Which is correct representation for function of variable? I have a function as
$$E=\int_\Omega -\log\big( p_i(x)\big) dx$$
where $p_i(x)$ is density distribution which estimated by Parzen window method. 
$$p_i(x)=\frac{1}{\Omega_i} \int_{\Omega_i}K_\sigma\big(I(x)-I(y)\big) dy$$
where $\Omega_i$ is the image domain, with 
$$K_\sigma=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(\frac{-z^2}{2\sigma^2}\right)$$ 
and $\sigma$ is scalar parameter. $I$ is an image $I(x):\Omega \to \mathbb{R}$, $i$ is given.

So, which ones is correct for left side: $E(\sigma),E(x),E(x,\sigma),E(p_i)$ or other form?

Update: I upload the original equation for better visualization
Based on your new explanations, I think the answer is just
$$
E_{u_i}
$$
given that what you're really trying to find a shorthand for is the expression inside the summation sign in (5) of the screenshot. That is:

$$
\min_{U} \left\{ \lambda \sum_{i=1}^N \int_\Omega r_i(\chi, p_i(\chi)) u_i(\chi) d\chi \right\} = \min_{U} \left\{ \lambda \sum_{i=1}^N E_{u_i} \right\}
$$

so that
$$
E_{u_i} = \int_\Omega r_i(\chi, p_i(\chi)) u_i(\chi) d\chi
$$



Original answer below for reference

As per @Fat32's comment, any of:


$E(\sigma, z, p_i)$
$E_{\sigma,p_i}(z)$
$E(z; \sigma,p_i)$


is acceptable.

But you need to tighten up your notation. What is $x$ vs $z$? How does $\Omega$ relate to them? What is $y$?