Two optimal double inequalities between power mean and logarithmic mean ?
For p?R p ? R the power mean Mp(a,b) M p ( a , b ) of order p p , the logarithmic mean L(a,b) L ( a , b ) and the arithmetic mean A(a,b) A ( a , b ) of two positive real values a a and b b are defined by In this article, we answer the questions: What are the greatest values p p and r r , and the least values q q and s s , such that the inequalities M p ( a , b ) ? 1 3 A ( a , b ) + 2 3 L ( a , b ) ? M q ( a , b ) and M r ( a , b ) ? 2 3 A ( a , b ) + 1 3 L ( a , b ) ? M s ( a , b ) hold for all a,b>0? a , b > 0 ?
power mean logarithmic mean arithmetic mean