E—凸函数的拓展研究(英文)

E—凸函数的拓展研究(英文)Title: E-Extensions of Convex Functions: A Comprehensive ReviewAbstract:Convex functio

E—() 凸函数的拓展研究英文 Title:E-ExtensionsofConvexFunctions:AComprehensive Review Abstract: Convexfunctionsareanessentialconceptinmathematics withnumerousapplicationsinvariousfieldssuchasoptimization, economics,andcomputerscience.Inrecentyears,researchers havebeenexploringandextendingtheconceptofconvex functionstoanewrealmoffunctionsknownasE-extensions.This paperaimstoprovideacomprehensivereviewofthelatest researchonE-extensionsofconvexfunctions,discussingtheir properties,applications,andpotentialfuturedirections. Introduction: Convexfunctionsplayafundamentalroleinoptimization problemsduetotheirnicemathematicalproperties.Traditionally, afunctionisconsideredconvexifitsatisfiesaspecificcondition knownasJensen'sinequality.However,inrecentyears, researchershavebeeninvestigatingtheconceptofE-extensions, whichgeneralizesconvexitytoabroaderclassoffunctions. DefinitionandPropertiesofE-Extensions: AnE-extensionisarelaxationofthenotionofconvexitythat allowsformoreflexibilityinmodelingcomplexphenomena. Unliketraditionalconvexfunctions,E-extensionsallowfor negativecurvatures,nonconstantslopeintervals,andbending behaviors.Despitethesenewproperties,E-extensionsstillretain somecorepropertiesofconvexfunctions,suchassubadditivity andmonotonicity.Thistrade-offbetweenflexibilityandretained propertiesmakesE-extensionsaversatiletoolformodeling real-worldproblems. ApplicationsinEconomics: TheconceptofE-extensionshasfoundsignificant applicationsineconomics,particularlyintheanalysisofrisk aversionanduncertainty.Traditionalutilityfunctionsineconomics oftenassumeconstantmarginalutilityofwealth,leadingtolinear riskpatterns.However,byincorporatingE-extensionsintoutility functions,economistscancaptureawiderrangeofrisk