If minimum value of $$f(x) = x^{2} + 2bx + 2c^{2}$$ is greater than the maximum value of $$g(x) = - x^{2} - 2cx + b^{2}$$, then for real value of x.

The minimum/maximum value of quadratic $$ax^2+bx+c=0$$ occurs at point $$x=-\frac{b}{2a}$$

So, for the quadratic, $$f(x) = x^{2} + 2bx + 2c^{2}$$, minimum value will occur at $$x=-\dfrac{2b}{2}=-b$$

Similarly, for the quadratic, $$g(x) = - x^{2} - 2cx + b^{2}$$, minimum value will occur at $$x=-\dfrac{-2c}{-2}=-c$$

Now, according to question,

$$\left(-b\right)^2+2b\left(-b\right)+2c^2>-\left(-c\right)^2-2c\left(-c\right)+b^2$$

or, $$b^2-2b^2+2c^2>-c^2+2c^2+b^2$$

or, $$-2b^2>-c^2$$

or, $$c^2>2b^2$$

or, $$\left|c\right|>\sqrt{\ 2}\left|b\right|$$