Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean

For r R , the Lehmer mean of two positive numbers a and b is defined by L r (a,b) = a r+1 + b r+1 a r + b r . In this paper, we establish two sharp inequalities as follows: I(a,b) > L -1 6 (a,b) and L(a,b) > L -1 3 (a,b) for all a,b > 0 with a = b . Here I(a,b) = 1 e b b a a 1 b-a and L(a,b) = b-a log b-log a denote the identric mean and logarithmic mean of two positive numbers a and b with a = b , respectively.