# Inequalities Of Lipschitz Type For Power Series In Banach Algebras

• Volume: 29, Issue: 1, page 61-83
• ISSN: 0860-2107

top

## Abstract

top
Let [...] f(z)=∑n=0∞αnzn $f\left(z\right)={\sum }_{n=0}^{\infty }{\alpha }_{n}{z}^{n}$ 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 $∥f\left(y\right)-f\left(x\right)∥\le ∥y-x∥{\int }_{0}^{1}{f}_{a}^{\text{'}}\left(∥\left(1-t\right)x+ty∥\right)dt$ where [...] fa(z)=∑n=0∞|αn| zn ${f}_{a}\left(z\right)={\sum }_{n=0}^{\infty }|{\alpha }_{n}|\phantom{\rule{0.277778em}{0ex}}{z}^{n}$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $∥f\left(x\right)f\left(y\right)-f\left(y\right)f\left(x\right)∥\le 2{f}_{a}\left(M\right){f}_{a}^{\text{'}}\left(M\right)∥y-x∥,$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

## How to cite

top

Sever S. Dragomir. "Inequalities Of Lipschitz Type For Power Series In Banach Algebras." Annales Mathematicae Silesianae 29.1 (2015): 61-83. <http://eudml.org/doc/276889>.

@article{SeverS2015,
abstract = {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 $\left\Vert \{f(y) - f(x)\} \right\Vert \le \left\Vert \{y - x\} \right\Vert \int \_0^1 \{f\_a^\prime \} (\left\Vert \{(1 - t)x + ty\} \right\Vert )dt$ 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‖, $\left\Vert \{f(x)f(y) - f(y)f(x)\} \right\Vert \le 2f\_a (M)f\_a^\prime (M)\left\Vert \{y - x\} \right\Vert ,$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.},
author = {Sever S. Dragomir},
journal = {Annales Mathematicae Silesianae},
keywords = {Banach algebras; Power series; Lipschitz type inequalities; Hermite-Hadamard type inequalities},
language = {eng},
number = {1},
pages = {61-83},
title = {Inequalities Of Lipschitz Type For Power Series In Banach Algebras},
url = {http://eudml.org/doc/276889},
volume = {29},
year = {2015},
}

TY - JOUR
AU - Sever S. Dragomir
TI - Inequalities Of Lipschitz Type For Power Series In Banach Algebras
JO - Annales Mathematicae Silesianae
PY - 2015
VL - 29
IS - 1
SP - 61
EP - 83
AB - 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 $\left\Vert {f(y) - f(x)} \right\Vert \le \left\Vert {y - x} \right\Vert \int _0^1 {f_a^\prime } (\left\Vert {(1 - t)x + ty} \right\Vert )dt$ 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‖, $\left\Vert {f(x)f(y) - f(y)f(x)} \right\Vert \le 2f_a (M)f_a^\prime (M)\left\Vert {y - x} \right\Vert ,$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.
LA - eng
KW - Banach algebras; Power series; Lipschitz type inequalities; Hermite-Hadamard type inequalities
UR - http://eudml.org/doc/276889
ER -

## References

top
1. [1] Azpeitia A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12. Zbl0832.26015
2. [2] Bhatia R., Matrix analysis, Springer-Verlag, New York, 1997. Zbl0863.15001
3. [3] Cheung W.-S., Dragomir S.S., Vector norm inequalities for power series of operators in Hilbert spaces, Tbilisi Math. J. 7 (2014), no. 2, 21–34. Zbl1320.47013
4. [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. Zbl0956.26015
5. [5] Dragomir S.S., A mapping in connection to Hadamard’s inequalities, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 128 (1991), 17–20. Zbl0747.26015
6. [6] Dragomir S.S., Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), 49–56. Zbl0758.26014
7. [7] Dragomir S.S., On Hadamard’s inequalities for convex functions, Math. Balkanica 6 (1992), 215–222. Zbl0834.26010
8. [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. Zbl0995.26009
9. [9] Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471–476.[Crossref] Zbl1113.26021
10. [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. Zbl1236.26020
11. [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. Zbl06082538
12. [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. [13] Dragomir S.S., Pearce C.E.M., Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, 2000. Available at
14. [14] Guessab A., Schmeisser G., Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), no. 2, 260–288. Zbl1012.26013
15. [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. Zbl1183.26025
16. [16] Matić M., Pečarić J., Note on inequalities of Hadamard’s type for Lipschitzian mappings, Tamkang J. Math. 32 (2001), no. 2, 127–130. Zbl0993.26019
17. [17] Merkle M., Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 113–117. Zbl0946.26016
18. [18] Mikusiński J., The Bochner integral, Birkhäuser Verlag, Basel, 1978. Zbl0369.28010
19. [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. Zbl0939.26009
20. [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. Zbl1067.26021
21. [21] Toader G., Superadditivity and Hermite–Hadamard’s inequalities, Studia Univ. Babeş-Bolyai Math. 39 (1994), no. 2, 27–32. Zbl0868.26012
22. [22] Yang G.-S., Hong M.-C., A note on Hadamard’s inequality, Tamkang J. Math. 28 (1997), no. 1, 33–37. Zbl0880.26019
23. [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. Zbl0939.26010

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.