The Pexider type generalization of the Minkowski inequality

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:393

Article

Matkowski, Janusz

关键词: Minkowski inequality;    Converse theorem for Minkowski inequality;    Pexider type of Minkowski inequality;    Measure space;    Convexity;    Geometrical convexity;   

DOI  :  10.1016/j.jmaa.2012.03.045

来源: Elsevier

【 摘 要 】

Let (Omega, Sigma, mu) be a measure space such that 0 < mu(A) < 1 < mu(B) < infinity for some A, B is an element of Sigma. The following converse Minkowski inequality theorem is proved in Matkowski (2008) [4]. If phi, psi, gamma : (0, infinity) -> (0, infinity) are bijective, phi is increasing, and phi(-i)(integral(Omega(x+y)) phi o (x+y)d mu) <= psi(-1) (integral(Omega(x)) psi o xd mu) + gamma(-1) (integral(Omega(y)) gamma o yd mu) for all nonnegative mu-integrable simple functions x, y : Omega -> R (where Omega(x) stands for the support of x), then there exists a real p >= 1 such that phi(t)/phi(1) = psi(t)/psi(1) = gamma(t)/gamma(1) = t(p). In the present paper we show that if, in the basic measure space, there is no A is an element of Sigma such that either 1 < mu(A) < infinity or 0 < mu(A) < 1, then there are some broad classes of non-power functions which satisfy the above Minkowski type inequality. Moreover we prove that, in the converse of the Minkowski inequality theorem, the assumption of the increasing monotonicity of phi is essential. (C) 2012 Elsevier Inc. All rights reserved.

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