14 3. PRELIMINARIES

Multiplying by ha(x), integrating with respect to νa = ν−af +g, and using the fact

that L−af +gq dνa = µa q dνa (see e.g. [PP]), gives

−i

ha(x)

ws(x)

⎛

⎝

σ(y)=x

f(y)

egs(y)

ws(y)⎠

⎞

dνa(x) =

∂

∂b

αs .

Hence

∂

∂b

αs ≤

⎛

⎝

σ(y)=x

f(y)

egs(y)

ws(y)

⎞

⎠

ha(x) dνa(x)

≤ |f|∞

⎛

⎝

σ(y)=x

eua(y)⎠

⎞

ha(x) dνa(x)

= |f|∞ ha(x) dνa(x) = |f|∞ .

Hence |αs − 1| = |αs − αa| ≤ |f|∞ |b|≤ |f|∞ |s − s0|, and therefore

|λs − λ| = |αs µa − µs0 | ≤ |αs − 1|· µa + |µa − µs0 | ≤ 2 |f|∞ |s − s0| λs0

e|f |∞ |s−s0|

.

This proves (3.17).