On a generalization of a function of J. Sándor

Abstract

Using a strictly increasing function $\alpha: \left [ 1,\infty \right )\rightarrow \left [ 1,\infty \right ),$ we define below (see(1.1) and (1.2)) two functions $S_{\alpha}:\left [ 1,\infty \right )\rightarrow \mathbb{N}$ and $S_{\alpha}^*:\left [ 1,\infty \right )\rightarrow \mathbb{N}$, where $\mathbb{N}$ is the set of all natural numbers. The functions $S_{\alpha}$ and $S_{\alpha}^*$ respectively generalize the functions $S$ and $S_{*}$ introduced and studied by J. Sándor [5] as well as the functions $G$ and $G_{*}$ considered by N. Anitha [1]. In this paper we obtain several properties of $S_{\alpha}$ and $S_{\alpha}^*$ - some of which give the results of Sándor [5] and of Anitha [1] as special cases.