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Proof.
By the previous Lemma,
(4.136) A − − ∫ T − z h ( u ) = 1 2 log ‖ S H ‖ + B T + ∫ z 0 h ( u ) 𝑑 u A_{-}-\int_{T_{-}}^{z}h(u)=\frac{1}{2}\log{\|S_{H}\|}+B_{T}+\int_{z}^{0}h(u)du
When | z | ≤ 1 |z|\leq 1
(4.137) ∫ 0 z h ( u ) 𝑑 u = B T + T z + 1 2 ( log ( r + z ) − log | y | ) \int_{0}^{z}h(u)du=B_{T}+Tz+\frac{1}{2}(\log(r+z)-\log|y|)
Since log ‖ S H ‖ = log | y | + B T \log{\|S_{H}\|}=\log|y|+B_{T}
(4.138) log r − = B T − T z + 1 2 log ( r − z ) \log r_{-}=B_{T}-Tz+\frac{1}{2}\log(r-z)
Similarly we get the estimate for log r + \log r_{+}