let $a,b,c,x,y,z>0$ such $x+y+z=a+b+c,abc=xyz$,and $a>\max\{x,y,z\}$,

I conjecture $$a^n+b^n+c^n\ge x^n+y^n+z^n,\forall n\in N^{+}$$

Maybe this kind of thing has been studied, like I found something relevant, but I didn't find the same one.[Schur convexity and Schur multiplicative convexity for a class of symmetric function]

let $a,b,c,x,y,z>0$ such $x+y+z=a+b+c,abc=xyz$,and $a>\max\{x,y,z\}$,

I conjecture $$a^n+b^n+c^n\ge x^n+y^n+z^n,\forall n\in N^{+}$$

Maybe this kind of thing has been studied, like I found something relevant, but I didn't find the same one.[Schur convexity and Schur multiplicative convexity for a class of symmetric function]

I found sometime maybe The sum of squared logarithms conjecture