BOUNDSFORPOISSONANDNEUTROSOPHICPOISSON DISTRIBUTIONSASSOCIATEDWITHCHEBYSHEV POLYNOMIALS
AbiodunTinuoyeOladipo
CommunicatedbyProfessorP.K.Banerji
MSC2010Classifications:30C45,30C50
Keywordsandphrases:analyticfunction,Poisson,univalentfunctions,distributionseries,subordination,Fekete-Szego. Theauthoracknowledgesandthanktherefereesfortheirusefulandvaluablesuggestions.
Abstract ThepaperinvestigatesPoissonandneutrosophicPoissondistributionseries.The firstfewcoefficientboundsforPoissondistributionwhoseparametertakesadefiniteanddeterminedvalueswerestudied,whilecoefficientboundsforneutrosophicPoissondistributionwhose parametertakesundeterminedvaluesorinaccuratestatisticaldatawereinvestigated.Examples todemonstrateourargumentforneutrosophicPoissondistributionwereprovided.
1Introduction
Let A denotetheclassoffunctionsoftheform f (z)= z + ∞ n=2 anzn (1.1)
whichareanalyticintheopenunitdisk U = {z : z ∈ C, |z| < 1},andlet S ∈ A consistingof univalentfunctionsin U normalizedwith f (0)= f (z) 1 = 0.. Afunction f oftheform(1.1)issaidtobestarlikeoforder α if Re zf (z) f (z) >α,z ∈ U andconvexoforder α if Re 1 + zf (z) f (z) >α,z ∈ U
andarerespectivelydenotedby S∗(α) and K(α) for0 ≤ α< 1.
Let P denotethewellknownclassofCaratheodoryfunctionswiththepositiverealpartin U Let P (pk), 0 ≤ k< ∞ denotethefamilyoffunctionsp,suchthat p ∈ P ,and p ≺ pk in U ; wherethefunction pk mapstheunitdiskconformallyontotheregion Ωk suchthat1 ∈ Ωk and δΩ = {u + iv : u 2 = k2(u 1)2 + k2 v 2}
Thedomain Ωk iselliptic,hyperbolic,parabolicandpossiblycoverstherighthalfplanewhenever k> 1, 0 ≤ k< 1,k = 1, and k = 0respectively.
Recently,severalauthorshaveinvestigatedvarioussublcassesofanalyticfunctionsassociated withChebyshevpolynomialsduetoitsimportanceinboththeoreticalandpracticalapplications (seefordetail[2,5,11])andthereferencestherein.Inthisworkweshallconcernourselveswith Chebyshevpolynomialofthefirstandsecondkinds.Inthecaseofrealvariable x ∈ [ 1, 1] we have Tn(x)= cosnθ,Un(x)= sin(n + 1)θ sinθ ,
PalestineJournalofMathematics
where n isthedegreeofpolynomialand x = cosθ.Itisobservedthatfor t = cosα,α ∈ ( π 3 , π 3 ), then H (z,t)= 1 1 2tz + z2 = 1 + ∞ n=1
sin(n + 1)α sinα zn
Thatis H (z,t)= 1 + 2cosαz +(3cos 2 α sin2 α)z 2 + ...,z ∈ U.
Theabovecanbefurtherexpressedasfollows
H (z,t)= 1 + U1(t)+ U2(t)z 2 + (z ∈ U,t ∈ [ 1, 1]) (1.2) Itisobservedthat U1(t)= 2t,U2(t)= 4t2 1,U3(t)= 8t3 4t,... (1.3)
ItisimportanttorecallthatChebyshevpolynomialsofthefirstkindandsecondkindareconnectedasfollows Tn(t) dt = nUn 1(t),Tn = Un(t) tUn 1(t), 2Tn(t)= Un(t) Un 2(t)
Recently,Poissondistributionseriesgatheredmomentumingeometricfunctiontheory.Porwal andSrivastava[12],Murugusundaramoorthyet.al[8],Porwal[9]SrivastavaandPorwal[3], extensivelystudiedPoissondistributionseriesbyestablishingvariousgeometricpropertiesin termsofunivalency,starlikeness,convexityandharmonicstructures.Detailcanbefoundinthe abovementionedliteraturesandthereferencestherein. Meanwhile,Poissondistributionseriesisdefinedas K(m,z)= z + ∞ n=2
mn 1 (n 1)! e mzn (1.4) andbyratiotesttheradiusofconvergenceoftheaboveseriesisinfinity. Theconvolutionoftwoseries f (z) and g(z)= z + ∞ n=2 bnzn isdefinedbypowerseriesgiven by (f ∗ g)(z)= z + ∞ n=2 anbnzn =(g ∗ f )(z); [1].
