Show that if $x,y>0$, $\left(\frac{x^3+y^3}{2}\right)^2≥\left(\frac{x^2+y^2}{2}\right)^3$ Through some rearrangement of the inequality and expansion, I have been able to show that the inequality is equivalent to
$$x^6-3x^4y^2+4x^3y^3-3x^2y^4+y^6≥0$$
However, I am not sure how to prove the above or if expansion and rearrangement are even correct steps.

Another longer way...
Let $x=r\cos\theta$ and $y=r\sin\theta$ where $0<\theta<\frac\pi 2$ and $r>0$. Then the expression becomes
$$|\sin(2\theta)-2||\sin(\theta+\frac\pi4)|\geq\frac{1}{\sqrt{2}}$$
which is true for $0<\theta<\frac\pi 2$.
