zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Generalizations and refinements of Hermite-Hadamard’s inequality. (English) Zbl 1096.26014
The Hermite-Hadamard inequality can be easily extended to the case of twice differentiable functions $f$ with bounded second derivative. Precisely, if $\gamma\leq f^{\prime\prime} \leq\Gamma,$ then $$ \frac{3S_{2}-2\Gamma}{24}(b-a)^{2}\leq\frac{1}{b-a}\int_{a}^{b}f\,dt-f\left( \frac{a+b}{2}\right) \leq\frac{3S_{2}-2\gamma}{24}(b-a)^{2} $$ and $$ \frac{3S_{2}-\gamma}{12}(b-a)^{2}\leq\frac{f(a)+f(b)}{2}-\frac{1}{b-a}\int _{a}^{b}f\,dt\leq\frac{\Gamma}{12}(b-a)^{2} $$ The paper under review contains extensions of the Hermite-Hadamard inequality to the context of functions with bounded derivatives of $n$th order. For example, if $f:[a,b]\rightarrow\Bbb{R}$ is an $n$-times differentiable function with $\gamma\leq f^{(n)}\leq\Gamma,$ then it is proved that $$ \frac{(b-a)^{n+1}}{n!2^{n}}\left[ S_{n}+\left( \frac{1+(-1)^{n}} {2(n+1)}-1\right) \Gamma\right]\leq (-1)^{n}\int_{a}^{b}f\,dt $$ $$ +\sum_{i=0}^{n-1}\frac{(b-a)^{n-i}}{(n-i)!}\frac{(-1)^{n+1}+(-1)^{i} }{2^{n-i}}f^{(n-i-1)}\biggl(\frac{a+b}{2}\biggr)\leq \frac{(b-a)^{n+1}}{n!2^{n}}\left[ S_{n}+\left( \frac{1+(-1)^{n} }{2(n+1)}-1\right) \gamma\right] $$ where $S_{n}=\frac{f^{(n-1)}(b)-f^{(n-1)}(a)}{b-a}.$ Further extensions are obtained via the concept of harmonic sequence of polynomials.

26D15Inequalities for sums, series and integrals of real functions
41A55Approximate quadratures
Full Text: DOI
[1] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions with formulas, graphs, and mathematical tables , National Bureau of Standards, Appl. Math. Series 55 , 4th printing, Washington, 1965, 1972.
[2] G. Allasia, C. Diodano and J. Pečarić, Hadamard-type inequalities for $(2r)$-convex functions with application , Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 133 (1999), 187-200. · Zbl 1035.26016
[3] P. Cerone and S.S. Dragomir, Midpoint-type rules from an inequality point of view , in Handbook of analytic-computational methods in applied mathematics (G. Anastassiou, ed.), CRC Press, New York, 20000.
[4] --------, Trapezoidal-type rules from an inequality point of view , Handbook of analytic-computational methods in applied mathematics (G. Anastassiou, ed.), CRC Press, New York, 2000.
[5] Lj. Dedić, C.E.M. Pearce and J. Pečarić, Hadamard and Dragomir-Argawal inequalities, higher order convexity and the Euler formula , J. Korean Math. Soc. 38 (2001), 1235-1243. · Zbl 1017.26019
[6] S.S. Dragomir and C.E.M. Pearce, Selected topics on Hermite-Hadamard type inequalities and applications , RGMIA Monographs, 2000. Available online at http://rgmia.vu.edu.au/monographs/hermite_hadamard.html.
[7] B.-N. Guo and F. Qi, Generalization of Bernoulli polynomials , Internat. J. Math. Ed. Sci. Tech. 33 (2002), 428-431. · Zbl 1021.11003 · doi:10.1080/002073902760047913
[8] Q.-M. Luo, B.-N. Guo, F. Qi and L. Debnath, Generalizations of Bernoulli numbers and polynomials , Internat. J. Math. Math. Sci. 2003 (2003), 3769-3776. RGMIA Res. Rep. Coll. 5 (2002), 353-359. Available online at http://rgmia.vu.edu.au/v5n2.html. · Zbl 1038.11013 · doi:10.1155/S0161171203112070 · eudml:50767
[9] Q.-M. Luo and F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials , Adv. Stud. Contemp. Math., Kyungshang, vol. 7, 2003, pp. 11-18. RGMIA Res. Rep. Coll. 5 (2002), 405-412. Available online at http://rgmia.vu.edu.au/v5n3.html. · Zbl 1042.11012
[10] F. Qi and B.-N. Guo, Generalized Bernoulli polynomials , RGMIA Res. Rep. Coll. 4 (2001), 691-695. Available online at http://rgmia.vu.edu.au/v4n4.html.
[11] N. Ujević, Some double integral inequalities and applications , Acta Math. Univ. Comenian. (N.S.) 71 (2002), 189-199. · Zbl 1053.26016 · emis:journals/AMUC/_vol-71/_no_2/_ujevic/ujevic.html · eudml:124036
[12] Zh.-X. Wang and D.-R. Guo, Tèshū Hánshù Gàilùn , Introduction to special function , in The Series of Advanced Physics of Peking University , Peking Univ. Press, Beijing, China, 2000 (in Chinese).