Letthelinearoperator I(m) : A → A defineby Φ(z)= K(m,z) ∗ f (z)= z + ∞ n=2
mn 1 (n 1)! e manzn (1.5)
Let f and g beanalyticin U ,then f issaidtobesubordinateto g writtenas f (z) ≺ g(z),ifthere existsafunction w(0)= 0and |w(z)| < 1suchthat f (z)= g(w(z))(z ∈ U ) [7]. Definition1.1 Let λ ≥ 0,t ∈ ( 1 2 , 1],m> 0,afunction f ∈ A oftheform(1.1)issaidtobelong totheclass M m λ (H (z,t)) ifthefollowingconditionholds (1 λ) zΦ (z) Φ(z) + λ 1 + zΦ (z) Φ (z) ≺ H (z,t)= 1 1 2tz + z2 , (z ∈ U )
TheauthorismotivatedbytheearlierworksofSrivastavaandPorwal[3],wemakeuseof Chebyshevpolynomialsexpansionstoestimatethefirstfewcoefficientsoffunctionsin M m λ (H (z,t)). Theauthorsin[3]investigatedcoefficientinequalitiesofPoissondistributionseriesinconicdomainrelatedtouniformlyconvex, k-spiralikeandstarlikefunctions.Intheirresults m isaprecise parameterforexamplewecansay m = 1,thatis m isaccuratelydefined.Sectin2ofthepresent investigationfurthertheworkin[3]byestimatingtheboundonthefirstfewcoefficientsand classicalFekete-SzegetheoremviasubordinationprincipleinconnectionwithChebyshevpolynomials.Insection3ofthepresentinvestigationtheworkin[3]isfurtherextendedtoasituation when m isnotpreciselydefined.Classicalprobabilitydistributionsonlydealswithspecifieddata anditsparametersarealwaysgivenwithaspecifiedvalue,whileneutrosophicprobabilitydistributiongivesamoregeneralandclearityofthestudyissueswhen m isaninterval.Summarily theworkin[3]andsection2ofthepresentinvestigationareparticularcasesoronesolutionof ourinvestigationinsection3.
170 AbiodunTinuoyeOladipo
2BoundsforPoissonDistributionSeries
Forthepurposeofourinvestigationthefollowinglemmashallbeemployed. Lemma2.1 [7]If ω ∈ Ω then ω2 tω2 1 ≤ max{1, |t|}
foranycomplexnumber t.Theresultissharpforthefunction ω(z)= z or ω(z)= z2,(seealso [4,6]).
NextweshallconsiderthefirstfewcoefficientboundsfortheclassdefinedinDefinition2.1. Theorem2.1 Letthefunction f givenby(1.1)beintheclass M m λ (H (z,t)), m> 0,λ ≥ 0.Then wehave |a2|≤ 2t m(1 + λ)e m |a3|≤ 2t m2(1 + 2λ)e m max 1, 4t2 1 2t + 2t(1 + 3λ) (1 + λ)2 a3 µa 2 2 ≤ 2t m2(1 + 2λ)e m max 1, 4t2 1 2t + 2t(e m µ) 2tλ(3e m 2µ) e m(1 + λ)2
Proof. If f ∈ M m λ (H (z,t)),and ω ∈ Ω suchthat (1 λ) zΦ (z) Φ(z) λ 1 + zΦ (z) Φ (z) = H (z,t). (2.1) Itisobservedfrom(1.3)and(2.1)that 1 +(1 + λ)me ma2z +[(1 + 2λ)m 2 e ma3 (1 + 3λ)m 2 e 2ma 2 2]z 2 + = 1 + U1(t)c1z +(U1(t)c2 + U2(t))z 2 + (2.2) Henceby(2.2)wehave a2 = U1(t)c1 (1 + λ)me m (2.3) and a3 = U1(t) m2(1 + 2λ)e m c2 + U2 U1 + (1 + 3λ)U1(t) (1 + λ)2 c 2 1 (2.4) By(2.3)and(2.4)wehave a3 µa 2 2 = U1 m2(1 + 2λ)e m c2 + ρc 2 1 where ρ = U2 U1 + (e m(1 + 3λ) µ(1 + 2λ))U1(t) (1 + λ)2
ThedesiredresultisobtainedbyapplyingLemma2.1 Corolllary2.1 k ∈ [0, ∞), 0 ≤ α< 1,λ = 0inTheorem2.1andif f ∈ M m 0 (H (z,t)) thenwe have |a2|≤ 2t me m |a3|≤ 2t m2e m max 1, 8t2 1 2t
BOUNDFORPOISSONANDNEUTROSOPHICPOISSON 171
3NeutrosophicPoissonDistributionSeries
Recently,preciselyin1995,Smarandacheintroducedtheconceptofneutrosophictheory.Itis anewbranchofphilosophyasageneralizationforthefuzzylogic,alsoasageneralizationof theintrinsticfuzzylogic.Thisnewbranchofphilosophyinthefuzzylogicprovidesanew foundationfordealingwithissuesthathaveindeterminatedata,whichmaybenumbers,(see [10]forneutrosophicnumbers)andthereferencestherein. Theuseofneutrosophiccrispsetstheorywiththeclassicalprobabiltydistributionsparticularly Poissondistribusion,exponentialdistibutionsanduniformdistributionsopensanewstairway fordealingwithissuesthatfollowtheclassicaldistributionsandatthesametimecontaindata notspecifiedaccurately.Theextensionofclassicaldistributionsaccordingtologic,meansthat parametersofclassicaldistributiontakeundeterminedvalues,whichallowsdealingwithevery situationsthatareencounteredwhiledealingwithstatisticaldata,especiallywhenworkingwith vagueandinaccuratestatisticaldata.
