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:

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

Proof

$\ds \forall x \in \R: \, $

$\ds 0$

$\le$

$\ds \paren {x \map f t + \map g t}^2$

$\ds 0$

$\le$

$\ds \int_a^b \paren {x \map f t + \map g t}^2 \rd t$

Relative Sizes of Definite Integrals

$\ds $

$=$

$\ds x^2 \int_a^b \paren {\map f t}^2 \rd t + 2 x \int_a^b \map f t \, \map g t \rd t + \int_a^b \paren {\map g t}^2 \rd t$

Linear Combination of Definite Integrals

$\ds $

$=$

$\ds A x^2 + 2 B x + C$

where:

$\ds A$

$=$

$\ds \int_a^b \paren {\map f t}^2 \rd t$

$\ds B$

$=$

$\ds \int_a^b \map f t \map g t \rd t$

$\ds C$

$=$

$\ds \int_a^b \paren {\map g t}^2 \rd t$

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

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.

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

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Schwarz's Inequality: $3.2.11$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: Cauchy-Schwarz Inequality for Integrals: $36.13$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.11$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.25$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 37$: Inequalities: Cauchy-Schwarz Inequality for Integrals: $37.13.$