I am trying to prove an equality in the form $(mA+nB+pC)>(mX+nY+pZ)$.

I can prove the inequality $(A+B+C)>(X+Y+Z)$ and I wonder if there is any condition,

under which we can deduce $(mA+nB+pC)>(mX+nY+pZ)$ from $(A+B+C)>(X+Y+Z)$ ?

And $A,B,C,X,Y,Z,m,n,p$ are all positive.

The first thing comes to mind is the Cauchy-Schwarz Inequality:

$$sqrt{left(m^{2}+n^{2}+p^{2}right)left(X^{2}+Y^{2}+Z^{2}right)}geleft(mX+nY+pZright)$$

Now it suffices to prove that

$$left(mA+nB+pCright)gesqrt{left(m^{2}+n^{2}+p^{2}right)left(X^{2}+Y^{2}+Z^{2}right)}$$

$$iffleft(mA+nB+pCright)^{2}geleft(m^{2}+n^{2}+p^{2}right)left(X^{2}+Y^{2}+Z^{2}right)$$

But I can’t find the condition under which we can deduce the above inequality from $(A+B+C)>(X+Y+Z)$.

Or using HÃ¶lder’s inequality:

$$left(mA+nB+pCright)^{frac{1}{2}}left(frac{1}{m}+frac{1}{n}+frac{1}{p}right)^{frac{1}{2}}ge A^{frac{1}{2}}+B^{frac{1}{2}}+C^{frac{1}{2}}$$

$$iffleft(mA+nB+pCright)^{frac{1}{2}}geleft(A^{frac{1}{2}}+B^{frac{1}{2}}+C^{frac{1}{2}}right)left(frac{1}{m}+frac{1}{n}+frac{1}{p}right)^{-frac{1}{2}}$$

And so on.