Optimal Inequalities For Two Toader-type Means

This paper introduces the construction of a new mean.Add any two means X(a,b)and Y(a,b)to two variables of Toader mean,which is known as Toader-type mean,denoted by T[X(a,b),Y(a,b)].It is well known that there are many classical means of any two positive numbers a and b,for example,which are arithmetic mean A(a,b),geometric mean G(a,b),harmonic mean H(a,b),contra-harmonic mean C(a,b),quadratic mean Q(a,b),logarithmic mean L(a,b),identric mean I(a,b),centroidal mean C(a,b),Seiffert mean of the first kind S(a,b),Seiffert mean of the second kind S(a,b)and so on.Obviously,we have the following unequal chain H(a,b)<G(a,b)<L(a,b)<S(a,b)<I(a,b)<A(a,b)<S(a,b)<C(a,b)<Q(a,b)<C(a,b),where a,b>0 and a ? b.Recently,Toader introduced a mean relative to the complete elliptic integral.In this paper,according to the location of the above inequalities,we establish optimal inequalities between two Toader-type means and many classical means or single parameter means.In 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 definition of binary mean,common means and the research results of some mean inequalities,especially the related concept of Toader mean and its related optimal inequalities.Finally,these inequal-ities of two Toader-type means and the innovation of this paper are presented.In the second chapter,in preparation for the main research results,and be-cause of Toader mean is related to complete elliptic integrals,we introduce the concept and properties of complete elliptic integrals in this paper.In the third chapter,the optimal inequalities for a Toader-type mean by vari-ous combinations of arithmetic and contra-harmonic means are discussed.There-fore,the optimal upper and lower bounds of complete elliptic integral of the second kind in a certain interval are obtained.In the forth chapter,the optimal inequality for a Toader-type mean by gener-alized Seiffert mean of the second kind is discussed.Therefore,the optimal upper and lower bounds of complete elliptic integral of the second kind in a certain interval are obtained.In the fifth chapter,summarize this paper and expound the problems to be solved in the future.