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Problem Of The Week # 310 - May 17, 2018

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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,198
Here is this week's POTW:

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Prove that, for any two bounded functions $g_1, g_2: \mathbb{R} \to [1, \infty)$, there exist functions $h_1, h_2: \mathbb{R} \to \mathbb{R}$ such that, for every $x \in \mathbb{R},$
$$
\sup_{s \in \mathbb{R}} (g_1(s)^x g_2(s)) = \max_{t \in \mathbb{R}} (x h_1(t) + h_2(t)).
$$

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,198
My apologies for not getting to this last week - I do have an excuse, actually: I was at NIWeek. In any case, no one answered last week's POTW, which was Problem B-5 in the 2012 Putnam Archive. The solution, attributed to Kiran Kedlaya and associates, follows.

Define the function
\[\newcommand{\RR}{\mathbb{R}}
f(x) = \sup_{s \in \RR} \{x \log g_1(s) + \log g_2(s)\}.
\]
As a function of $x$, $f$ is the supremum of a collection of affine functions, so it is convex. The function $e^{f(x)}$ is then also convex, as may be checked directly from the definition: for $x_1, x_2 \in \RR$ and $t \in [0,1]$, by the weighted AM-GM inequality
\begin{align*}
t e^{f(x_1)} + (1-t) e^{f(x_2)}&\geq e^{t f(x_1) + (1-t)f(x_2)} \\
&\geq e^{f(t x_1 + (1-t)x_2)}.
\end{align*}
For each $t \in \RR$, draw a supporting line to the graph of $e^{f(x)}$ at $x=t$; it has the form $y = x h_1(t) + h_2(t)$ for some $h_1(t), h_2(t) \in \RR$. For all $x$, we then have
\[
\sup_{s \in \RR} \{g_1(s)^x g_2(s) \} \geq x h_1(t) + h_2(t)
\]
with equality for $x = t$. This proves the desired equality (including the fact that the maximum on the right side is achieved).
 
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