Continuously differentiable means
- PDF / 258,441 Bytes
- 15 Pages / 467.717 x 680.315 pts Page_size
- 44 Downloads / 253 Views
We consider continuously differentiable means, say C 1 -means. As for quasi-arithmetic means Q f (x1 ,...,xn ), we need an assumption that f has no stationary points so that Q f might be continuously differentiable. Introducing quasi-weights for C 1 -means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of C 1 -means, we observe that a skew power mean Mt is a composition of power means if t is an integer. Copyright © 2006 Jun Ichi Fujii et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let M(x1 ,...,xn ) be a continuously differentiable n-variable positive function on (0, ∞)n . Then, throughout this paper, M is called a continuously differentiable mean, or shortly C 1 -mean if M satisfies (i) M is monotone increasing in each term; (ii) M(a,...,a) = a for all positive numbers a. A mean M is called homogeneous if M satisfies
M ax1 ,...,axn = aM x1 ,...,xn
(1.1)
for all a,xk > 0. Almost all classical means are homogeneous C 1 -ones. The Kubo-Ando (operator) means in [6] and chaotic ones in [2] are C 1 -means. Here note that (numerical) Kubo-Ando means K f (a,b) are defined by
K f (a,b) = a f
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 75941, Pages 1–15 DOI 10.1155/JIA/2006/75941
b a
(1.2)
2
Continuously differentiable means
for positive operator monotone functions f , which form a special class of numerical means. Let f be a continuously differentiable monotone function on (0, ∞) with no stationary points, that is, f (x) = 0 for all x > 0. In this case, f −1 is also continuouslydifferentiable. Let w = {wk } be a weight, that is, a set of nonnegative numbers wk with k wk = 1. For such f and a weight w, it follows that a quasi-arithmetic mean Q f ,w defined by
Q f ,w x1 ,...,xn = f −1
n
wk f xk
(1.3)
k =1
is a typical C 1 -mean. As we will see later in the next section, the assumption that f has no stationary points is necessary for continuous differentiability. Our main interest in this paper is when integral functions
ᏹ f ,P x1 ,...,xn = f −1
∞ 0
f (x)dPx1 ,...,xn (x)
(1.4)
are C 1 -means, where Px1 ,...,xn is a probability measure on (0, ∞) for each xk . Note that these functions differ from the continuous quasi-arithmetic means, cf. [4, 5], but they include the above discrete quasi-arithmetic ones Q f ,w . In fact, for a convex combination n for Dirac measures Px1 ,...,xn = k=1 wk δxk , we have M f ,P = Q f ,w . In this paper, we discuss continuous differentiability of such integral functions as means, and observe when ᏹ f ,P is a C 1 -mean, particularly as 2-variable functions. Many mathematicians have been interested in means of positive numbers. But, even in a quasiarithmetic mean, odd properties appear as we will see in some examples later. We noticed that the
Data Loading...
