ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5210 on D2455: Real geometric mean
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}
Proofs
Proof 0
Let $x, y \in [0, \infty)$ each be an D4767: Unsigned real number.
This result is a particular case of R5182: Tight upper bound to a finite product of unsigned real numbers. $\square$