Minkowski-type inequalities for means generated by two functions and a measure

• Autores: László Losonczi, Zsolt Páles
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 78, Fasc. 3-4, 2011, págs. 743-753
• Idioma: inglés
• DOI: 10.5486/pmd.2011.5017
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• Resumen

  • Given two continuous functions f; g : I ! R such that g is positive and f=g is strictly monotone, and a probability measure  on the Borel subsets of [0; 1], the two variable mean Mf;g; : I2 ! I is de ned by Mf;g;(x; y) := ( f g )..1 (�ç �ç 1 0 f ( tx + (1 .. t)y ) d(t) �ç �ç 1 0 g ( tx + (1 .. t)y ) d(t) ) (x; y 2 I):

    The aim of this paper is to study Minkowski-type inequalities for these means, i.e., to nd conditions for the generating functions f0; g0 : I0 ! R, f1; g1 : I1 ! R, . . . , fn; gn : In ! R, and for the measure  such that Mf0;g0;(x1 +
    
    
    + xn; y1 +
    
    
    + yn)  [] Mf1;g1;(x1; y1) +
    
    
    +Mfn;gn;(xn; yn) holds for all x1; y1 2 I1, . . . , xn; yn 2 In with x1 +
    
    
    + xn; y1 +
    
    
    + yn 2 I0.

    The particular case when the generating functions are power functions, i.e., when the means are generalized Gini means is also investigated.