# infimum and supremum of sum and product

Suppose that the real functions $f$ and $g$ are defined on an interval $\Delta$.  Then on this interval

• $\inf(f\!+\!g)\;\geqq\;\inf f+\inf g$

• $\sup(f\!+\!g)\;\leqq\;\sup f+\sup g$

If $f$ and $g$ are also nonnegative on $\Delta$, we can write

• $\inf(fg)\;\geqq\;\inf f\cdot\inf g$

• $\sup(fg)\;\leqq\;\sup f\cdot\sup g$

Title infimum and supremum of sum and product InfimumAndSupremumOfSumAndProduct 2013-03-22 19:01:25 2013-03-22 19:01:25 pahio (2872) pahio (2872) 4 pahio (2872) Theorem msc 06A05 msc 26D15 ProductAndQuotientOfFunctionsSum