ON THE CLASS GAMMA AND
RELATED CLASSES OF FUNCTIONS

Edward Omey

Abstract: 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 $(f(x+yg(x))-f(x))/a(x)\to\alpha y$. 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.

Keywords: Beurling variation; the class gamma; local uniform convergence; remainder terms; differential equations; growth of functions

Classification (MSC2000): 26A12, 33B99, 39B22, 34D05

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