If a and b = 1 then ((a+b)^2)/(4*a*b) = 1 (not > )
(a+b)^2 = a^2 + 2ab + b^2 4ab = 2(2ab) subtract 2ab from both numerator and denominator a^2 + b^2 ~ 2ab If a = b then a^2 = b^2 so the numerator becomes 2a^2 and the denominator becomes 2aa = 2a^2 so the quotient is 1.
If a and b are not equal then one is greater than the other by a positive amount say d.
This is symmetrical so let's make a = b + d
now we have (b + d)^2 + b^2 > 2(b+d)b b^2 +2bd + d^2 + b^2 > 2b^2 + 2bd and d^2 is > 0 as defined so the left is > the right It is true when not equal a and b