# A Class of Exponential Inequalities

Borwein, Jonathan M. and Girgensohn, Roland (2003) A Class of Exponential Inequalities. Mathematical Inequalities and Applications, 6 . pp. 397-411.

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We prove that for reals $x_i$ with $\sum x_i \ge 0$, the estimate $\sum x_i\,e^{x_i} \ge \frac{C_N}N \sum x_i^2$ holds, where $C_N = \max\{2,e\left(1-1/N\right)\}$. We also prove analogues for the $1$-norm and for Lebesgue-integrable functions.