In the diagram the measurements are:
PQ = a
PS = b
PM = PO = (a + b)/2
MQ = SO = TN = (a - b)/2

The blue rectangle measures (a - b)/2 by b, the same as the orange rectangle.

Rectangle PQRS, with area ab, is made up of the green plus the orange rectangle..
The square MNOP, with area [ (a + b)/2] ² , is made up of the green plus the blue rectangles plus the yellow square.

area of MNOP - area PQRS = the area of the yellow square = [ (a - b)/2] ².

Hence the area of PQRS < area MNOP, that is ab < [ (a + b)/2] ².