```markdown
### Self-concordant functions

#### Definition
- A function \( f : \mathbb{R} \to \mathbb{R} \) is self-concordant if \( |f'''(x)| \leq 2 f''(x)^{3/2} \) for all \( x \in \text{dom } f \).
- A function \( f : \mathbb{R}^n \to \mathbb{R} \) is self-concordant if \( g(t) = f(x + tv) \) is self-concordant for all \( x \in \text{dom } f \), \( v \in \mathbb{R}^n \).

#### Examples on \( \mathbb{R} \)
- Linear and quadratic functions
- Negative logarithm \( f(x) = -\log x \)
- Negative entropy plus negative logarithm: \( f(x) = x \log x - \log x \)

#### Affine invariance
If \( f : \mathbb{R} \to \mathbb{R} \) is self-concordant, then \( \tilde{f}(y) = f(ay + b) \) is self-concordant:
- \( \tilde{f}'''(y) = a^3 f'''(ay + b) \)
- \( \tilde{f}''(y) = a^2 f''(ay + b) \)
```
