Bunyakovskii inequality

An equality in mathematical analysis, established by V.Ya. Bunyakovskii [1] 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

[1]	W. [V.Ya. Bunyakovskii] Bounjakowsky, "Sur quelques inegalités concernant les intégrales aux différences finis" Mem. Acad. Sci. St. Petersbourg (7) , 1 (1859) pp. 9

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.

References

[a1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)