Inequality for $s\in(1,2)$

Question

Let $p>1$ be a real number and $1<s<2$ be a real number. Then does the following inequality hold: $ (s^p-1)-(s-1)^p\leq c(p)(s-1) $ where $c(p)$ is a constant depending on $p$.

Answer 1

Disclaimer: I don't have a complete solution yet. But I did some numerical and graphical experimentations that suggest that the claim is true.

First, I substituted $s=1+x$, so the desired inequality becomes $(1+x)^p-1-x^p\le c(p)\cdot x$.

Let's consider the function $\displaystyle f(x)=\frac{(1+x)^p-1-x^p}{x}$ on $0<x<1$ with $p>1$.

By graphing, I could see that there's a critical value of $p=2$:

• for $1<p<2$, $f(x)$ is decreasing, attaining a maximum as $x\to0$, and the maximum is always less than $2$, so $c(p)=2$ would work.
• for $p=2$, $f(x)=2$ is a constant function, so $c(p)=2$ works again.
• for $p>2$, $f(x)$ is increasing, attaining a maximum as $x\to1$. That maximum value is $c(p)=f(1)=2^p-2$.

Of course, as I said, this is not a complete proof yet, because I haven't worked out how to prove that $f(x)$ is decreasing or increasing in those two cases. But I hope this helps and you can take it from here.