content_image/325632.jpg content_image/325633.jpg content_image/325639.jpg content_image/325640.jpg
where 

<img src='content_image/325632.jpg'>

Hence, if x ∈ D, ( ξ $^{2}$, ..., ξ $^{n}$) ∈ G ′ , t ∈ [0 , d$_{0}$ ] are fixed, then applying the mean value theorem (e.g., [13], p.186) to the function . z k ( x, 1 | ξ $^{′}$| θ , θξ 2 | ξ $^{′}$| θ , · · · , θξ n | ξ $^{′}$| θ , t ) with respect to θ on the interval [0 , µ ] , from the equality . z k ( x, ν $^{0}$, t ) = 0 for 2 ≤ k ≤ n , we have 

<img src='content_image/325633.jpg'>

where ∂$_{µ}$$_{0}$ . z k is the derivative of the function . z k ( x, 1 | ξ $^{′}$| θ , θξ | ξ $^{′}$| θ , · · · , θξ | ξ $^{′}$| θ , t ) with re- spect to θ at a point θ = µ$_{0}$. It is worth to note here that . z 1 ( x, ν $^{0}$, t = 1 and equality (A.2) is not valid for k = 1 . 

Since . z k ( x, ν, t ) ∈ C $^{5}$(Ω( d$_{0}$ )) and ( 1 | ξ $^{′}$| µ , µξ 2 | ξ $^{′}$| µ , · · · , µξ n | ξ $^{′}$| µ ) ∈ S $^{n}$( x ) , the function ∂$_{µ}$ . z k ( x, 1 | ξ $^{′}$| µ , µξ 2 | ξ $^{′}$| µ , · · · , µξ n | ξ $^{′}$| µ , t ) is bounded on Ω ( d$_{0}$ ). Here we note that Ω ( d$_{0}$ ) is closed and bounded. Therefore, by (A.1) and (A.2), since the vector ν = ξ/ | ξ | ∈ S $^{n}$( x ) tends to ν 0 = (1 , 0 , ..., 0) ∈ R n as ξ 1 −→ + ∞ , we have 

<img src='content_image/325639.jpg'>

for 2 ≤ k ≤ n in the set Ω , where K$_{1}$ > 0 is independent of ( x, ξ ) ∈ D × G , but depends on the norm of the vector function . z ( x, ν, t ) in C 1 (Ω ( d$_{0}$ )) and the diameter of G ′ . In the same way as above, we can prove the last inequality for the case ξ 1 −→ −∞ . 

It is not difficult to verify the following equalities 

<img src='content_image/325640.jpg'>

