<s>[INST] Turkevicius inequality From Zdravko Cvetkovski's Inequalities — Theorems, Techniques and Selected:

Let $a$, $b$, $c$ and $d$ be positive real numbers. Prove that:
$$a^4+b^4+c^4+d^4+2abcd \geq a^2b^2+ a^2c^2+ a^2d^2+ b^2c^2+ b^2d^2+ c^2d^2.$$


Please suggest how to prove this inequality using basic methods. No logarithms.
 [/INST] Assume $a\geq b\geq c\geq d.$ Let $f(a) = a^4+b^4+c^4+d^4+2abcd - a^2b^2-a^2c^2-a^2d^2-b^2c^2-b^2d^2-c^2d^2.$ Then, $f'(a) = 4a^3 -2a(b^2+c^2+d^2)+2bcd$ and $f''(a) = 12a^2 - 2b^2-2c^2-2d^2\geq 0.$ Hence, $f'$ is increasing and $f'(b) = 2(b^3-b(c^2+d^2)+bcd) \geq 2b(b-c)(b-d)\geq 0.$ Thus, $f(a)$ is increasing and therefore:
$$f(a)\geq f(b) = b^4+c^4+d^4+2b^2cd-2b^2c^2-2b^2d^2-c^2d^2 = $$$$b^4-2b^2(c^2+d^2-cd)+c^4+d^4-c^2d^2 = (b^2 - c^2-d^2+cd)^2+2cd(c-d)^2\geq 0.$$
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