grandes-ecoles 2011 Q1

grandes-ecoles · France · centrale-maths1__pc Proof Direct Proof of an Inequality

Let $\lambda$ be a real number in the interval $]0,1[$, and let $a$ and $b$ be two non-negative real numbers. Show that $$\lambda a + (1-\lambda) b \geq a^{\lambda} b^{1-\lambda}$$ (one may introduce a certain auxiliary function and justify its concavity). Moreover, show that for all real $u > 1$, $$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

Let $\lambda$ be a real number in the interval $]0,1[$, and let $a$ and $b$ be two non-negative real numbers. Show that
$$\lambda a + (1-\lambda) b \geq a^{\lambda} b^{1-\lambda}$$
(one may introduce a certain auxiliary function and justify its concavity). Moreover, show that for all real $u > 1$,
$$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

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