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Inequalities Of Lipschitz Type For Power Series In Banach Algebras

100%

Dragomir S. S.

Annales Mathematicae Silesianae

2015

tom 29

nr 1

61-83

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.

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On Kaluza’s sign criterion for reciprocal power series

100%

Baricz A., Vesti J., Vuorinen M.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2011

tom 65

nr 2

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied.

Ograniczanie wyników

1 Annales Mathematicae Silesianae

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Baricz A.

1 Dragomir S. S.

1 Vesti J.

1 Vuorinen M.

1 2015

1 2011

Wyniki wyszukiwania

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

On Kaluza’s sign criterion for reciprocal power series