# Understanding the proof of Fenchel-Rockafellar Theorem In Brezis

I'm reading the proof of Fenchel-Rockafellar's theorem in ''Functional Analysis, Sobolev Spaces and Partial Differential Equations'' by Haim Brezis.

Let $$E$$ be a topological vector space and $$\varphi, \psi: E \to \mathbb R \cup \{+\infty\}$$ convex functions. Assume there exists $$x_{0} \in E$$ such that $$\varphi(x_0) < +\infty, \psi(x_0) < +\infty$$, and $$\varphi$$ is continuous at $$x_{0}$$. Then \begin{aligned} \inf _{x \in E}\{\varphi(x)+\psi(x)\} &=\sup _{f \in E^{\star}}\left\{-\varphi^{\star}(-f)-\psi^{\star}(f)\right\} \\ &=\max _{f \in E^{\star}}\left\{-\varphi^{\star}(-f)-\psi^{\star}(f)\right\}. \end{aligned}

In the proof, I read page 16

From (16) and (17) we obtain $$\varphi^{\star}(-\frac{f}{k})\leq -\frac{\alpha}{k}$$ $$\psi^{\star}(\frac{f}{k}) \leq \frac{\alpha}{k} - a$$

I don't understand how do we get there.

For instance the first inequality since $$\forall (x, \lambda) \in epi \varphi$$ $$<\frac{f}{k},x> + \lambda \geq \frac{\alpha}{k}$$ And $$\varphi(x) \leq \lambda$$ We have $$<-\frac{f}{k},x> - \lambda +\varphi(x) -\varphi(x) \leq -\frac{\alpha}{k}$$ Then $$\varphi^{\star}(-\frac{f}{k}) - \lambda +\varphi(x) \leq -\frac{\alpha}{k}$$ But I don't see how I can conclude

Is anyone able to help please ?