# inequality – Sum of the power tower that tends to \$ 1 \$

Inspired by an inequality of Vasile Cirtoaje I have this:

Leave $$0 and define the function:
$$f (y) = x ^ {1-x + y} quad operator name {y} quad g (y) = (1-x) ^ {x + y}$$
Y $$h (x) = x ^ {2 (1-x)}$$
Now denotes by:
$$f ^ n (y) = f (f (f (f cdots (y) cdots))$$ Y $$g ^ n (y) = g (g (g (g cdots (y) cdots))$$

Where we compose $$n$$ times the functions $$f (y), g (y)$$ with himself.$$n$$ It is an even number.

Then :
$$f ^ n (h (x)) + g ^ n (h (1-x)) leq1$$

## Example:

by $$n = 2$$ we have :

$$f (f (f (h (x))) = x ^ {1-x + x ^ {1-x + x ^ {1-x + h (x)}}}$$

Y

$$g (g (g (h (1-x)))) = (1-x) ^ {x + (1-x) ^ {x + (1-x) ^ {x + h (1-x )}}}$$

I try to use the Lambert function but it doesn't solve my energy tower.

The main comment is when $$n$$ tends to infinity, the terms of inequality tend to $$x$$ Y $$1-x$$

If you have a good idea, thank you in advance for sharing it.