INEQUALITIES AND MONOTONICITYPROPERTIES FOR SOME SPECIAL FUNCTIONS不等式与单调性 一些特殊函数地性质.pdf

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1、JournalofMathematicalInequalitiesVolume3,Number1(2009),79–91INEQUALITIESANDMONOTONICITYPROPERTIESFORSOMESPECIALFUNCTIONSCHAO-PINGCHEN(communicatedbyG.Alassia)Abstract.Themonotonicity,convexity,log-convexityandcompletemonotonicitypropertiesforsomespecialfunctio

2、nsareproved,andsomeinequalitiesareestablished.1.IntroductionAfunctionfissaidtobecompletelymonotoniconanintervalI,iffhasderiva-tivesofallordersonIandsatisfies(−1)nf(n)(x)0forallx∈Iandn=0,1,2,....(1)Iftheinequality(1)isstrict,thenfissaidtobestrictlycompletelymon

3、otoniconI.Itisknown(Bernstein’sTheorem)thatfiscompletelymonotonicon(0,∞)ifandonlyif∞f(x)=e−xtdμ(t),0whereμisanonnegativemeasureon[0,∞)suchthattheintegralconvergesforallx>0,see[13,p.161].Adetailedcollectionofthemostimportantpropertiesofcompletelymonotonicfunct

4、ionscanbefoundin[13,ChapterIV].Asequence{an}∞n=1ofrealnumbersiscalledstrictlyconvex(concave),ifan+2−2an+1+an>(<)0foralln1.Asequence{an}∞ofrealnumbersiscalledstrictlylog-convex(log-concave),ifitisn=1positiveanda2<(>)an+1nan+2forallintegersn1.Bythearithmetic-g

5、eometricmeaninequality,thelog-convexityimpliestheconvexity,andtheconcavityimpliesthelog-concavity.Mathematicssubjectclassification(2000):33B15,26A48.Keywordsandphrases:Monotonicity,convexity,log-convexity,completemonotonicity,psifunction,inequality.Thisworkwass

6、upportedbyNaturalScientificResearchPlanProjectofEducationDepartmentofHenanProvince(2008A110007),byProjectofthePlanofScienceandTechnologyofEducationDepartmentofHenanProvince(2007110011).c,Zagreb79PaperJMI-03-0780CH.-P.CHENTheclassicalgammafunction∞Γ(x)=tx−1e

7、−tdt(x>0)0isoneofthemostimportantfunctionsinanalysisanditsapplications.Thepsiordigammafunction,thelogarithmicderivativeofthegammafunction,andthepolygammafunctionscanbeexpressed[1,pp.259-260]asΓ(x)∞e−t−e−xtψ(x)==−γ+dt,(2)Γ(x)01−e−t∞n(n)n+1t−xtψ(x)=(−1)edt(3)

8、01−e−tforx>0andn∈N,whereγ=0.57721566490153286...istheEuler-Mascheroniconstantdefinedbyn1γ=limDn,whereDn=∑−logn.(4)n→∞kk=1ItisalsoknownastheEuler-Mascheroniconstant.AccordingtoGlaish