Cauchy-Bunyakovsky-Schwarz Inequality/Indefinite Integrals

Theorem

Let $f$ and $g$ be continuous real functions.


Then, where all these primitives exist:

$\ds \paren {\int \map f t \, \map g t \rd t}^2 \le \int \paren {\map f t}^2 \rd t \int \paren {\map g t}^2 \rd t$


Proof


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Also known as

The Cauchy-Bunyakovsky-Schwarz Inequality in its various form is also known as:

the Cauchy-Schwarz-Bunyakovsky Inequality
the Cauchy-Schwarz Inequality
Schwarz's Inequality or the Schwarz Inequality
Bunyakovsky's Inequality or Buniakovski's Inequality.

For brevity, it is sometimes referred to by the abbreviations CS inequality or CBS inequality.


Source of Name

This entry was named for Augustin Louis Cauchy, Karl Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(1)$
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(1)$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality for integrals
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality for integrals
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