On Convergence Rates of Some Limits

On Convergence Rates of Some Limits

In 2019 Seneta has provided a characterization of slowly varying functions <i>L</i> in the Zygmund sense by using the condition, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn&...

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Main Authors: Edward Omey, Meitner Cadena
Format: Article
Language:English
Published: MDPI AG 2020-04-01
Series:Mathematics
Subjects:
slowly varying
monotony in the Zygmund sense
class Γ<sub><i>a</i></sub>(<i>g</i>)
self-neglecting function
convergence rates
Online Access:https://www.mdpi.com/2227-7390/8/4/634
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author Edward Omey
Meitner Cadena
author_facet Edward Omey
Meitner Cadena
author_sort Edward Omey
collection DOAJ
description In 2019 Seneta has provided a characterization of slowly varying functions <i>L</i> in the Zygmund sense by using the condition, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>)</mo> </mrow> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>. Very recently, we have extended this result by considering a wider class of functions <i>U</i> related to the following more general condition. For each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, for some functions <i>r</i> and <i>g</i>. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.
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spelling doaj.art-e2268309144246b38ae270e38f89ec7a2023-11-19T22:14:45ZengMDPI AGMathematics2227-73902020-04-018463410.3390/math8040634On Convergence Rates of Some LimitsEdward Omey0Meitner Cadena1Research Centre for Mathematics, Education, Econometrics and Statistics (MEES), Catholic University Leuven at Campus Brussels, Warmoesberg 26, 1000 Brussels, BelgiumDepartamento de Ciencias Exactas, Universidad de las Fuerzas Armadas, Sangolqui 171103, EcuadorIn 2019 Seneta has provided a characterization of slowly varying functions <i>L</i> in the Zygmund sense by using the condition, for each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>)</mo> </mrow> <mrow> <mi>L</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>. Very recently, we have extended this result by considering a wider class of functions <i>U</i> related to the following more general condition. For each <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>g</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mi>U</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mfrac> <mo>−</mo> <mn>1</mn> </mfenced> <mo>→</mo> <mn>0</mn> <mspace width="1.em"></mspace> <mi>as</mi> <mspace width="1.em"></mspace> <mi>x</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, for some functions <i>r</i> and <i>g</i>. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.https://www.mdpi.com/2227-7390/8/4/634slowly varyingmonotony in the Zygmund senseclass Γ<sub><i>a</i></sub>(<i>g</i>)self-neglecting functionconvergence rates
spellingShingle Edward Omey
Meitner Cadena
On Convergence Rates of Some Limits
Mathematics
slowly varying
monotony in the Zygmund sense
class Γ<sub><i>a</i></sub>(<i>g</i>)
self-neglecting function
convergence rates
title On Convergence Rates of Some Limits
title_full On Convergence Rates of Some Limits
title_fullStr On Convergence Rates of Some Limits
title_full_unstemmed On Convergence Rates of Some Limits
title_short On Convergence Rates of Some Limits
title_sort on convergence rates of some limits
topic slowly varying
monotony in the Zygmund sense
class Γ<sub><i>a</i></sub>(<i>g</i>)
self-neglecting function
convergence rates
url https://www.mdpi.com/2227-7390/8/4/634
work_keys_str_mv AT edwardomey onconvergenceratesofsomelimits
AT meitnercadena onconvergenceratesofsomelimits

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