I have the following inequality:

$$ \left(\frac{1}{3} \left( \left(\frac{a+b}{2}\right)^3 + \left(\frac{a+c}{2}\right)^3 + \left(\frac{b+c}{2}\right)^3 \right) \right)^\frac{1}{3} \leq \left(\frac{a^p+b^p+c^p}{3}\right)^\frac{1}{p}$$ which is true $\forall a,b,c \in \mathbb{R}^+\cup\{0\} $.

I would like to show that this inequality above holds if and only if $ p \geq \frac{3}{2} $.

i.e. the inequality holds for all non-negative reals $a,b,c $ as long as $ p \geq \frac{3}{2}$.

Any help whatsoever would be appreciated, so please comment even if you don't have a full solution (even proving 1 direction of the implication would be very helpful).

P.S. I'm not sure exactly what to tag this, I've tagged it as convex-optimisation for now, as one of the things I've tried unsuccessfully is maximising/minimising one the sides subject to the constraint of the other side being held constant. If this is tagged wrong, please let me know the right tag(s).