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猜想：设 $\alpha=(\alpha_1,\cdots,\alpha_m)\in \mathbb{R}_{+}^m, \beta=(\beta_1,\cdots,\beta_m)\in \mathbb{R}_{+}^m$ 满足
$\alpha_1\geq \cdots\geq \alpha_m, \beta_1\geq \cdots \geq \beta_m$ 且 $\alpha_1+\cdots+\alpha_m=\beta_1+\cdots+\beta_m$.
如果对任意的正实数 $x$ 都有下面的不等式成立
\[
(x^{\alpha_1}+1)\cdots (x^{\alpha_m}+1)\geq (x^{\beta_1}+1)\cdots (x^{\beta_m}+1),
\]
则有 $\alpha \succeq \beta$, 即
\begin{align*}
\alpha_1&\geq \beta_1\\
\alpha_1+\alpha_2&\geq \beta_1+\beta_2\\
\cdots\\
\alpha_1+\cdots+\alpha_{m-1}&\geq \beta_1+\cdots+\beta_{m-1}
\end{align*} |
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