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infimum and supremum of sum and product
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(Theorem)
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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
- $\inf(fg) \;\geqq\; \inf f\cdot\inf g$
- $\sup(fg) \;\leqq\; \sup f\cdot\sup g$
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"infimum and supremum of sum and product" is owned by pahio.
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Cross-references: interval, real functions
This is version 1 of infimum and supremum of sum and product, born on 2009-09-03.
Object id is 11895, canonical name is InfimumAndSupremumOfSumAndProduct.
Accessed 502 times total.
Classification:
| AMS MSC: | 06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order) | | | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
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Pending Errata and Addenda
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