Geodesic
The Index Function Operator for O-regularly Varying Functions
Kragujevac Journal of Mathematics, Tome 47 (2023) no. 7, p. 1040 .

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The paper examines the functional transformation $K$ of the class $ORV_{\varphi}$ (see \cite{ex3}) into the class of positive functions on interval $(0,+ \infty)$ defined as follows: \begin{equation} K(f)=k_f, abel{eq:1} \end{equation} where \begin{equation*} k_f (ambda)= imsup_{x o +ıfty}\frac{f( ambda x)}{f(x)}, \quad ambda ı (0, + ıfty), \end{equation*} and $f \in ORV_{\varphi}$. Let $f \in IRV_{\varphi}$ or $SO_{\varphi}$ (see \cite{ex4}), $K$ be the transformation \eqref{eq:1} and for any $n \in \mathbb{N}$, $K_n(f)= \underbrace{ K(K\cdots(K}_\text{n}(f))\cdots),$ then the function $p(s)=\lim_{ n \to +\infty}K_n(f)(s)$, $s>0$, is $IRV_{\varphi}$ (and continuous) and $SO_{\varphi}$, respectively.
DOI : 10.46793/KgJMat2307.1041DJ
Classification : 26A12
Keywords: O-regularly varying function, index function 
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Dragan Djurčić; Danica Fatić; Nebojša Elez. The Index Function Operator for O-regularly Varying Functions. Kragujevac Journal of Mathematics, Tome 47 (2023) no. 7, p. 1040 . doi : 10.46793/KgJMat2307.1041DJ. http://geodesic.mathdoc.fr/articles/10.46793/KgJMat2307.1041DJ/

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