| Ψ$_{3}$ | < C$_{1}$ ∣ ∮ K$_{i}$$_{2}$∪ ¯ K$_{i}$$_{2}$ e − nf$_{i}$ ( z $^{)}$g ( z ) d z ∣ ∣$^{,}$ (46) | Ψ$_{2}$ | < C$_{1}$e − $^{nf$_{i}$$_{0}$}$e − $^{nC$_{2}$}$h − 1 i sup z ∈K$_{i}$$_{1}$∪ ¯ K$_{i}$$_{1}$ | g ( z ) | , and (45) | Ψ$_{1}$ | < C$_{1}$e − $^{nf$_{i}$$_{0}$}$h − 1 i n − 3 / $^{2}$sup z ∈ ¯ B | g ( z ) | , (44) where C$_{1}$ and C$_{2}$ are some positive constants, and ¯ B is a closed ball with center at z$_{i}$$_{0}$ and radius r$_{i}$/ 2. Now, let g ( z ) = g$_{j}$ ( z ) = z j − $^{1}$exp {− Δ$_{p}$ ( z ) } . Lemma A2 in OMH implies that sup$_{z}$$_{∈}$ ¯ 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), with Turning to the analysis of Ψ$_{3}$, note that by definition of f$_{i}$ ( z ) and g ( z ) , For z ∈ K i 2 ∪ ¯ K i $_{2}$, we have ∣ ( z − λ$_{j}$ ) − $^{1}$∣ ∣ < (3 z$_{i}$$_{0}$ ) − $^{1}$, and ∣ z ( z − λ$_{j}$ ) − $^{1}$∣ ∣ < 2 , for any j = 1 , ..., p . Therefore, using (48), we get On the other hand, for any h$_{i}$ ∈ [ 0 , ¯ h ] , h$_{i}$ < √ c$_{p}$ for sufficiently large n and p, and 29 ∣ ∮ K$_{i}$$_{2}$∪ ¯ K$_{i}$$_{2}$ e − nf$_{i}$ ( z $^{)}$g ( z ) d z ∣ < 2 j − $^{1}$(3 z$_{i}$$_{0}$ ) − p + j − 1 ∮ K$_{i}$$_{2}$∪ ¯ K$_{i}$$_{2}$ ∣ e n h i 1+ h $^{$_{i}$z}$d z ∣ = 2 j (3 z$_{i}$$_{0}$ ) − p + j − 1 ( n h$_{i}$ 1 + h$_{i}$ ) − 1 e n h i 1+ h $_{i}$z$_{i}$$_{0}$ = 2 j (3 z$_{i}$$_{0}$ ) j − 1 ( n h$_{i}$ 1 + h$_{i}$ ) − 1 e − n ( c$_{p}$ ln(3 z$_{i}$$_{0}$ ) − h i 1+ h $_{i}$z$_{i}$$_{0}$ ) = 2 j (3 z$_{i}$$_{0}$ ) j − 1 n h$_{i}$ 1 + h$_{i}$ − 1 3 − $^{p}$e − n ( c$_{p}$ ln( z$_{i}$$_{0}$ ) − h$_{i}$ − c$_{p}$ $^{)}$. Ψ$_{1}$ + Ψ$_{2}$ = e − $^{nf$_{i}$$_{0}$}$h − j i n − 3 / $^{2}$O$_{p}$ (1) . (47) e − nf$_{i}$ ( z $^{)}$g ( z ) = e n h i 1+ h $^{$_{i}$z}$z j − 1 p ∏ j =1 ( z − λ$_{j}$ ) − $^{1}$. (48)