Inequality properties between their multiplications

Question

If $A>B>C>D$ then will $$(A/B * A/C * A/D) > (B/A * B/C * B/D) > (C/A * C/B * C/D) > (D/A * D/B * D/C)$$ be always true? If not, in what intervals will it not be true?

Obviously $A,B,C,D \geq 0$ and they take values in the interval of ${\Bbb R}_+$, that is positive real numbers only.

Answer 1

First of all, we're dividing by $A,B,C,D$ so none can be $0$. Hence, in this answer, we assume $A,B,C,D>0$.

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Anyway, since $A>B>C>D>0$, we know that

$$A^4>B^4>C^4>D^4$$

and now since $ABCD>0$, we may divide by $ABCD$ without flipping any signs:

$$\frac{A^3}{BCD}>\frac{B^3}{ACD}>\frac{C^3}{ABD}>\frac{D^3}{ABC}$$

which is the exact same as the thing you were trying to prove.

Answer 2

Yes---compare term by term e.g. $A/B > B/A$, $A/C > B/C$, $A/D > B/D$ which will give you the first result, and so on.