Let $x$ be a strictly positive real number, $\beta$ a real number such that $0 < \beta < 1$. Prove that: $x ^ { \beta } \leqslant \beta x + 1 - \beta$.
Let $x$ be a strictly positive real number, $\beta$ a real number such that $0 < \beta < 1$.
Prove that: $x ^ { \beta } \leqslant \beta x + 1 - \beta$.