📝 题目
例 15 设 ${f}^{\prime \prime }\left( x\right) \geq 0\left( {-\infty < x < + \infty }\right) ,u\left( x\right) \in C\left( {-\infty , + \infty }\right)$ . 求证: 对 $\forall a > 0$ ,有
$$ \frac{1}{a}{\int }_{0}^{a}f\left\lbrack {u\left( x\right) }\right\rbrack \mathrm{d}x \geq f\left( {\frac{1}{a}{\int }_{0}^{a}u\left( x\right) \mathrm{d}x}\right) . $$
💡 答案与解析
证 令 $A = \frac{1}{a}{\int }_{0}^{a}u\left( x\right) \mathrm{d}x$ . 根据 $f\left( x\right)$ 的凹性,有
$$ f\left( t\right) \geq {f}^{\prime }\left( A\right) \left( {t - A}\right) + f\left( A\right) \;\left( {\forall t \in \left( {-\infty , + \infty }\right) }\right) . $$
代入 $t = u\left( x\right)$ ,并对 $x$ 从 0 到 $a$ 积分,得
$$ \frac{1}{a}{\int }_{0}^{a}f\left( {u\left( x\right) }\right) \mathrm{d}x \geq {f}^{\prime }\left( A\right) \left( {\frac{1}{a}{\int }_{0}^{a}u\left( x\right) \mathrm{d}x - A}\right) + f\left( A\right) = f\left( A\right) , $$
即
$$ \frac{1}{a}{\int }_{0}^{a}f\left( {u\left( x\right) }\right) \mathrm{d}x \geq f\left( {\frac{1}{a}{\int }_{0}^{a}u\left( x\right) \mathrm{d}x}\right) . $$