partial derivative notation question

partial derivative notation question

I'm reading a book called Correlated Data Analysis, Analytics, and
Applications and I simply don't understand some notation. The author says,
in ch2 pg 26:
a unit deviance is called regular if function $d(y;u)$ is twice
continuously differentiable with respect to $(y, u)$ on $\Omega \times
\Omega$ and satisfies $$ \frac{ \partial^2 d}{\partial u^2}(y;y) = \frac{
\partial^2 d}{\partial u^2}(y;u) \Bigg|_{u=y} >0, \forall y \in \Omega$$
Um... what does that notation mean? On the left, I guess I take the second
partial derivative then evaluate with y as both variables? In the right
case, I have no idea...
Could someone share an example of a function where this would not be true?
Thank you.
edit: you can view this in place by downloading this sample pdf
http://www.springer.com/cda/content/document/cda_downloaddocument/9780387713922-c2.pdf
on page logical 26, pdf 4