THE PACE OF SCIENCE -- THE DEVELOPMENT OF EXTENSIONS
Augustin-Louis Cauchy (1789-1857) published his famous inequality in 1821 in the second of two notes on the theory of inequalities that formed the final part of his book Cours d'Analyse Algébrique, a volume which was perhaps the world's first rigorous calculus text. Oddly enough, Cauchy did not use his inequality in his text, except in some illustrative exercises. The first time Cauchy's inequality was applied in earnest by anyone was in 1829, when Cauchy used his inequality in an investigation of Newton's method for the calculation of the roots of algebraic and transcendental equations. This eight-year gap provides an interesting gauge of the pace of science; now, each month, there are hundreds--perhaps thousands--of new scientific publications where Cauchy's inequality is applied in one way or another.
A great many of those applications depend on a natural analog of Cauchy's inequality where sums are replaced by integrals,
$$ \int_{a}^{b} f(x)g(x) \, dx \leq \left(\int_{a}^{b} f^{2}(x)\, dx\right)^{\frac12}\left(\int_{a}^{b} g^{2}(x)\, dx\right)^{\frac12} \quad ... (*)$$
This bound first appeared in print in a Mémoire by Victor Yacovlevich Bunyakovsky which was published by the Imperial Academy of Sciences of St. Petersburg in 1859. Bunyakovsky (1804-1889) had studied in Paris with Cauchy, and he was quite familiar with Cauchy's work on inequalities; so much so that by the time he came to write his Mémoire, Bunyakovsky was content to refer to the classical form of Cauchy's inequality for finite sums simply as well-known. Moreover, Bunyakovsky did not dawdle over the limiting process; he took only a single line to pass from Cauchy's inequality for finite sums to his continuous analog in $(*)$. By ironic coincidence, one finds that this analog is labelled as inequality $(\mathbf{C})$ in Bunyakovsky's Mémoire, almost as though Bunyakovsky had Cauchy in mind.
Bunyakovsky's Mémoire was written in French, but it does not seem to have circulated widely in Western Europe. In particular, it does not seem to have been known in Göttingen in 1885 when Hermann Amandus Schwarz (1843-1921) was engaged in his fundamental work on the theory of minimal surfaces.
In the course of this work, Schwarz had the need for a a two-dimensional integral analog of Cauchy's inequality. In particular, he needed to show that if $S \subseteq \mathbb{R}^{2}$ and $f \colon S \to \mathbb{R}$ and $g \colon S \to \mathbb{R}$, then the double integrals
$$ A = \iint_{S} f^{2} \, dxdy, \quad B = \iint_{S} fg \, dxdy \quad C = \iint_{S} g^{2} \, dxdy$$
must satisfy the inequality
$$ |B| \leq \sqrt{A} \cdot \sqrt{C}, $$
and Schwarz also needed to know that the inequality is strict unless the functions $f$ and $g$ are proportional.
An approach to this result via Cauchy's inequality would have been problematical for several reasons, including the fact that the strictness of a discrete inequality can be lost in the limiting passage to integrals. Thus, Schwarz had to look for an alternative path, and, faced with necessity, he discovered a proof whose charm has stood the test of time.
Schwarz based his proof on one striking observation. Specifically, he noted that the real polynomial
$$ p(t) = \iint_{S} \left(tf(x,y)+g(x,y)\right)^{2} \, dxdy = At^{2}+2Bt+C$$
is always nonnegative, and, moreover, $p(t)$ is strictly positive unless $f$ and $g$ are proportional. The binomial formula then tells us that the coefficients must satisfy $B^{2}\leq AC$, and unless $f$ and $g$ are proportional, one actually has the strict inequality $B^{2} < AC$. Thus, from a single algebraic insight, Schwarz found everything he needed to know.
Schwarz's proof requires the wisdom to consider the polynomial $p(t)$, but, granted that step, the proof is lightning quick. Moreover, ... Schwarz's argument can be used almost without change to prove the inner product form of Cauchy's inequality, and even there Schwarz's argument provides one with a quick understanding of the case of equality. Thus, there is a little to reason to wonder why Schwarz's argument has become a textbook favorite, even though it does require one to pull a rabbit--or at least a polynomial--out of a hat.
THE NAMING OF THINGS -- ESPECIALLY INEQUALITIES
In light of the clear historical precedence of Bunyakovsky's work over that Schwarz, the common practice of referring to the bound $(*)$ as Schwarz's inequality may seem unjust. Nevertheless, by modern standards, both Bunyakovsky and Schwarz might count themselves lucky to have their names so closely associated with such a fundamental tool of mathematical analysis. Except in unusual circumstances, one garners little credit nowadays for crafting a continuous analog to a discrete inequality, or vice versa...
Ultimately, one sees that inequalities get their names in a great variety of ways. Sometimes the name is purely descriptive, such as one finds with the triangle inequality... Perhaps, more often, an inequality is associated with the name of a mathematician, but even then there is no hard-and-fast rule to govern that association. Sometimes the inequality is named after the first finder, but other principles may apply--such as the framer of the final form, or the provider of the best known application.
If one were to insist on the consistent use of the rule of the first finder, then Hölder's inequality would become Roger's inequality, Jensen's inequality would become Hölder's inequality, and only riotous confusion would result. The most practical rule--and the one used here--is simply to use the traditional names. Nevertheless, from time to time, it may be scientifically informative to examine the roots of those traditions.
