Optimal inequalities between harmonic,geometric,logarithmic,and arithmetic-geometric means

We find the least values p,q,and s in (0,1/2) such that the inequalities h (pa + (1 - p)b,pb + (1 - p) a) > a g (a,b),g (q a + (1 - q) b,q b + (1 - q) a) > a g (a,b),and l (s a + (1 - s) b,s b + (1 - s) a) > a g (a,b) hold for all a,b > 0 with a ≠ b,respectively. here a g (a,b),h (a,b),g (a,b),and l (a,b) denote the arithmetic-geometric,harmonic,geometric,and logarithmic means of two positive numbers a and b,respectively. © copyright 2011 yu-ming chu and miao-kun wang.