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

Result R5210 on D2455: Real geometric mean
Subresult of R5182: Tight upper bound to a finite product of unsigned real numbers

Tight upper bound to a product of two unsigned real numbers

Formulation 0

Let $x, y \in [0, \infty)$ each be an D4767: Unsigned real number.

Then

(1)	\begin{equation} x y \leq \left( \frac{x + y}{2} \right)^2 \end{equation}
(2)	\begin{equation} x y = \left( \frac{x + y}{2} \right)^2 \quad \iff\quad x = y \end{equation}

Subresults

▶	R5209: Of all rectangles with a given perimeter, the square maximizes the area

Proofs

Proof 0

Let $x, y \in [0, \infty)$ each be an D4767: Unsigned real number.