If $f\left(x\right)=x^2+2bx+2c^2$  and

 $g\left(x\right)=-x^2-2cx+b^2$   are such that 

min  $f\left(x\right)>$max $g\left(x\right)$ , then relation between b and c, is 

$\Rightarrow \min f\left(x\right)=2c^2-b^2$

$f\left(x\right)={\left(x+b\right)}^2+2c^2-b^2$

Also  $g\left(x\right)=-x^2-2cx+b^2=b^2+c^2-{\left(x+c\right)}^2$

$\Rightarrow \max g\left(x\right)=b^2+c^2$

As min $f\left(x\right)>\max g\left(x\right)$  we get 

  $2c^2-b^2>b^2+c^2$

$\Rightarrow c^2>2b^2\Rightarrow \left|c\right|>\sqrt{2}\left|b\right|$