TITLE: Proof by induction for an exponential inequality: $(2^r-1)(1-x)x^{2^{r}-2}+x^{2^{r}-1}>x^{2^{r}-r-1}$
QUESTION [5 upvotes]: How do I prove
$(2^r-1)(1-x)x^{2^{r}-2}+x^{2^{r}-1}>x^{2^{r}-r-1}$ 
for $\frac{1}{2}<x<1$ and $r \in \mathbb N$ $(r\neq 1)$?

REPLY [3 votes]: The assertion holds for every $x$ in $(1/2,1)$ and every real number $r>1$. 
To see this, one can begin like Moron and try to show that $f(y)<f(2)$ with $y=1/x$ and
$$
f(y)=(y^r-1)/(y-1).
$$
Since $y<2$, it is enough to prove that $f$ is nondecreasing. The sign of $f'(y)$ is the sign of $g(y)$ with
$$
g(y)=(r-1)y^r-ry^{r-1}+1.
$$
Now, $g'(y)=r(r-1)y^{r-2}(y-1)$ is positive on $y>1$ because $r>1$ hence $y\mapsto g(y)$ is increasing on $y>1$, and $g(1)=0$ hence $g(y)>0$ for $y>1$. Finally, $y\mapsto f(y)$ is increasing on $y>1$ hence we are done.