infimum and supremum of sum and product

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

• $inf(f+g)\geqq inff+infg$

• $sup(f+g)\leqq supf+supg$

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

• $inf(fg)\geqq inff\cdot infg$

• $sup(fg)\leqq supf\cdot supg$

Title	infimum and supremum of sum and product
Canonical name	InfimumAndSupremumOfSumAndProduct
Date of creation	2013-03-22 19:01:25
Last modified on	2013-03-22 19:01:25
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	4
Author	pahio (2872)
Entry type	Theorem
Classification	msc 06A05
Classification	msc 26D15
Related topic	ProductAndQuotientOfFunctionsSum