Inequality curve and inequality index based on the ratios between lower and upper arithmetic means

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$L$-functions, processes, and statistics in measuring economic inequality and actuarial risks (2009, methodology): “This is the index that Zenga [94] introduced, and which since then has been called Zenga’s (new) index of economic inequality.”
Approximation and decomposition of Gini, Pietra–Ricci and Theil inequality measures (2012, background): “…index based on the ratios between lower and upper arithmetic means, and its group decomposition obtained by Radaelli, are worth mentioning (Zenga 2007; Radaelli 2008), amongst the most meaningful applications and discussions of group decomposition, at least the research by Quintano et…”
On the Decomposition by Subgroups of the Gini Index and Zenga's Uniformity and Inequality Indexes (2010, methodology): “In order to measure the point inequality between these two groups, Zenga (2007) proposes to consider the ratio of the correspondent arithmetic means − Mi = Qi Ni = 1 Ni i∑ t=1 xt nt · i = 1, . . . , r (31) + Mi = ⎧⎨ ⎩ T − Qi N − Ni for i = 1 . . . , r − 1, x∗r+1 for i = r , where x∗r+1 xr may be…”
Zenga's New Index of Economic Inequality, Its Estimation, and an Analysis of Incomes in Italy (2010, unknown): “For mathematical and historical details on the Vervaat process, see Zitikis (1998), Greselin et al.”