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We have that 

<img src='content_image/306681.jpg'>

A third-order Taylor expansion of b$_{0}$ ↦→ exp( − b 2 $_{0}$) about b$_{0}$ = b yields 

<img src='content_image/306682.jpg'>

where the magnitude of the O (( b$_{0}$ − b ) $^{4}$) term is uniformly bounded above by C ( b$_{0}$ − b ) 4 for some constant C > 0 when b$_{0}$ and b fall in [ − 1 , 1]. For the second-order term, we have 

<img src='content_image/306683.jpg'>

Thus we have that 

<img src='content_image/306684.jpg'>

A Taylor expansion of f$_{1}$ ( z ) = 2 ze − z 2 shows that there exists a ˜ B$_{1}$ ( o$_{1}$,o$_{2}$ ) that falls between 

