; = 1( = 0,1,,), the Theorems established i the reset aer readily yield the results due to aa ad Jai []. 4.3 The results due to Chaurasia ad Srivastava [4], Dixit ad Verma [5] ad Shamugam et.al. [14] also follow as articular cases of our mai results. V. CKNOWLEDGEMENT The authors are grateful to Professor H.M. Srivastava, Uiversity of Victoria, Caada for his kid hel ad valuable suggestios i the rearatio of this aer. REFERENCES RÉFÉRENCES REFERENCIS 1. l-refai, Olah ad Darus, Maslia, Extesio to the Owa-Srivastava Fractioal Oerator with licatios to Parabolic Starlike ad Uiformly Covex Fuctios, Iteratioal Joural of Differetial Euatios, Volume (009), rticle ID 5979, 18 Pages.. aa, I.. ad Jai, Nidhi,Parabolic Starlike ad Uiformly Covex Fuctios, It. Joural of Math. alysis, Vol. 4, (010), o. 39, Chaurasia, V.. L. ad Kumawat, Yaghvedra,Uiformly Starlike d Uiformly Covexity Proerties For Certai Secial Fuctios, Global Joural of Sciece ad Frotier Research, vol. 11,(011). 4. Chaurasia, V..L. ad Srivastava, mber, Uiformly starlike ad uiformly covex fuctios ertaiig to secial fuctios, J. Ieual. Pure ad l. Math., 9(1) (008), rt. 30, Dixit, K.K. ad Verma, V., Uiformly starlike ad uiformly covexity roerties for hyergeometric fuctios, ull. Cal. Math. Soc., 93(6) (001), Goodma,.W., O uiformly covex fuctios,. Polo. Math., 56 (1991), Goodma,.W., O uiformly starlike fuctios,j. Math. al. ad l., 155(1991), Guta, K.C., Jai, R. ad Sharma,., study of uified fiite itegral trasforms with alicatios, J. Ra. cad. Phy. Sci., (4) (003), Kaas, S. ad Roig, F., Uiformly starlike ad covex fuctios ad other related classes of uivalet fuctios,. Uiv. Mariae Curie Ma, W. ad Mida, D., Uiformly Covex Fuctios,. olo. Math., 57 (199), Roig,F.,Some radius results for uivalet fuctios, Joural of Mathematical alysis licatios, Vol.194, No.1, (1995), Roig, F., survey o uiformly covex ad uiformly starlike fuctios, ales Uiv. Mariae Curie-Sklodowska Sectio, Vol.47, (1993), Roig, F., Uiformly Covex Fuctios ad a Corresodig Class of Starlike Fuctios, Proc. mer. Math. Soc., 118(1993), Shamugam,T.N.,Ramachadra,C.,Sivasubramai a S. ad Gagadhara,.,Geeralized hyergeo - metric fuctios associated with uiformly starlike ad uiformly covex fuctios, cta Ciecia Idica, XXXIM() (005), Silverma, H., Class of Starlike Fuctios Defied by the Dziok Srivastava Oerator, Kyugook Math. J. 49 (009), October 31 Global J oural of Sciece Frotier Research Volume XI Issue VII v v v v v v v v v v v v v v V ersio I

9 Global Joural of Sciece Frotier Research Volume XI Issue VII Versio I October Uiformly Starlike ad Uiformly Covexity Proerties Pertaiig to Certai Secial Fuctios 16. Srivastava, H.M. ad Mishra,.K., licatios of fractioal calculus to arabolic starlike ad uiformly covex fuctios, Comuter Math. l., 39 (000), Srivastava, H.M., Mishra,.K. ad Das, M.K., class of arabolic starlike fuctios defied by meas of certai fractioal derivative oerator,fract.calc. l.al.,6 (003), Swamiatha,., Hyergeometric Fuctios i the Parabolic Domai,Tamsui Oxford Joural of Mathematical Scieces, 0(1) (004)1-16.