Local and global comparison of nonsymmetric generalized Bajraktarević means

Abstract

The purpose of this paper is to investigate the local and global comparison of two n-variable nonsymmetric generalized Bajraktarević means, i.e., to establish necessary as well as sufficient conditions in terms of the unknown functions f,g,p1,…,pn,q1,…,qn:I→R for the comparison inequalityf−1(p1(x1)f(x1)+⋯+pn(xn)f(xn)p1(x1)+⋯+pn(xn))≤g−1(q1(x1)g(x1)+⋯+qn(xn)g(xn)q1(x1)+⋯+qn(xn)) in local and global sense. Here I is a nonempty open real interval, x1,…,xn∈I, and f,g are assumed to be continuous, strictly monotone and p1,…,pn,q1,…,qn:I→R+ are positive valued. Concerning the global comparison problem, the main result of the paper states that if f,g are differentiable functions with nonvanishing first derivatives and, for all i∈{1,…,n},pip0=qiq0andp0(x)(f(x)−f(y))p0(y)f′(y)≤q0(x)(g(x)−g(y))q0(y)g′(y)(x,y∈I) are satisfied (where p0:=p1+…+pn and q0:=q1+…+qn), then the above comparison inequality holds for all x1,…,xn∈I.