While solving a bigger problem, I've reduced it to an inequality $$\left(1+2^{b^\frac{1}{b-1}-1}\right)^b < 1+2^{b^\frac{b}{b-1}-1}$$for $b>2$, which looks plausible when looking at the plots. I've tried using Jensen inequality here with $x\mapsto x^{b-1}$, but not much luck.
I've also checked that the inequality works empirically with Wolfram Alpha.
from Hot Weekly Questions - Mathematics Stack Exchange
Jakobian
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