If $x+y=1$ Prove that $\left(1+1/x^n\right) \left(1+1/y^n\right) \ge (1+2^n)^2$

Question

Let $x+y=1$, where $x,y\in \mathbb R^+$ Prove that $\forall x\in \mathbb N$ $$P(n):\left(1+\frac{1}{x^n}\right)\left(1+\frac{1}{y^n}\right)\ge (1+2^n)^2$$

It's clear that the best method for proving it is induction.

$P(0)$ is true, so assume that $P(k)$ is true for some $k\ge 0$, but the problem occurs here because I can't use the fact $x+y=1$ properly in my proof. I tried expanding the RHS and the LHS but it didn't work.

I don't see why induction is clearly the best method. My method of choice would be to prove that $f(x) = \log(1 + \frac1{x^n})$ is convex, but which method you'd want to use depends on the tools you know for proving inequalities. — Misha Lavrov, Nov 04 '21 at 14:07
Sorry but this is from my homework, so you can just use Induction, Transitivity or split the problem into multiple cases (we didn't study logarithms yet)@MishaLavrov — PNT, Nov 04 '21 at 14:10

score 2 · Answer 1 · edited Nov 04 '21 at 14:24

2

Since $x+y =1$ we get $xy\leq 4^{-1} $ hence

$$\left(1+\frac{1}{x^n } \right)\left(1+\frac{1}{y^n } \right)=1 +\frac{1}{(xy)^n} +\frac{1}{x^n}+\frac{1}{y^n}\geq 1+\frac{1}{(xy)^n} +\frac{2}{\sqrt{xy}^n}\geq 1+\frac{1}{(\frac{1}{4})^n} +\frac{2}{\sqrt{\frac{1}{4}}^n} =(1+2^n )^2$$

edited Nov 04 '21 at 14:24

PNT

4,164

answered Nov 04 '21 at 14:11

score 0 · Accepted Answer · answered Nov 04 '21 at 14:10

Here is a direct proof using only AM-GM inequality :

First, by AM-GM inequality, on has $$2\sqrt{xy} \leq x+y=1, \quad \quad \text{so} \quad xy \leq \frac{1}{4}$$

You deduce that $$\frac{1}{(xy)^n} \geq 4^n \quad \quad \quad \quad \quad (1)$$

Secondly, by AM-GM inequality again, one has $$\frac{1}{x^n} + \frac{1}{y^n} \geq 2 \sqrt{\frac{1}{(xy)^n}}$$

so using the inequality $(1)$, you deduce $$\frac{1}{x^n} + \frac{1}{y^n} \geq 2 \sqrt{4^n} = 2^{n+1} \quad \quad \quad \quad (2)$$

Now, summing $(1)$ and $(2)$, you deduce that $$\frac{1}{x^n} + \frac{1}{y^n} + \frac{1}{(xy)^n} \geq 4^n + 2^{n+1} = (2^n)^2 + 2^{n+1}$$

so $$1 + \frac{1}{x^n} + \frac{1}{y^n} + \frac{1}{(xy)^n} \geq 1+ (2^n)^2 + 2^{n+1}$$

i.e. finally $$\boxed{\left(1+\frac{1}{x^n}\right)\left(1+\frac{1}{y^n}\right)\ge (1+2^n)^2}$$

I would use the cauchy- schwarz inequality This would avoid the expansion.Nevertheless this is effective — Albus Dumbledore, Nov 04 '21 at 14:12

If $x+y=1$ Prove that $\left(1+1/x^n\right) \left(1+1/y^n\right) \ge (1+2^n)^2$

2 Answers2