An optimal double inequality between geometric and identric means

We find the greatest value pp and least value qq in (0,1/2)(0,1/2) such that the double inequality G(pa+(1−p)b,pb+(1−p)a)0a,b>0 with a≠ba≠b. Here, G(a,b)G(a,b), and I(a,b)I(a,b) denote the geometric, and identric means of two positive numbers aa and bb, respectively.