Prove the following inequality: $\frac{a^3b}{(3a+b)^p} + \frac{b^3c}{(3b+c)^p} + \frac{c^3a}{(3c+a)^p} \ge \frac{a^2bc}{(2a+b+c)^p} + \frac{b^2ca}{(2b+c+a)^p} + \frac{c^2ab}{(2c+a+b)^p}$ For all positive real numbers $a,b,c,p$.
What I tried is to use the power mean inequality but it has no effect because p is any real rumber. I haven't gotten any results with CSB either. Maybe Holders inequality could work but I have no idea what to put in the other brackets.
