An optimal double inequality between geometric and identric means

We find the greatest value p and least value q in ( 0 , 1 / 2 ) such that the double inequality G ( p a + ( 1 − p ) b , p b + ( 1 − p ) a ) < I ( a , b ) < G ( q a + ( 1 − q ) b , q b + ( 1 − q ) a ) holds for all a , b > 0 with a ≠ b . Here, G ( a , b ) , and I ( a , b ) denote the geometric, and identric means of two positive numbers a and b , respectively.