Karamata 不等式

在数学，Karamata不等式，后命名乔文·卡拉马塔，也被称为优化不等式，用于处理定义在实数区间的凸和凹的实值函数。它推广了Jensen不等式的离散形式，并反过来推广了Schur-凸函数的概念。
最近处理不等式的时候反复碰到，故归纳之.

定理（英文描述)
Let $I$ be an interval of the real line and let $f$ denote a real-valued, convex function defined on $I$. If $x_1, . . . , x_n$ and $y_1, . . . , y_n$ are numbers in $I$ such that $(x_1, . . . , x_n)$ majorizes $(y_1, . . . , y_n)$, then
${\displaystyle f(x_{1})+\cdots +f(x_{n})\geq f(y_{1})+\cdots +f(y_{n}).}$ (1)

Here majorization means that $x_1, . . . , x_n$ and $y_1, . . . , y_n$ satisfies ${\displaystyle x_{1}\geq x_{2}\geq \cdots \geq x_{n}}$ and ${\displaystyle y_{1}\geq y_{2}\geq \cdots \geq y_{n}}$
and we have the inequalities
${\displaystyle x_{1}+\cdots +x_{i}\geq y_{1}+\cdots +y_{i}}$ for all $i ∈ {1, . . . , n − 1}$.
and the equality ${\displaystyle x_{1}+\cdots +x_{n}=y_{1}+\cdots +y_{n}}$
If $f$ is a strictly convex function, then the inequality (1) holds with equality if and only if we have $x_i = y_i$ for all $i ∈ {1, . . . , n}$.

一道题：
$-x \left(x^a-(x-1)^a-\left(x-\frac{1}{2}\right)^a+\left(x-\frac{3}{2}\right)^a\right)-(x-1)^a \\+\left(x-\frac{3}{2}\right)^a+\left(1-2^{a-1}\right) \left(\left(x-\frac{3}{2}\right)^a-\left(x-\frac{1}{2}\right)^a\right)>0$
此时有 $x\ge 2$ 和 $-1<a<0$ .

证明：
The inequality is written as $f(a, x) + g(a, x) > 0$ where
$f(a, x) = - x^{1+a} + (x-1)^{1+a} + (x-1/2)^{1+a} - (x-3/2)^{1+a}$and
$g(a, x) = (1/2 - 2^{a-1})((x-3/2)^a - (x-1/2)^a).$Clearly, $g(a, x) > 0$. It suffices to prove that $f(a, x) \ge 0$.
Since $x \mapsto -x^{1+a}$ is convex on $(0, \infty)$, and $(x, x - 3/2)$ majorizes $(x-1/2, x-1)$, by using Karamata inequality, we have $f(a, x) \ge 0$. We are done.
2020年4月22日最后修改