Bunyakovskii inequality

An inequality in mathematical analysis, established by V.Ya. Bunyakovskii for square-integrable functions $ f $ and $ g $ :

\[ \left[ \int_{a}^{b}f(x)g(x)\,dx\right]^2 \le \int_{a}^{b}f^2(x)\,dx \int_{a}^{b}g^2(x)\,dx. \]

This inequality is analogous to Cauchy's algebraic inequality

\[ (a_1 b_1 + \dots + a_n b_n)^2 \le (a_1^2 + \dots + a_n^2)(b_1^2 + \dots + b_n^2). \]

The Bunyakovskii inequality is also known as the Schwarz inequality; however, Bunyakovskii published his study as early as 1859, whereas in H.A. Schwarz' work this inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).

References

2. W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)

Comments

In Western literature this inequality is often called the Cauchy inequality, or the Cauchy Schwarz inequality. Its generalization to a function $ f $ in $ L_p $ and a function $ g $ in $ L_q $, $ 1/p + 1/q = 1 $, is called the Hölder inequality.

Cauchy's algebraic inequality stated above holds for real numbers $ a_i, b_i, \quad i = 1, \dots, n $. For complex numbers $ a_i, b_i, \quad i = 1, \dots, n$, it reads

\[ \left| a_1 \overline{b_1} + \dots + a_n \overline{b_n}\right|^2 \le (|a_1^2| + \dots + |a_n^2|) \cdot (|b_1^2| + \dots + |b_n^2|). \]

It has a generalization analogous to the Hölder inequality.