Publ. Math. Debrecen 78/3-4 (2011), 743–753 DOI: 10.5486/PMD.2011.5017 Minkowski-type inequalities for means generated by two functions and a measure By L ´ ASZL ´ O LOSONCZI (Debrecen) and ZSOLT P ´ ALES (Debrecen) Abstract. Given two continuous functions f,g : I → R such that g is positive and f/g is strictly monotone, and a probability measure μ on the Borel subsets of [0, 1], the two variable mean M f,g;μ : I 2 → I is defined by M f,g;μ (x, y) :=  f g  -1      1 0 f ( tx + (1 - t)y ) dμ(t)  1 0 g ( tx + (1 - t)y ) dμ(t)     (x, y ∈ I ). The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to find conditions for the generating functions f 0 ,g 0 : I 0 → R, f 1 ,g 1 : I 1 → R, ..., fn,gn : In → R, and for the measure μ such that M f 0 ,g 0 ;μ (x 1 + ··· + x n ,y 1 + ··· + y n ) ≤ [≥] M f 1 ,g 1 ;μ (x 1 ,y 1 )+ ··· + M f n ,g n ;μ (x n ,y n ) holds for all x1,y1 ∈ I1, ..., xn,yn ∈ In with x1 + ··· + xn,y1 + ··· + yn ∈ I0. The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated. Mathematics Subject Classification: Primary: 39B12, 39B22. Key words and phrases: generalized Cauchy means, equality and homogeneity problem. This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK81402 and by the T ´ AMOP 4.2.1./B-09/1/KONV-2010-0007 project implemented through the New Hungary Development Plan co-financed by the European Social Fund, and the Euro- pean Regional Development Fund.