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2015 | 29 | 1 | 61-83
Tytuł artykułu

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let [...] f(z)=∑n=0∞αnzn $f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that [...] ‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt $$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$ where [...] fa(z)=∑n=0∞|αn| zn $f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.
Wydawca
Rocznik
Tom
29
Numer
1
Strony
61-83
Opis fizyczny
Daty
wydano
2015-09-01
otrzymano
2014-10-21
poprawiono
2015-03-04
online
2015-09-30
Twórcy
  • Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia, sever.dragomir@vu.edu.au
  • School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, URL:
Bibliografia
  • [1] Azpeitia A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12.
  • [2] Bhatia R., Matrix analysis, Springer-Verlag, New York, 1997.
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  • [4] Dragomir S.S., Cho Y.J., Kim S.S., Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. Math. Anal. Appl. 245 (2000), no. 2, 489–501.
  • [5] Dragomir S.S., A mapping in connection to Hadamard’s inequalities, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 128 (1991), 17–20.
  • [6] Dragomir S.S., Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), 49–56.
  • [7] Dragomir S.S., On Hadamard’s inequalities for convex functions, Math. Balkanica 6 (1992), 215–222.
  • [8] Dragomir S.S., An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Art. 35.
  • [9] Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471–476.[Crossref]
  • [10] Dragomir S.S., Gomm I., Bounds for two mappings associated to the Hermite–Hadamard inequality, Aust. J. Math. Anal. Appl. 8 (2011), Art. 5, 9 pp.
  • [11] Dragomir S.S., Gomm I., Some new bounds for two mappings related to the Hermite–Hadamard inequality for convex functions, Numer. Algebra Cont Optim. 2 (2012), no. 2, 271–278.
  • [12] Dragomir S.S., Milośević D.S., Sándor J., On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math. 4 (1993), 21–24.
  • [13] Dragomir S.S., Pearce C.E.M., Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, 2000. Available at
  • [14] Guessab A., Schmeisser G., Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), no. 2, 260–288.
  • [15] Kilianty E., Dragomir S.S., Hermite–Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl. 13 (2010), no. 1, 1–32.
  • [16] Matić M., Pečarić J., Note on inequalities of Hadamard’s type for Lipschitzian mappings, Tamkang J. Math. 32 (2001), no. 2, 127–130.
  • [17] Merkle M., Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 113–117.
  • [18] Mikusiński J., The Bochner integral, Birkhäuser Verlag, Basel, 1978.
  • [19] Pearce C.E.M., Rubinov A.M., P-functions, quasi-convex functions, and Hadamard type inequalities, J. Math. Anal. Appl. 240 (1999), no. 1, 92–104.
  • [20] Pečarić J., Vukelić A., Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, in: Functional equations, inequalities and applications, Kluwer Acad. Publ., Dordrecht, 2003, pp. 105–137.
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  • [23] Yang G.-S., Tseng K.-L., On certain integral inequalities related to Hermite–Hadamard inequalities, J. Math. Anal. Appl. 239 (1999), no. 1, 180–187.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_amsil-2015-0006
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