The power mean and the logarithmic mean

Abstract

In a very interesting and recent note, Tung-Po Lin [1] obtained the least valueqand the greatest valuepsuch thatMp<L<Mqis valid for all distinct positive numbersxandywhereMs=(xs+ys2)1s   and   L=x−yIn x-In yThe object of this paper is to give a simpler proof than Lin's of a more general result. More precisely, the author obtained the classes of functionsfαandhα,α∈Rsuch thatIn fα(t)−hα(t)[t1/α+1]−α>0,   t>1.