1991 Putnam B6

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in

terms of $a$ and $b$, such that

$$

a^x b^{1-x} \leq a \frac{\sinh ux}{\sinh u} + b \frac{\sinh u(1-x)}{\sinh u}

$$

for all $u$ with $0 < |u| \leq c$ and for all $x$, $0 < x < 1$. (Note:

$\sinh u = (e^u - e^{-u})/2$.)

令 $a$ 和 $b$ 为正数。找到最大的数字$c$，

$a$ 和 $b$ 项，使得

$$

a^x b^{1-x} \leq a \frac{\sinh ux}{\sinh u} + b \frac{\sinh u(1-x)}{\sinh u}

$$

对于所有 $u$ 且 $0 < |u| \leq c$ 并且对于所有 $x$，$0 < x < 1$。 (注：

$\sinh u = (e^u - e^{-u})/2$。)