An inequality $(a+b+c)^p-a^p-b^p-c^p \le C \sum\limits_{\mathrm{cyc}} [(a + b)^p - a^p - b^p]$

Question

Does anyone know how to prove the following inequality?

There exists some constant $C=C(p)$ such that $$(a+b+c)^p-a^p-b^p-c^p\leq C[(a+b)^p-a^p-b^p+(a+c)^p-a^p-c^p+(b+c)^p-b^p-c^p] $$ for any $a,b,c>0$ and $p>1$.

Answer 1

Some estimate:

If $a=b=c=x$ we get $$(3x)^p-3x^p \leq C\Big(3(2x)^p-6x^p\Big)$$

so if divide it by $x^p$ we get $$3^p-3 \leq C\Big(3\cdot 2^p-6\Big)$$

so $$C \geq {3^p-3 \over 3\cdot 2^p-6}$$

So try if $C = \displaystyle{3^p-3 \over 3\cdot 2^p-6}$ works. You can also assume that $a+b+c=1$, when trying to prove it.