content_image/300745.jpg content_image/300744.jpg content_image/300743.jpg content_image/300754.jpg content_image/300755.jpg content_image/300753.jpg
with 

where C$_{1}$ and C$_{2}$ are some positive constants, and ¯ B is a closed ball with center at z$_{i}$$_{0}$ and radius r$_{i}$/ 2. 

<img src='content_image/300745.jpg'>

<img src='content_image/300744.jpg'>

<img src='content_image/300743.jpg'>

Now, let g ( z ) = g$_{j}$ ( z ) = z j − $^{1}$exp {− Δ$_{p}$ ( z ) } . Lemma A2 in OMH implies that sup$_{z}$$_{∈}$ ¯ B ∪K$_{i}$$_{1}$∪ ¯ K$_{i}$$_{1}$ | g ( z ) | = h 1 − j i O$_{p}$ (1) uniformly in h$_{i}$ ∈ ( 0 , ¯ h ] . Therefore, by (44) and (45), 

<img src='content_image/300754.jpg'>

Turning to the analysis of Ψ$_{3}$, note that by definition of f$_{i}$ ( z ) and g ( z ) , 

<img src='content_image/300755.jpg'>

For z ∈ K i 2 ∪ ¯ K i $_{2}$, we have ∣ ( z − λ$_{j}$ ) − $^{1}$∣ ∣ < (3 z$_{i}$$_{0}$ ) − $^{1}$, and ∣ z ( z − λ$_{j}$ ) − $^{1}$∣ ∣ < 2 , for any j = 1 , ..., p . Therefore, using (48), we get 

<img src='content_image/300753.jpg'>

On the other hand, for any h$_{i}$ ∈ [ 0 , ¯ h ] , h$_{i}$ < √ c$_{p}$ for sufficiently large n and p, and 

