Let $1/2 < s< 1$ be a real number. How does one prove that $1-s \leq d(1-s^{1/d})$ for any positive integer $d$?
I can see the equality for $d=1$. I thought about showing that the derivative of $d(1-s^{1/d})$ with respect to $d$ is strictly positive, but didn't succeed. That is, I couldn't show that $d-ds^{1/d} +\log(s)s^{1/d} >0$ for every positive integer $d$.
I recall once seeing somewhere that $s^{1/d} \approx 1-s/d$ . If one makes this precise, one can prove the above inequality, I believe.