<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></msubsup><mrow><msub><mi>α</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup></mrow></mstyle></math><tex-math>$f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$</tex-math></alternatives></inline-formula>
 be a function defined by power series with complex coefficients and convergent on the open disk <italic>D</italic> (0, <italic>R</italic>) ⊂ ℂ, <italic>R</italic> > 0. For any <italic>x</italic>, <italic>y</italic> ∈ ℬ, a Banach algebra, with ‖<italic>x</italic>‖, ‖<italic>y</italic>‖ &lt; <italic>R</italic> we show among others that
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>‖</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo>‖</mo></mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msubsup><mi>f</mi><mi>a</mi><mo>′</mo></msubsup></mrow></mrow></mstyle><mo stretchy="false">(</mo><mrow><mo>‖</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mi>x</mi><mo>+</mo><mi>t</mi><mi>y</mi></mrow><mo>‖</mo></mrow><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></math><tex-math>$$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$</tex-math></alternatives></disp-formula>
where
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i3.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></msubsup><mrow><mo>|</mo><msub><mi>α</mi><mi>n</mi></msub><mo>|</mo></mrow></mstyle><mtext> </mtext><msup><mi>z</mi><mi>n</mi></msup></math><tex-math>$f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$</tex-math></alternatives></inline-formula>
. Inequalities for the commutator such as
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i4.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>‖</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>‖</mo></mrow><mo>≤</mo><mn>2</mn><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><msubsup><mi>f</mi><mi>a</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mrow><mo>‖</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo>‖</mo></mrow><mo>,</mo></math><tex-math>$$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$</tex-math></alternatives></disp-formula>
if ‖<italic>x</italic>‖, ‖<italic>y</italic>‖ ≤ <italic>M</italic> &lt; <italic>R</italic>, as well as some inequalities of Hermite–Hadamard type are also provided.</p></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$$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$
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.

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia

School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, URL:

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Inequalities Of Lipschitz Type For Power Series In Banach Algebras"}

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...

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Publié en ligne: 30 sept. 2015

Pages: 61 - 83

Reçu: 21 oct. 2014

DOI: https://doi.org/10.1515/amsil-2015-0006

© Sever S. Dragomir

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.