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Picture below is from AN UPPER BOUND TO THE SPECTRUM OF $\Delta$ ON A MANIFOLD OF NEGATIVE CURVATURE .

First, how to get (6) from (5) ? The (6) implies $f^{old}=f^{new}\sqrt{g}$. But I can't get it from $f'^{~new}=\sqrt{g}f'^{~old}$. Because $\sqrt{g}$ and $f$ are functions of $r$. Then integrate them , there will be not $f^{old}=f^{new}\sqrt{g}$ .

Second, How to get (7) from (4) ?

Thanks for any help.

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