No it does not, and you don't need calculus to find the answer:
Let's say you've got a square initially with side "a".
Then, you look at the degenerate rectangle with two sides 3a, and the other two length 0.
The perimeter of the degenerate rectangle is 3a+0+3a+0=6a, that is, greater than your original square's 4a, yet the rectangle's area is zero..