论文标题
詹森（Jensen）不平等现象背后的宏伟图片
The grand picture behind Jensen's inequality
论文作者
论文摘要
令$ i $和$ j $为两个间隔，让$ f，g：i \ rightarrow \ mathbb {r} $。如果对于任何点$ a $ a $ in $ i $以及任何正数$ p $和$ q $，以便$ p + q = 1 $，我们有\ begin {align} \ nonumber p f（a） + q f（a） + q f（b） + g（pa + qb（pa + qb）\ in J，in \ end end {align} in n in n option {正数$λ_{1}，\ ldots，λ_{n} $，以至于$ \ sum_ {i = 1}^{n}λ_{i} = 1 $，我们有\ begin {align} \ nonumber \ sum__ \ sum_ {i = 1}^{n}λ_{i} x_ {i} \ right）\ in J. \ end end {align}如果我们服用$ g = -f $和$ j = [0， +\ \ infty）$，那么詹森的不平等。结论只是本文中詹森（Jensen）不平等现象背后的宏伟图片的一小段瞥见。
Let $I$ and $J$ be two intervals, and let $f, g: I \rightarrow \mathbb{R}$. If for any points $a$ and $b$ in $I$ and any positive numbers $p$ and $q$ such that $p + q = 1$, we have \begin{align} \nonumber p f(a) + q f(b) + g(pa + qb) \in J, \end{align} then for any points $x_{1}, \ldots, x_{n}$ in $I$ and any positive numbers $λ_{1}, \ldots, λ_{n}$ such that $\sum_{i=1}^{n}λ_{i} = 1$, we have \begin{align} \nonumber \sum_{i=1}^{n}λ_{i} f(x_{i}) + g\left( \sum_{i=1}^{n}λ_{i}x_{i} \right) \in J. \end{align} If we take $g = -f$ and $J = [0, +\infty)$, then the Jensen's inequality. The conclusion is only a short glimpse of the grand picture behind Jensen's inequality shows in this paper.
