Prove this power inequality $ (1 - F + x)^{s-1} \geq \frac{(1 - F + x)^s - (1 - F)^s}{sx}$

Question

I want to show that $$ (1 - F + x)^{s-1} \geq \frac{(1 - F + x)^s - (1 - F)^s}{sx}$$ for $x \in [0, F]$, $F\in (0, 1]$, and $s$ is a positive integer larger than one.

I can't find any obvious identity to be used here. The only strategy I could come up with was that for $x\to 0$, the right-hand side becomes identical to the left-hand side. So then it's sufficient to show that

$$ \frac{\partial}{\partial x} (1 - F + x)^{s-1} \geq \frac{\partial}{\partial x}\frac{(1 - F + x)^s - (1 - F)^s}{sx} \qquad \forall x \in (0, F)$$

Which appears to be true numerically. However, the derivatives look even more messy:

$$ (s-1)(1 - F + x)^{s-2} \geq \frac{s(1-F+x)^{s-1} - \frac{s}{x}\left[(1-F+x)^s - (1-F)^s\right]}{sx}.$$

Is there any path forward here?

Answer 1

$\frac{(1 - F + x)^s - (1 - F)^s}{sx} = (\frac{(1-F+x)-(1-F)}{sx})\sum_{k=0}^{s-1}(1-F+x)^k(1-F)^{s-1-k}=\frac{1}{s}\cdot\sum_{k=0}^{s-1}(1-F+x)^k(1-F)^{s-1-k} \leq \frac{1}{s}\cdot\sum_{k=0}^{s-1}(1-F+x)^k(1-F+x)^{s-1-k}=\frac{1}{s}\cdot\sum_{k=0}^{s-1}(1-F+x)^{s-1}=\frac{s(1-F+x)^{s-1}}{s} =(1-F+x)^{s-1} $

Some of the minute details maybe need to be shown, but I'll leave that up to you. Sorry about the spacing. I haven't been on here in a while.

If it's still not clear, rewrite it with $A = 1-F$ and $n=s-1$.

Hope this helps.

Answer 2

Let $C(x)=1-F+x$ and let $H(x)=xsC(x)^{s-1}-C(x)^s+(1-F)^s.$

For $x>0$ and $s>0$ the inequality is equivalent to $H(x)\ge 0.$

We have $H(0)=0.$

For a fixed $F$ and fixed $s\ge 2$ we have $C'(x)=1$ so $$H'(x)=sC(x)^{s-1}+xs(s-1)C(x)^{s-2}-sC(x)^{s-1}=xs(s-1)C(x)^{s-2}$$ which is positive when $0<x<F\le 1.$

Answer 3

Apply the Mean Value Theorem to the function $f(t)=t^s$, with $t_1= 1-F$ and $t_2=1-F +x$,

$$\frac{(1 - F + x)^s - (1 - F)^s}{x} = f'(c)=sc^{s-1} \le s(1 - F + x)^{s-1}$$

where used in the last step are the facts that $t^{s-1}$ of $s>1$ monotonically increases and $c \le 1 - F + x$. Thus,

$$ (1 - F + x)^{s-1} \geq \frac{(1 - F + x)^s - (1 - F)^s}{sx}$$