Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of t...
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| Date: | 2011 |
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Інститут математики НАН України
2011
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| Series: | Український математичний журнал |
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| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/166346 |
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| Cite this: | Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation / М.Н. Нооshmаnd // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 281–288. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1663462020-02-20T01:26:24Z Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation Нооshmаnd, M.H. Статті We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x))=α(x)+1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm f-type function is its unique solution. We also show that the infralogarithm f-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential f-type functions and their functional equation β(x)=f(β(x−1)) which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them. Запропоновано узагальненi форми ультраекспоненцiальних та iнфралогарифмiчних функцiй, що були введенi i вивченi автором ранiше, та наведено два класи спецiальних функцiй — ультраекспоненцiального та iнфралогарифмiчного f-типу. В результатi дослiджень отримано загальний розв’язок рiвняння Абеля α(f(x))=α(x)+1 за певних умов для реальної функцiї f i доведено нову цiлком iншу теорему єдиностi для рiвняння Абеля з твердженням про те, що функцiя iнфралогарифмiчного f-типу є єдиним розв’язком цього рiвняння. Також показано, що функцiя iнфралогарифмiчного f-типу є суттєво єдиним розв’язком рiвняння Абеля. Подiбнi теореми доведено для функцiй ультраекспоненцiального f-типу та їх функцiонального рiвняння β(x)=f(β(x−1)), яке можна вважати дуальним для рiвняння Абеля. Також розв’язано задачу, що не була розв’язана до теперiшнього часу, вивчено властивостi двох розглядуваних функцiональних рiвнянь та деякi спiввiдношення мiж ними. 2011 Article Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation / М.Н. Нооshmаnd // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 281–288. — Бібліогр.: 5 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166346 517.9 en Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Статті Статті |
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Статті Статті Нооshmаnd, M.H. Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation Український математичний журнал |
| description |
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x))=α(x)+1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm f-type function is its unique solution. We also show that the infralogarithm f-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential f-type functions and their functional equation β(x)=f(β(x−1)) which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them. |
| format |
Article |
| author |
Нооshmаnd, M.H. |
| author_facet |
Нооshmаnd, M.H. |
| author_sort |
Нооshmаnd, M.H. |
| title |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_short |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_full |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_fullStr |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_full_unstemmed |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_sort |
functions of ultraexponential and infralogarithm types and general solution of the abel functional equation |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166346 |
| citation_txt |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation / М.Н. Нооshmаnd // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 281–288. — Бібліогр.: 5 назв. — англ. |
| series |
Український математичний журнал |
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