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Given any parameter $g > 1$, I can show that the following equation has a unique solution $r^{*} \in (\frac 1 g, \infty) $,

$$∫_0^1(x- \frac 1 g )x^{ \frac 1{gr-1} }\frac {\ln x}{(1+xgr)^2}dx = 0$$

However, even after spending many hours, I could not verify that the solution $r^{*}$ is increasing in $g$ for $g$ large enough --- which seems to be true from some extensive computation. Can anyone help?

Update 1: If useful, I would be happy to add graphs etc. The proof that the equation has a unique solution is quite long and not very instructive, so I did not include it.

Update 2: Is this something where it would be useful to post a bond?

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