Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir

Annales Mathematicae Silesianae (2015)

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

Abstract

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Let [...] f(z)=∑n=0∞αnzn f ( z ) = n = 0 α 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 ( y ) - f ( x ) y - x 0 1 f a ' ( ( 1 - t ) x + t y ) d t where [...] fa(z)=∑n=0∞|αn| zn f a ( z ) = n = 0 | α n | z n . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, f ( x ) f ( y ) - f ( y ) f ( x ) 2 f a ( M ) f a ' ( M ) y - x , if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

How to cite

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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

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