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Omni-MATH / Let R_(>0) be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions f:R_(>0)→R such that there exists a two-variable polynomial …

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problem

Let R>0\mathbb{R}_{>0} be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions f:R>0→Rf:\mathbb{R}_{>0} \to \mathbb{R} such that there exists a two-variable polynomial P(x,y)P(x, y) with real coefficients satisfying f(xy)=P(f(x),f(y)) f(xy)=P(f(x), f(y)) for all x,y∈R>0x, y\in\mathbb{R}_{>0}.\\
Plain-text mathematical notation (without MathML)
Let R_(>0) be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions f:R_(>0)→R such that there exists a two-variable polynomial P(x,y) with real coefficients satisfying
f(xy)=P(f(x),f(y))
for all x,y∈R_(>0).\\
Original LaTeX notation
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying
$$
f(xy)=P(f(x), f(y))
$$
for all $x, y\in\mathbb{R}_{>0}$.\\

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Source and history

task_id
eb936a96-2611-5b2d-a398-fa929304555f
task_key
test--eb936a96-2611-5b2d-a398-fa929304555f
task_revision_id
1
config / split
/ test

Official source

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