ThmDex – An index of mathematical definitions, results, and conjectures.

Real exponentiation function with positive exponent is strictly isotone on positive reals

Formulation 0

Let $f : (0, \infty) \to (0, \infty)$ be a D4364: Real function such that

(i)	$p \in (0, \infty)$ is a D5407: Positive real number
(ii)	\begin{equation} f(t) = t^p \end{equation}

Let $x, y \in (0, \infty)$ each be a D5407: Positive real number such that

(i)	\begin{equation} x < y \end{equation}

Then \begin{equation} f(x) < f(y) \end{equation}

Formulation 1

Let $f : (0, \infty) \to (0, \infty)$ be a D4364: Real function such that

(i)	$p \in (0, \infty)$ is a D5407: Positive real number
(ii)	\begin{equation} f(t) = t^p \end{equation}

Let $x, y \in (0, \infty)$ each be a D5407: Positive real number.

Then \begin{equation} x < y \quad \implies \quad f(x) < f(y) \end{equation}