Convexity And Concavity Of The Modified Bessel Functions With Respect To H?lder Means

As we know,study on the concavity and convexity of a function in the tradi-tional sense can derive the binary inequalities of variables in some range.Clearly.H?lder means of two positive numbers is a generalization of the classical arith-metic mean.This enable us to generalize the definition of concavity and convexity in the traditional sense,and then study the concavity and convexity with respect to H?lder means.Recently,the concavity and convexity of a function with re-spect to H?lder means has been attracted extensively by many mathematicians.for example,complete elliptic integral,error function,zero-balanced hypergeo-metric functions,Grotzsch function and deviation function and so on have been studied[16-23].This main goal of this thesis is to study the concavity and con-vexity of the n order and n +1/2 order modified Bessel functions with respect to H?lder mean,where n is a non-negative integerIn the first chapter,firstly,we introduce the research significance and history of the subject,exposing the long history of its development,the wide influence and the key of the role.Then we introduce the binary average and two kinds of modified Bessel functions,secondly,convexity and concavity of the elliptic integrals of the first and second kinds and alternate function with respect to H?lder mean.Finally.the conclusion and innovation the major achievements and innovations about two kinds of modified Bessel functions with respect to H?lder meanIn the second chapter,firstly,we introduce the knowledge background of the first kind of modified Bessel function.Secondly,give some formulas of derivatives Finally,give some important lemmas of series,in particular,give a monotonicity criterion for two series quotientIn the third chapter,we mainly study the convexity and concavity of the non-negative integer order modified Bessel functions of the first kind with respect to H?lder means.Firstly,several important lemmas will be presented,which are used to proof the main results.Secondly,we prove the concavity and convexity of modified Bessel functions for the cases n = 0 and n?1 respectively by the above lemmas,and draw a graph of parameter curve by software.In the forth chapter,we mainly study the convexity and concavity of the non-negative integer order modified spherical Bessel functions of the first kind with respect to H?lder means.Firstly,several important lemmas will be presented,which are used to proof the main results.Secondly,we prove the concavity and convexity of modified Bessel functions for the cases n = 0 and n?1 respectively by the above lemmas,and draw a graph of parameter curve by software.In the fifth chapter,summarize this paper and expound the problems to be solved in the future.