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 ?



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