On Weighted Extensions of Bajraktarević Means

Abstract

For real functions $f,g,\alpha,\beta$ defined in an interval $I$ we introduce a mean $B_{\alpha,\beta}^{[f,g]}.$ which extends the Bajraktarevi\'{c} mean $B^{[f,g]}$ in $I$. The problem of symmetry of $B_{\alpha,\beta}^{[f,g]}$, leading to a functional with two unknown functions, is solved. We show that, under some conditions, every Bajraktarevi\'{c} mean $B^{[f,g]}$ in $(0,\infty)$ can be embedded in a two-parameter family of means $\big\{ B_{a,b}^{[f,g]}:a,b>0\big\}.$ As a special case a new family of means $\big\{ B_{t}^{[p,q]}:t>0\big\},$ which can be treated as the weighted Gini means, is constructed. As an application, the pairs of the these means which leave the geometric mean invariant are indicated, the effective limits of the sequence of iterates of the relevant mean-type mappings are given, as well as some functional equations are solved.

2000 Mathematics Subject Classification. Primary: 26E30, 39B22