ACTA issues

Minkowski's inequality for two variable Gini means

László Losonczi, Zsolt Páles

Acta Sci. Math. (Szeged) 62:3-4(1997), 413-425

Abstract. We study Minkowski's inequality $$S_{ab}(x_1+x_2, y_1+y_2)\le S_{ab}(x_1, y_1)+S_{ab}(x_2,y_2) \quad(x_1,x_2, y_1,y_2\in{\msbm R}_+)$$ and its reverse where $S_{ab}$ is the two variable Gini mean defined by $$S_{ab}(x,y)=\cases{ \left(\frac{x^a+y^a}{x^b+y^b}\right)^{\frac1{a-b}} & if a-b\ne0,\cr\exp\left(\frac{x^a\ln x+y^a\ln y}{x^a+y^a}\right) & if a-b=0. }$$ Generalizing results of Beckenbach~[\B], Dresher~[\Dr], Danskin~[\Da], Losonczi~[\Ltwo], we give necessary and sufficient conditions (concerning the parameters $a,b$) for the inequality above to hold. For the reverse inequality we give necessary conditions (which are not sufficient), sufficient conditions (which are not necessary) and also a conjecture concerning the necessary and sufficient condition.

AMS Subject Classification (1991): 26D15, 26D07

Keyword(s): Gini means, Minkowski's inequality

Received March 22, 1996. (Registered under 6125/2009.)