JSON source · original UTF-8 line locators · 2.6 KiB
{
"paper_id": "10.0000/fixture.1975.1.01",
"source_pages": 2,
"license_notice": "Translated from Russian. Original: A. B. Ivanov, V. G. Petrov, “Об одном классе интегральных неравенств”, Proceedings of an Invented Academy. Physics. Mathematics, 1975, vol. 24 no. 1, pp. 1–2. Published by Estonian Academy Publishers; license CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/) as deposited with Crossref by the publisher (the publisher's policy page names CC BY-NC 4.0). Changes: translated into English.",
"title_en": "On a class of integral inequalities",
"title_ru": "Об одном классе интегральных неравенств",
"authors": [
"A. B. Ivanov",
"V. G. Petrov"
],
"abstract_en": "The note studies nonnegative summable functions bounded by a power of x on [0,1]. It proves a sharp estimate for their primitives and shows that the constant is best possible.",
"abstract_source": "generated",
"body_md": "<!-- source-page: 001 -->\n\nUDC 517.5\n\nA. B. IVANOV, V. G. PETROV\n\n# ON A CLASS OF INTEGRAL INEQUALITIES\n\nIn this note we consider nonnegative functions \\(f\\) that are summable on the segment \\([0,1]\\) and satisfy the condition \\(f(x)\\le C x^{\\alpha}\\) for some constant \\(C>0\\) and \\(0<\\alpha<1\\). Inequalities of this type arise in the study of boundary value problems for ordinary differential equations, and they were considered under stronger assumptions in [1, 2].\n\nDenote by \\(K_\\alpha\\) the class of all such functions. For \\(f\\in K_\\alpha\\) we set\n\n\\[\nF(x)=\\int_0^x f(t)\\,dt .\n\\tag{1}\n\\]\n\n**Theorem 1.** *If \\(f\\in K_\\alpha\\), then for all \\(x\\in[0,1]\\) the estimate \\(F(x)\\le C x^{1+\\alpha}/(1+\\alpha)\\) holds, and the constant cannot be improved.*\n\n<!-- source-page: 002 -->\n\n**Proof.** The estimate follows by integrating the condition on \\(f\\) over \\([0,x]\\). The function \\(f(x)=Cx^{\\alpha}\\) belongs to \\(K_\\alpha\\) and turns the estimate into an equality, so the constant is sharp.\n\nInstitute of Cybernetics, Academy of Sciences of the Estonian SSR. Received May 4, 1974\n\n## References\n\n1. A. A. Author, Mat. Sb. 1 (1960) (Russian).\n2. B. B. Author, Uspekhi Mat. Nauk 2 (1965) (Russian).\n\n## Summary (Estonian)\n\nOn a class of integral inequalities. The note proves a sharp estimate for the primitives of nonnegative summable functions bounded by a power of x.\n",
"translator_notes": [
{
"page": 2,
"note": "The source writes the exponent as α throughout."
}
],
"produced_with": "Fixture: written by hand."
}