# Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals

## Theorem

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.

Then:

- $\displaystyle \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$

## Proof

\(\displaystyle \forall x \in \R: \ \ \) | \(\displaystyle 0\) | \(\le\) | \(\displaystyle \paren {x \map f t + \map g t}^2\) | ||||||||||

\(\displaystyle 0\) | \(\le\) | \(\displaystyle \int_a^b \paren {x \map f t + \map g t}^2 \rd t\) | Relative Sizes of Definite Integrals | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^2 \int_a^b \paren {\map f t}^2 \rd t + 2 x \int_a^b \map f t \, \map g t \rd t + \int_a^b \paren {\map g t}^2 \rd t\) | Linear Combination of Integrals | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle A x^2 + 2 B x + C\) |

where:

- $\displaystyle A = \int_a^b \paren {\map f t}^2 \rd t$

- $\displaystyle B = \int_a^b \map f t \, \map g t \rd t$

- $\displaystyle C = \int_a^b \paren {\map g t}^2 \rd t$

The quadratic equation $A x^2 + 2 B x + C$ is non-negative for all $x$.

It follows (using the same reasoning as in Cauchy's Inequality) that the discriminant $\paren {2 B}^2 - 4 A C$ of this polynomial must be non-positive.

Thus:

- $B^2 \le A C$

and hence the result

$\blacksquare$

## Also known as

This theorem is also known as the **Cauchy-Schwarz inequality**.

Some sources give it as the **Cauchy-Schwarz-Bunyakovsky inequality**.

## Source of Name

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

## Historical Note

The Cauchy-Bunyakovsky-Schwarz Inequality for Definite Integrals was first stated in this form by Bunyakovsky in $1859$, and later rediscovered by Schwarz in $1888$.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples: Example $2.2.11$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.25$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Cauchy-Schwarz inequality for integrals**