Two optimal double inequalities between power mean and logarithmic mean

For p 2 R the power mean Mp.a; b/ of order p, the logarithmic mean L.a; b/ and the arithmetic mean A.a; b/ of two positive real values a and b are defined by Mp.a; b/ D 8>< >: ap C bp 2 1 p ; p 6D 0; p ab; p D 0; L.a; b/ D 8< : b 􀀀 a log b 􀀀 log a ; a 6D b; a; a D b and A.a; b/ D aCb 2 , respectively. In this article, we answer the questions: What are the greatest values p and r, and the least values q and s, such that the inequalities Mp.a; b/ 1 3 A.a; b/ C 23 L.a; b/ Mq.a; b/ and Mr .a; b/ 2 3 A.a; b/ C 1 3 L.a; b/ Ms.a; b/ hold for all a; b > 0?