On Mulholland's Inequality (original) (raw)

On Minkowski's inequality and its application

Journal of Inequalities and Applications, 2011

In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes.

Alternative Proof of Mulholland's Theorem and New Solutions to Mulholland Inequality

2013 IEEE 43rd International Symposium on Multiple-Valued Logic, 2013

Mulholland inequality is a real functional inequality presented in 1950 in a paper by Mulholland as a generalization of the Minkowski inequality. In his paper, Mulholland has also provided a sufficient condition for the inequality to be satisfied. However, until now, it has remained an open problem whether this sufficient condition is also necessary. This paper investigates a geometric interpretation of Mulholland inequality and offers a class of functions satisfying the inequality which is strictly larger compared to the class delimited by the Mulholland's condition. Thus, it is proven that the condition is not necessary.

The Converse of the Minkowski's Inequality Theorem and its Generalization

Proceedings of the American Mathematical Society, 1990

Let (£2, X, p.) be a measure space with two sets A, B el. such that 0 < p(A) < 1 < p(B) < oo , and let ?>:R+ -+ R+ be bijective and (¡> continuous at 0. We prove that if for all //-integrable step functions JCy:fi->R, </>~X (I <po\x+y\dp) <y~X (j <po\x\dp) +<p~* ( tpo\y\dpAj then tp(t) = <p{l)tp for some p > 1 . In the case of normalized measure we prove a generalization of Minkowski's inequality theorem. The suitable results for the reversed inequality are also presented.

On a generalization of mulholland’s inequality

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1993

Let (f2, E, #) be a measure space such that #(A) = 0 or #(A) > 1 for every A c E. We prove among others that if q~ is a convex homeomorphism of [0, oo) such that the function log o~b o exp is convex in R then q~-l(~olx_.}_yld,u) < -q~-l(f~q~olx, d~ ) q_~-l(~q~o,yld/t ) for all nonnegative/~-integrable step functions x, y:f~ ---, R.

The converse theorem for Minkowski's inequality

Indagationes Mathematicae, 2004

Let (s2, ~, #) be a measure space and ~, ~ : (0, co) ~ (0, ec) some bijective functions. Suppose that the functional P~,~ defined on class of #-integrable simple functions x : S? --, [0, ec), #({w : x(w) > 0} > 0, by the formula satisfies the triangle inequality. We prove that if there are A, B C C such that 0 < #(A) < 1 </~(B) < ee, the function ~ o ~p is superadditive, and limit0 ~(t) = 0 then there is ap > 1 such that

On a characterization of LsppL\sp pLspp-norm and a converse of Minkowski's inequality

Hiroshima Mathematical Journal

Let C be a cone in a linear space. Under some weak regularity conditions we show that every subadditive function p: C -> R such that p(rx) < rp(x) for some re(0,1) and all xeC must be positively homogenous. As an application we obtain a new characterization of ZΛnorm. This permits us to prove among other things the following converse of Minkowski's inequality.