ACTA issues

Minkowski-type inequalities for two variable Stolarsky means

Péter Czinder, Zsolt Páles

Acta Sci. Math. (Szeged) 69:1-2(2003), 27-47

Abstract. We study the following Minkowski-type inequality \[D_{a_0,b_0}(x_1+y_1, x_2+y_2)\le D_{a_1,b_1}(x_1, x_2)+D_{a_2,b_2}(y_1,y_2)\qquad(x_1,x_2, y_1,y_2\in{\msbm R}_+)\] and also its reverse for the two variable Stolarsky (or difference) means that are defined (in the case $ab(a-b)(x-y)\ne0$) by \[D_{a,b}(x,y)=\left(\frac{x^a-y^a}{a}\frac{b}{x^b-y^b}\right)^{\frac{1}{a-b}},\] for $a,b\in{\msbm R}$, $x,y\in{\msbm R}_+$. The results obtained extend that of the paper [14] by Losonczi and the second author concerning the case $a_0=a_1=a_2$, $b_0=b_1=b_2$.

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

Keyword(s): Minkowski inequality, two variable homogeneous means, Stolarsky means, Minkowski separator

Received December 27, 2001. (Registered under 2882/2009.)