The Power Mean and the Logarithmic Mean

In a very interesting and recent note, Tung-Po Lin [I] obtained the least value q and the greatest value p such that M <L<M P q is valid for all distinct positive numbers x and y where M (x +.y.)s and L s 2 In x-ln Y The 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 functions f and ha, agR such that In f (t) a (t) [tl/ + I] > O, t > I.