Any natural number can be decomposed into a sum of potencies of 2, that is, for any $n\in \mathbb{N}$ exists a sequence $\{b_i\}_{i\leq k}$ of $0$'s and $1$'s such that $$ n = \sum_{i=0}^k 2^ib_i$$
We define then the following function: Given $0<\varepsilon<1$, $$\varphi(\varepsilon,n) = \frac{\ln\left(\sum_{i=0}^k(2^ib_i)^{1-\varepsilon}\right)}{\ln\left(\sum_{i=0}^k(2^ib_i)\right)}$$ with the appropriate sequence $\{b_i\}_{i\leq k}$ for every $n$.
Here's an example of the behavior of this function for a fixed $\varepsilon=0.01$
where the red lines are the values with $n=2^t, t\in \mathbb{N}$
I'd like to show that, for a fixed value of $\varepsilon$ this function finds it's maximun on $n=3$. To do that, my attempt was to prove the following properties I believe this function has, but wasn't able to show yet:
- If $n =\sum_{i=0}^k 2^ib_i$ and there's a $j<k$ such that $b_j=0$, then $$\varphi(n,\varepsilon)\leq \varphi(n+2^j,\varepsilon)$$
- $\varphi(2^k,\varepsilon)>\varphi(2^{k+1},\varepsilon), \; \forall k \in \mathbb{N}$
The fist property is saying that, by "completing" a number adding the potencies of 2 that it didn't have, the function $\phi$ grows. Looking at the plot of the function, this proccess would be like jumping from a point $n$ the highest value before the red line, which is the highest point of the function inbetween red lines.
The second property states that these closest points to the left of the red lines are in a decreasing fashion, therefore every number between red lines has image lower then the one closest to the line, which is less than $\varphi(\varepsilon,3)$.
Now, how do I do that? For the first property I attempted to show that
$\frac{ln(a+x^{1-\varepsilon})}{\ln(b+x)}\geq \frac{\ln(a)}{\ln(b)}$ for every $b>a$ and $x<b$ but it didn't work. Calculating the derivative with respect to x wasn't useful, nor trying to bound parts of the inequality.
Another thing I thought might work would be finding some $\psi:(0,1)\times\mathbb{R}\to \mathbb{R}$ such that $\psi(\varepsilon,3) = \varphi(\varepsilon,3)$ and $\psi(\varepsilon, x)\geq\varphi(\varepsilon,x) \forall x \in \mathbb{N}$ but I have no clue what function it could be.
So any ideas on how to show this are appreciated. Thanks in advance.