Partial proof by induction of the inequality : $(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1$

Question

Claim

Let $0.5\leq x<1$ and $n\geq 2$ a natural number then we have : $$(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1\quad (I)$$

Sketch\Partial (of) proof .

We use a form of the Young's inequality or weighted Am-Gm :

Let $a,b>0$ and $0<v<1$ then we have :

$$av+b(1-v)\geq a^vb^{1-v}$$

Taking account of this theorem and putting :

$a=x^{(2(1-x))^{n-1}}$$\quad$$b=1$$\quad$$v=2(1-x)$ we get $0.5\leq x<1$:

$$x^{(2(1-x))^n}\leq x^{(2(1-x))^{n-1}}2(1-x)+1-2(1-x)$$

Now the idea is to show :

Let $$(1-x)^{(2x)^n}\leq 1-\Big(x^{(2(1-x))^{n-1}}2(1-x)+1-2(1-x)\Big)$$

Or :

$$(1-x)^{(2x)^n}\leq2(1-x)(1-x^{(2(1-x))^{n-1}})$$

Or: $$(1-x)^{(2x)^n-1}+2x^{(2(1-x))^{n-1}}\leq 2\quad (0)$$

Now we want to show that :

$$(1-x)^{(2x)^n-1}\leq 2(1-x)^{(2x)^{n-1}}\quad(1)$$

For that we need a lemma :

Lemma :

Let $0.5\leq x<1$ and $n\geq 2$ a natural number then we have :

$$f(n)=\ln(1-x)((2x)^n-1-(2x)^{n-1})\leq f(n-1)=\ln(1-x)((2x)^{n-1}-1-(2x)^{n-2})$$

It's true because it's equivalent to :

$$(2x)^{n-2}(2x-1)^2\geq 0$$

So we have :

$$f(n)\leq f(2)$$

We can show also that on $[0.61,1)$ :

$$f(n)\leq f(2)\leq \ln(2)$$

Wich is equivalent to $(1)$

So we have from $(1)$ and $(0)$ we need to show :

$$2x^{(2(1-x))^{n-1}}+2(1-x)^{(2x)^{n-1}}\leq 2$$

Or :

$$x^{(2(1-x))^{n-1}}+(1-x)^{(2x)^{n-1}}\leq 1$$

So now we can use induction to prove $(I)$.

My questions :

It is good ?

How to write the proof properly using induction on the interval $[0.61,1)$ ?

Thanks in advance

Max

Answer 1

Your idea looks good to me.

I think that the following inductive step works on the interval $[\alpha,1)$ where $\alpha\approx 0.6046$ is a root of $f(2)=\ln(2)$.

Inductive step :

Suppose that $$(1-x)^{(2x)^{n-1}}+x^{(2(1-x))^{n-1}}\leq 1\tag2$$

You've already proved $$(1-x)^{(2x)^n-1}\leq 2(1-x)^{(2x)^{n-1}}\tag1$$ on the interval $[\alpha,1)$.

So, $2\times (2)+(1)$ gives $$2x^{(2(1-x))^{n-1}}+(1-x)^{(2x)^n-1}\le 2\tag0$$ which is equivalent to $$(1-x)^{(2x)^n}\leq 1-\Big(x^{(2(1-x))^{n-1}}2(1-x)+1-2(1-x)\Big)\tag3$$

You've already got $$x^{(2(1-x))^n}\leq x^{(2(1-x))^{n-1}}2(1-x)+1-2(1-x)\tag4$$ Finally, $(3)+(4)$ gives $$(1-x)^{(2x)^{n}}+x^{(2(1-x))^{n}}\leq 1$$