This problem that I have been unable to solve is from the book "Introduction to Inequalities" by Beckenbach and Bellman, chapter 2, page 22, problem 4.
Problem 4. Show that $$(a^2-b^2)(a^4-b^4)\le (a^3-b^3)^2$$ and $$(a^2+b^2)(a^4+b^4)\ge (a^3+b^3)^2$$ for all $a,b$.
My try for first part of the problem, just so you know that I have done some work, is this:
We have $$(a^2-b^2)(a^4-b^4)\le (a^3-b^3)^2$$ by factoring out, we are left with $$(a-b)(a+b)(a-b)(a+b)(a^2+b^2)\le (a^3-b^3)^2,$$ $$(a-b)^2(a+b)^2(a^2+b^2)\le (a-b)^2(a^2+ab+b^2)^2,$$ $$a^4+2a^2b^2+2a^3b+2ab^3+b^4\le a^4+2a^3b+ab^3+2a^2b^2+2ab^3+b^4,$$we get $$ab^2\ge 0$$which is true but for only $a,b\ge 0$, so my solution is false, can anyone hint me to correct way of solving this?