<s>[INST] inequality $(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$ I am trying to prove the inequality:
$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$$ given $a,b,c,d$ are positive and unequal.
starting from LHS 
since AM of mth power > mth power of AM 
$$(a^3+b^3+c^3+d^3)/4 > ((a+b+c+d)/4)^3$$
multiplying both sides by $(a+b+c+d)$
$$(a+b+c+d)(a^3+b^3+c^3+d^3)/4 > (a+b+c+d)^4/(4^3)$$
$\implies$
$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a+b+c+d)^4/16$$
now taking expression on the RHS and using AM of mth power > mth power of AM 
$$(a^2+b^2+c^2+d^2)/4 > ((a+b+c+d)/4)^2$$
squaring both sides 
$$(a^2+b^2+c^2+d^2)^2/16 > (a+b+c+d)^4/(4^4)$$
$$\implies (a^2+b^2+c^2+d^2)^2 > ((a+b+c+d)^4)/16$$
Now I have proved that both LHS and RHS are greater than  $$((a+b+c+d)^4)/16$$
still unable to prove LHS > RHS. Please help with this.
 [/INST] Hint: You will get the left-hand side minus the right-hand side:
$${a}^{3}b+{a}^{3}c+{a}^{3}d-2\,{a}^{2}{b}^{2}-2\,{a}^{2}{c}^{2}-2\,{a}^
{2}{d}^{2}+a{b}^{3}+a{c}^{3}+a{d}^{3}+{b}^{3}c+{b}^{3}d-2\,{b}^{2}{c}^
{2}-2\,{b}^{2}{d}^{2}+b{c}^{3}+b{d}^{3}+{c}^{3}d-2\,{c}^{2}{d}^{2}+c{d
}^{3}
$$ Now you need to combine like terms:
$$a^3b-2a^2b^2+ab^2=ab(a^2-2ab+b^2)$$ and so on.
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