NeutrosophicPoissondistributionofadiscretevariable X isaclassicalPoissondistributionof x,butitsparameterisimprecise.ForexampleandfromourearlierdiscussioninSection2 m can besetwithtwoormoreelements.Themostcommonofsuchdistributioniswhen m isinterval. Let NP (x = k)= e mN . (mN )k k! ,k = 0, 1, 2... where mN isthedistributionparameterandisequaltotheexpectedvalueandthevariance.That is NE(x)= NV (x)= mN and N = d + I isaneutrosophicstatisticalnumber(see[10])andthereferencestherein. Nowwemodifiy(1.4)asfollows K(mN ,z)= z + ∞ n=2
(mN )n 1 (n 1)! e mN zn (3.1)
Ourquestinthissectionistoinvestigatethepossibilityofhavingarangeofvaluesforthefirst fewcoefficientsforneutrosophicPoissondistributionserieswhoseparameterisindeterminate. WeshalluseexamplestodemonstratetheconceptofneutrosophicPoissondistributionusing Theorem2.1. Theorem3.1 Letthefunction f givenby(1.1)beintheclass M m λ (H (z,t)), m ∈ [1, ∞].Then wehave |a2|≤ 2t mN (1 + λ)e mN |a3|≤ 2t m2 N (1 + 2λ)e mN max 1, 4t2 1 2t + 2t(1 + 3λ) (1 + λ)2
172 AbiodunTinuoyeOladipo
a3 µa 2 2 ≤ 2t m2e m max 1, 4t2 1 2t + 2t(e m µ) e m Corollary2.2 For k ∈ [0, ∞), 0 ≤ α< 1,λ = 1,t = 1 2 inTheorem3.1,andif f ∈ M m 1 (H (z, 1 2 )) thenwehave |a2|≤ em 2m |a3|≤ em 3m2 a3 µa 2 2 ≤ em 3m2 max 1, µ 2e m 4e m
a3 µa 2 2 ≤ 2t m2 N (1 + 2λ)e mN max 1, 4t2 1 2t + 2t(e mN µ) 2tλ(3e mN 2µ) e mN (1 + λ)2
Exampleforcasestudy
Inacompany,phoneemployeereceivesphonecalls,thecallsarrivewithrateof[1,3]callsper minute,wewillcalculatetheprobabilitythattheemployeewillnotreceiveanycallwithina minute.
Corollary3.1 Letthefunction f givenby(1.1)beintheclass M m λ (H (z,t)), m ∈ [1, 3].Then wehave |a2|≤ 2t [1, 3](1 + λ)e [1,3] |a3|≤ 2t [1, 3]2(1 + 2λ)e [1,3] max 1, 4t2 1 2t + 2t(1 + 3λ) (1 + λ)2 a3 µa 2 2 ≤ 2t [1, 3]2(1 + 2λ)e [1,3] max 1, 4t2 1 2t + 2t(e [1,3] µ) 2tλ(3e [1,3] 2µ) e [1,3](1 + λ)2 .
Corollary3.2 Letthefunction f givenby(1.1)beintheclass M m 0 (H (z,t)), m ∈ [1, 3] and λ = 0.Thenwehave |a2|≤ 2t [1, 3]e [1,3] |a3|≤ 2t [1, 3]2e [1,3] max 1, 4t2 1 2t + 2t a3 µa 2 2 ≤ 2t [1, 3]2e [1,3] max 1, 4t2 1 2t + 2t(e [1,3] µ) e [1,3]
Conclusion: Weconcludethattheneutrosophicprobabilitydistributiongivesamoregeneral andclaritystudyoftheproblemunderinvestigation.Thatistheclassicalprobabilitydistribution discussedinsection2isonesolutionamongthesolutionsresultingfromthestudyinsection3, thisisasaresultofgivingthedistributionparametersseveraloptionspossiblewhichdoesnot remainlinkedtoasinglevalue.Wehopethisworkwillserveasastairwaytostudyothertypes ofprobabilitydistributionsaccordingtotheneutrosophiclogic.
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Authorinformation
AbiodunTinuoyeOladipo,DepartmentofPureandAppliedMathematics,LadokeAkintolaUniversityof Technology,Ogbomoso,P.M.B.4000Ogbomoso,Nigeria.
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174 AbiodunTinuoyeOladipo
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