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From time to time one sees insistence that the inequality name "Cauchy-Schwarz" should include Buniakovsky.

This is based on a paragraph in a note to the St Petersburg Academy from 1859, where Buniakovsky takes Cauchy's inequality to be well known ("la relation bien connue", Cauchy is not named or cited), and fills it with values of continuous functions to get, in the limit, the analogous inequality $\int f(x)^2 dx \int g(x)^2 dx \geq (\int f(x)g(x) dx)^2$ for functions on an interval.

By today's standards the entire paper of Buniakovsky contains nothing more than the idea that a limit of inequalities on finite sums gives an inequality on integrals. He derives the integral formula for the geometric mean of a function on an interval this way, and introduces $G$ and $M$ as notation for functionals that compute the geometric and arithmetic means of a function on an interval. As definitions and notation those may have been novel in some way at the time, but as a contribution to the theory of the Cauchy-Schwarz inequality, Buniakovsky's paper did not introduce any new method or result that would not have been obvious to Cauchy (or any mathematician of the times knowing Cauchy's inequality).

The question:

  • is the tendency to attach Buniakovsky's name to the inequality particular to the Russian sphere of influence (mathematical, linguistic, political) or is also attested in independent Western sources

  • was Buniakovsky's paper the first to treat nonlinear integral inequalities as a stand-alone subject?

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    $\begingroup$ I've never encountered inclusion of Bunyakovski's name in U.S. math circles and vaguely remember "Cauchy-Bunyakovski" from my olympiad days in Ukraine. Russian wikipedia shows that Russians customarily drop Schwartz and include Bunyakovski instead. $\endgroup$ – A.S. Mar 3 '16 at 23:15
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    $\begingroup$ Never encountered it either in France. Looks very much like Popov's theorem (any well-known theorem was already proved by a Russian mathematician named Popov…) $\endgroup$ – Bernard Mar 3 '16 at 23:21
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    $\begingroup$ From time to time one sees insistence on overlooking Bunyakovsky's work. :D Yes, it seems that Bunyakovsky was the first to consider integrals. You should also notice that Riemann had just introduced the Riemann integral; perhaps Bunyakovsky didn't even know about it. So, by modern standards, Bunyakovsky's work has to be considered important. $\endgroup$ – John B Mar 3 '16 at 23:37
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    $\begingroup$ @grand_chat, the Wikipedia page on Buniakovsky has the exaggerated statement that he "is credited with an early discovery of the Cauchy–Schwarz inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's works on the subject." Wikipedia's article on the Cauchy-Schwarz inequality is more precise: it says that B was C's student, who "noted that by taking limits one can obtain an integral form of Cauchy's inequality." $\endgroup$ – zyx Mar 4 '16 at 0:15
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    $\begingroup$ I asked a Russian mathematician once what is Buniakovski's inequality. His idea: Cauchy is for sums, Buniakovski is for integrals, Schwarz is for bilinear forms. $\endgroup$ – Gerald Edgar Mar 7 '16 at 4:03
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Some reflections of J. Michael Steele (cf. The Cauchy-Schwarz Master Class. Cambridge University Press, 2004, pp. 10-12) on this matter:

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.

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    $\begingroup$ J.H.S., the author of this passage is careful to respect a certain symmetry with respect to Bunyakovsky and Schwarz, but the small print seems to indicate that unlike Bunyakovsky, Schwarz did provide a proof. Nuance! $\endgroup$ – Mikhail Katz Mar 9 '16 at 10:10
  • $\begingroup$ The transition from a uniformly-spaced Riemann sum to an integral was not something in need of proof by the standards of the time, and I don't think Buniakovsky can be faulted for treating it superficially. What needs proof is his later assertion in the 1859 paper, that the equality condition is the same for the integral form of the inequality. He could have done this by taking the limit of some steps in Cauchy's proof, ending up with an integral form of the sum-of-squares identity. If that too is taken as obvious then it's even less clear what was the contribution to the inequality. @katz $\endgroup$ – zyx Mar 9 '16 at 16:36
  • $\begingroup$ @zyx, I didn't study any of Buniakovski's work beyond his translations of Cauchy (by which I was not impressed to say the least). I gather your impression is that his mathematical contribution to the Cauchy-Buniakovski inequality was minimal. Did I understand you correctly? $\endgroup$ – Mikhail Katz Mar 9 '16 at 17:09
  • $\begingroup$ From today's point of view it's fair to call the 1859 note minimal (with non-minimal contributions to other things than the inequality). The correct historical assessment is a different question that I do not have enough information to answer. For the Cauchy-Schwarz inequality Buniakovsky could be seen as packaging a fact about collections of $n$ numbers as a statement about familiar individual objects, the integrals. I don't know what else he might have done on the same subject in other papers prior to Schwarz. For the translations, his French was bad, or he got the math wrong? @katz $\endgroup$ – zyx Mar 9 '16 at 17:24
  • $\begingroup$ @zyx, the translation looks like a machine translation. If he had access to it I would have guessed he fed Cauchy into Google Translate. We find a stream of Russian words that parallels the French words but crucial logic behind the sentences is messed up in such a way as to make it unreadable unless you already know the material. $\endgroup$ – Mikhail Katz Mar 9 '16 at 17:42
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Maybe it is interesting to note that the term "l’inégalité de Schwarz" was coined by Poincaré in an 1896 paper in Acta Mathematica 20, p. 73, and was used in the French and German literature for the integral inequality until well into the 20th century. https://archive.org/stream/actamathematica20upps#page/73/mode/1up

The term "Cauchy-Schwarz inequality" was used for the algebraic inequality by Hardy and Littlewood in a paper in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1920, pp. 38-39. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002505649&physid=PHYS_0045

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This is the elephant phenomenon that Russian mathematicians like to joke about. It is widely known that Russia is the homeland of the elephants ("Rossia--rodina slonov"). Similarly any important mathematical result ultimately must have a Russian source and since Buniakovski translated some of Cauchy's work into Russian, he was a natural choice for promotion.

Incidentally I had occasion to study Buniakovski's translations of Cauchy and they are uniformly atrocious.

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