## New results on slowly varying functions in the Zygmund sense.(English)Zbl 1443.26002

A positive and measurable $$g$$ is self-neglecting (notation: $$g \in SN$$) if it satisfies $\frac{g(x+yg(x))}{g(x)} \to 1, \forall y \in \mathbb{R}$ and locally uniformly in $$y$$.
A positive and measurable function $$a$$ is in the class $$\Gamma _{\alpha} (g)$$ if $$g \in SN$$ and if $\frac{a(x+yg(x))}{a(x)} \to e ^{\alpha y}, \forall y \in \mathbb{R}.$ A positive and measurable function $$U(x)$$ is in the class $$Z(g,a)$$ if it satisfies: $\lim _{x \to \infty} \frac{a(x)}{g(x)}\left(\frac{U(x+yg(x))}{U(x)}-1\right)=0, \forall y \in \mathbb{R}.$ In this paper, the authors extend Senata’s characterization of slowly varying functions $$L$$ in the Zygmund sense by considering a wider class of functions and a more general condition than for each $$y > 0$$, $x\left(\frac{L(x+y)}{L(x)}-1 \right) \to 0, \,\,\, x \to \infty.$ Specifically, the authors prove the following theorems.
Theorem 1. Assume that $$g \in SN$$, $$a \in \Gamma _{0}(g)$$ and $$g(x)=o(a(x))$$. Let $$y >0$$. The inequalities, for $$x$$ large and $$\epsilon >0$$, $U(x+yg(x)) \times \exp \epsilon \int _{x^{o}}^{x+yg(x)} a^{-1}(t)dt \geq U(x) \times \epsilon \int _{x^{o}}^{x} a^{-1}(t)dt$ and $U(x+yg(x)) \times \exp -\epsilon \int _{x^{o}}^{x+yg(x)} a^{-1}(t)dt \leq U(x) \times -\epsilon \int _{x^{o}}^{x} a^{-1}(t)dt$ holds if and only if $$U \in Z(g,a)$$.
Theorem 2. Assume that $$g \in SN$$, $$a \in \Gamma _{0}(g)$$, $$g(x)=o(a(x))$$ and $$g(x) >0$$. Assume that $$U \in Z(g,a)$$. Then, for some function $$f$$ and for $$x \geq A$$ for some $$A >0$$, $U(x)=\exp c + \int_{A}^{x} f(t)dt$ where $$a(x)f(x) \to 0$$.

### MSC:

 26A12 Rate of growth of functions, orders of infinity, slowly varying functions 28A10 Real- or complex-valued set functions 45M05 Asymptotics of solutions to integral equations 60G70 Extreme value theory; extremal stochastic processes
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### References:

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