How to prove $a^3+b^3+c^3+d^3+3\left(a+b+c+d\right) \geq 14+2abcd$ when $a^2+b^2+c^2+d^2=4$

Question

This is a problem from AoPS I can't solve:

Let $a,b,c,d\geq0$ with $a^2+b^2+c^2+d^2=4$. How can I prove: $$a^3+b^3+c^3+d^3+3\left(a+b+c+d\right) \geq 14+2abcd$$

My attempt:

I try setting $a=2\cos(x), b=2\sin(x)\cos(y), c=2\sin(x)\sin(y)\cos(z),d=\sin(x)\sin(y)\sin(z)$ for $x,y,z\in[0,\frac\pi2[$.

Then indeed $a^2+b^2+c^2+d^2=4$ but I would have to prove $$2 \sin (x) \sin (y) \left(4 \sin ^2(x) \sin ^2(y) \sin ^3(z)+4 \sin ^2(x) \sin ^2(y) \cos ^3(z)+3 \sin (z)+3 \cos (z)\right)+\cos (x) \left(6-8 \sin ^3(x) \sin (y) \sin (2 y) \sin (2 z)\right)+8 \sin ^3(x) \cos ^3(y)+6 \sin (x) \cos (y)+8 \cos ^3(x)-14\geq0$$ which I don't know how to do.

Also I tried using Lagrange but the stationary points have no closed form.

Answer 1

We need to prove that: $$4\sum_{cyc}a^3+3\sum_{cyc}a^2\sum_{cyc}a\geq\frac{7}{2}\left(\sum_{cyc}a^2\right)^2+8abcd$$ or $$\sum_{cyc}a^2\left(4\sum_{cyc}a^3+3\sum_{cyc}a^2\sum_{cyc}a\right)^2\geq\left(7\left(\sum_{cyc}a^2\right)^2+16abcd\right)^2.$$ Now, let $a=\min\{a,b,c,d\}$, $b=a+u,$ $c=a+v$ and $d=a+w.$

Thus, $$\frac{1}{2}\left(\sum_{cyc}a^2\left(4\sum_{cyc}a^3+3\sum_{cyc}a^2\sum_{cyc}a\right)^2-\left(7\left(\sum_{cyc}a^2\right)^2+16abcd\right)^2\right)=$$ $$=512\sum_{cyc}(3u^2-2uv)a^6+128\sum_{cyc}\left(19u^3+5u^2v+5u^2w-\frac{34}{3}uvw\right)a^5+$$ $$+64\sum_{cyc}(31u^4+22u^3v+u^3w+22u^2v^2-30u^2vw)a^4+$$ $$+4\sum_{cyc}(263u^5+223u^4v+223u^4w+282u^3v^2+282u^3w^2-24u^3vw-142u^2v^2w)a^3+$$ $$+4\sum_{sym}\left(40.5u^6+101u^5v+57u^4v^2+85u^3v^3+12.5u^4vw+54u^3v^2w-\frac{76}{3}u^2v^2w^2\right)a^2+$$ $$+\sum_{sym}(31.5u^7+69u^6v+51u^5v^2+89u^4v^3+73u^5vw-33u^4v^2w+162u^3v^3w-39u^3v^2w^2)a+$$ $$+\sum_{sym}(21u^7v-48u^6v^2+79u^5v^3-48u^4v^4+4.5u^6vw+51u^5v^2w+39u^4v^3w-108u^4v^2w^2+59u^3v^3w^2).$$ Now, all coefficients before $a^k$ for any $k\in\{1,2,3,4,5,6\}$ are non-negative by Muirhead.

Also, by Muirhead and AM-GM we obtain: $$\sum_{sym}(21u^7v-48u^6v^2+79u^5v^3-48u^4v^4+4.5u^6vw+51u^5v^2w+39u^4v^3w-108u^4v^2w^2+59u^3v^3w^2)$$ $$\geq\sum_{sym}(21u^7v-48u^6v^2+31u^5v^3+4.5u^6vw+51u^5v^2w+39u^4v^3w-108u^4v^2w^2+59u^3v^3w^2)\geq$$ $$\geq uvw\sum_{sym}(4.5u^5+51u^4v+39u^3v^2-108u^3vw+59u^2v^2w)\geq$$ $$\geq uvw\sum_{sym}(51u^4w-108u^3vw+59u^2v^2w)\geq$$ $$\geq uvw\sum_{sym}(2\sqrt{51\cdot59u^4w\cdot u^2v^2w}-108u^3vw)=2u^4v^2w^2(\sqrt{51\cdot59}-54)\geq0$$ and we are done!