We denote $p_{+} = \mathbb{P}(X' \in C_{+1})$ and $p_{-} = \mathbb{P}(X' \in C_{-1})$, with $p_{+} \geqslant p_{-}$. Deduce that
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8}d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = -1\right) \leqslant \exp\left(\frac{\lambda^{2}}{2}\right) \left(\mathbb{E}\left(\exp\left(\frac{1}{8}d(X', C_{-1})^{2}\right)\right)\right)^{1-\lambda} \cdot \left(\mathbb{E}\left(\exp\left(\frac{1}{8}d(X', C_{+1})^{2}\right)\right)\right)^{\lambda}$$