If $abc=1$ ,and $n$ is a natural number,prove $ \frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)} \geqslant \frac{n(n-1)}{2} $

Question

If $a, b, c$ are distinct positive real numbers such that $abc=1$,and $n$ is a natural number,prove $$ \frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)} \geqslant \frac{n(n-1)}{2} $$ I know for $n=3$,the answer is here,and how to go further?

Answer 1

As can be found here and here , we have that

$$ P_n = \frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)} = \displaystyle\sum_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3} $$ Now by AM-GM, it follows that

$$ P_n = \displaystyle\sum_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3} \geq Q [\displaystyle\prod_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3} ]^{1/Q} $$

where $Q$ is the number of terms in the sum $\sum_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3}$.

Now note that by symmetry,

$$ P_n \geq Q [\displaystyle\prod_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3} ]^{1/Q} = Q [a b c ]^{\frac{Q(n-2)}{3 Q}} = Q $$

where the last one is true by the condition $abc =1 $. So it remains to find $Q$. This derives from the sum's conditions: we can select $n_1$ from $0$ to $n-2$, i.e. $n-1$ possibilites. Then we have that $n_2$ can be taken from $0$ to $n-2-n_1$, i.e. $n-1-n_1$ possibilites. So $Q = \sum_{n_1 =0}^{n-2} (n-1-n_1) = \frac12 n (n-1)$ which proves the claim. $\qquad \Box$

Some examples, using AM-GM directly:

$$P_2 = 1 \geq 1 = \frac12 \cdot 2 \cdot (1)\\ P_3 = a + b +c \geq 3 [abc]^{\frac{1}{3}} = 3 = \frac12 \cdot 3 \cdot(2)\\ P_4 = a^2 + b^2 + c^2 + ab + bc + ac \geq 6 [abc]^{\frac{2}{3}} = 6 = \frac12 \cdot 4 \cdot(3) $$

Answer 2

Hint :

Using Fuchs's inequality wich is an extension of Karamata's inequality one can show that for ($f(x)=x^n$),$a\geq b \geq c>0$ such that $abc=1$:

$$\left(\frac{a^{n}}{\left(a+c\right)\left(a+c\right)}-1\right)a\cdot\frac{1}{\left(a-b\right)\left(a-c\right)}\geq \left(\frac{b^{n}}{\left(b+c\right)\left(b+a\right)}-1\right)\cdot\frac{b}{\left(b-a\right)\left(b-c\right)}\geq \left(\frac{c^{n}}{\left(a+c\right)\left(b+c\right)}-1\right)\cdot\frac{c}{\left(c-a\right)\left(c-b\right)}$$

We have the inequalities :

$$\left(\frac{a^{n}}{\left(a+c\right)\left(a+c\right)}-1\right)a\cdot\frac{1}{\left(a-b\right)\left(a-c\right)}\ge 0$$

$$\left(\frac{a^{n}}{\left(a+c\right)\left(a+c\right)}-1\right)a\cdot\frac{1}{\left(a-b\right)\left(a-c\right)}+\left(\frac{b^{n}}{\left(b+c\right)\left(b+a\right)}-1\right)\cdot\frac{b}{\left(b-a\right)\left(b-c\right)}\ge 0$$

And :

$$\left(\frac{a^{n}}{\left(a+c\right)\left(a+c\right)}-1\right)a\cdot\frac{1}{\left(a-b\right)\left(a-c\right)}+ \left(\frac{b^{n}}{\left(b+c\right)\left(b+a\right)}-1\right)\cdot\frac{b}{\left(b-a\right)\left(b-c\right)}+ \left(\frac{c^{n}}{\left(a+c\right)\left(b+c\right)}-1\right)\cdot\frac{c}{\left(c-a\right)\left(c-b\right)} \ge 0$$

So using Fuchs's inequality we have shown that :

$$\frac{a^{n}}{(a-b)(a-c)}+\frac{b^{n}}{(b-a)(b-c)}+\frac{c^{n}}{(c-a)(c-b)}\le \frac{a^{2n}}{(a^{2}-b^{2})(a^{2}-c^{2})}+\frac{b^{2n}}{(b^{2}-a^{2})(b^{2}-c^{2})}+\frac{c^{2n}}{(c^{2}-a^{2})(c^{2}-b^{2})}$$

Now conclude is easy in fact .

Hope it helps .