Proving an inequality with power

Question

I want to prove that there exist a constant $c>0$ such that

$$\left(a^\frac43+b^\frac43\right)\le c\Big(\left(1+a^2\right)^\frac23+\left(1+b^2\right)^\frac23\Big)$$

for all $a,b\ge 0$

Thanks for help

Answer 1

$\left(1+a^2\right)^\frac23+\left(1+b^2\right)^\frac23 > \left(a^2\right)^{\frac23} + \left(b^2\right)^{\frac23} = a^{\frac43} + b^{\frac43}.$

Answer 2

The inequality with $c=1$ is trivial and such inequality is optimal/sharp by considering $(a,b)=(M,M)$ with $M\to +\infty$.