Dependence on the Parameter of Generalized Grotzsch Ring and Generalized Hubner Functions

For a is an element of(0, 1/2] and r is an element of (0, 1), let mu(a)(r) and m(a)(r) be the generalized Grotzsch ring function and generalized Hubner function, respectively. In this paper, the authors mainly study the dependence on the parameter a of mu(a)(r) and m(a)(r), and present several properties of mu(a)(r) and m(a)(r), as functions of a is an element of (0, 1/2] for arbitrarily given r is an element of(0, 1). The obtained results improve and extend some previously well-known results.