Geodesic
Chebyshev--Gr\"uss-type inequalities via discrete oscillations
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2014), pp. 63-89.

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The classical form of Grüss' inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In the subsequent years, many variants of this inequality appeared in the literature. The aim of this paper is to introduce a new approach, presenting a new Chebyshev–Grüss-type inequality and applying to different well-known linear, not necessarily positive, operators. Some conjectures are presented. We also compare the new inequalities with some older results. In some cases this new approach gives better estimates than the ones already known. 
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Heiner Gonska; Ioan Raşa; Maria-Daniela Rusu. Chebyshev--Gr\"uss-type inequalities via discrete oscillations. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2014), pp. 63-89. http://geodesic.mathdoc.fr/item/BASM_2014_1_a4/

[1] Ukrainian Math. J., 63 (2011), 843–864 | DOI | MR | Zbl

[2] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications, de Gruyter, New York, 1994 | MR | Zbl

[3] Andrica D., Badea C., “Grüss' inequality for positive linear functionals”, Periodica Mathematica Hungarica, 19:2 (1988), 155–167 | DOI | MR | Zbl

[4] Berdysheva E., “Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions”, J. Approx. Theory, 149:2 (2007), 131–150 | DOI | MR | Zbl

[5] Brutman L., “Lebesgue functions for polynomial interpolation – a survey”, Ann. Numer. Math., 4 (1997), 111–127 | MR | Zbl

[6] Chebyshev P. L., “O priblizhennyh vyrazhenijah odnih integralov cherez drugie”, Soobschenija i Protokoly Zasedanij Mathematicheskogo Obschestva pri Imperatorskom Khar'kovskom Universitete, 2, 1882, 93–98; Polnoe Sobranie Sochinenii P. L. Chebysheva, v. 3, Moskva–Leningrad, 1978, 128–131

[7] Oeuvres, 2 (1907), 716–719

[8] Cheney W., Light W., A Course in Approximation Theory, AMS, Providence, RI, 2000

[9] Elsner C., Prévost M., “Expansion of Euler's constant in terms of Zeta numbers”, J. Math. Anal. Appl., 398:2 (2013), 508–526 | DOI | MR | Zbl

[10] Favard J., “Sur les multiplicateurs d'interpolation”, J. Math. Pures Appl., 23:9 (1944), 219–247 | MR | Zbl

[11] Gavrea I., Ivan M., On a conjecture concerning the sum of the squared Bernstein polynomials, Submitted

[12] Gonska H., Heilmann M., Lupaş A., Raşa I., On the composition and decomposition of positive linear operators. III: A non-trivial decomposition of the Bernstein operator, 2012, arXiv: 1204.2723v1

[13] Gonska H., Piţul P., “Remarks on an article of J. P. King”, Comment. Math. Univ. Carolinae, 46 (2005), 645–652 | MR | Zbl

[14] Grüss G., “Über das Maximum des absoluten Betrages von $\frac1{b-a}\int^b_a{f(x)g(x)\,dx}-\frac1{(b-a)^2}\int^b_a{f(x)\,dx}\cdot\int^b_a{g(x)\,dx}$”, Math. Z., 39 (1935), 215–226 | DOI | MR

[15] Hermann T., Vértesi P., “On an interpolatory operator and its saturation”, Acta Math. Hungar., 37:1–3 (1981), 1–9 | DOI | MR | Zbl

[16] King J. P., “Positive linear operators which preserve $x^2$”, Acta Math. Hungar., 99:3 (2003), 203–208 | DOI | MR | Zbl

[17] Korovkin P. P., Linear Operators and Approximation Theory, Hindustan Publ. Corp., Delhi, 1960 | MR

[18] Mercer A. Mc. D., Mercer P. R., “New proofs of the Grüss inequality”, Australian J. Math. Anal. Appl., 1:2 (2004), Article 12, 6 pp. | MR | Zbl

[19] Mirakjan G. M., “Approximation of continuous functions with the aid of polynomials”, Dokl. Akad. Nauk SSSR, 31 (1941), 201–205 (Russian)

[20] Mitrinović D. S., Pečarić J. E., Fink A. M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht et al., 1993 | MR | Zbl

[21] Nikolov G., Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa, 2014, arXiv: 1402.6539v1 | MR

[22] Pǎltǎnea R., “Representation of the $K-$functional $K(f,C[a,b],C^1[a,b],\cdot)$ – a new approach”, Bull. Transilv. Univ. Braşov, Ser. III, Math. Inform. Phys., 3:52 (2010), 93–100 | MR

[23] Rivlin T. J., An Introduction to the Approximation of Functions, Blaisdell Pub. Co., Waltham, Mass.; Dover, New York, 1961 | MR

[24] Rusu M.-D., “On Grüss-type inequalities for positive linear operators”, Stud. Univ. Babeş-Bolyai Math., 56:2 (2011), 551–565 | MR

[25] Smith S. J., “Lebesgue constants in polynomial interpolation”, Annales Mathematicae et Informaticae, 33 (2006), 109–123 | MR | Zbl

[26] Szabados J., Vértesi P., Interpolation of Functions, World Scientific, Singapore, 1990 | MR | Zbl

[27] Szász O., “Generalization of S. Bernstein's polynomials to the infinite interval”, J. Res. Nat. Bur. Standards, 45 (1950), 239–245 | DOI | MR