Geodesic
On the Class Gamma and Related Classes of Functions
Publications de l'Institut Mathématique, _N_S_93 (2013) no. 107, p. 1

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The gamma class $\Gamma_{\alpha}(g)$ consists of positive and measurable functions that satisfy $f(x+yg(x))/f(x)\to\exp(\alpha y)$. In most cases the auxiliary function $g$ is Beurling varying and self-neglecting, i.e., $g(x)/x\to0$ and $g\in\Gamma_0(g)$. Taking $h=\log f$, we find that $h\in E\Gamma_{\alpha}(g,1)$, where $E\Gamma_{\alpha}(g,a)$ is the class of positive and measurable functions that satisfy $(f(x+yg(x))-f(x))/a(x)olpha y$. In this paper we discuss local uniform convergence for functions in the classes $\Gamma_{\alpha}(g)$ and $E\Gamma_{\alpha}(g,a)$. From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class $\Gamma_{\alpha}(g)$ and related classes. Two applications are given.
Classification : 26A12 33B99 39B22 34D05
Keywords: Beurling variation, the class gamma, local uniform convergence, remainder terms, differential equations, growth of functions 
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     title = {On the {Class} {Gamma} and {Related} {Classes} of {Functions}},
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Edward Omey. On the Class Gamma and Related Classes of Functions. Publications de l'Institut Mathématique, _N_S_93 (2013) no. 107, p. 1 . http://geodesic.mathdoc.fr/item/PIM_2013_N_S_93_107_a0/