G) = i. - American Mathematical Society

proceedings of the american mathematical

society

Volume 76, Number 1, August 1979

NEAR-DERIVATIONSAND INFORMATION FUNCTIONS1 JOHN LAWRENCE2, GEOFF MESS AND FRANK ZORZITTO3 Abstract. Near-derivations y satisfy the conditions y(xy) = xy(y) + yy(x) for x,y £ R, y(x + y) > y(x) + y(y) for xy > 0, y(x) = 0 for x £ Q. Existence of near-derivations other than derivations is tied in with that of nonnegative information functions and an example of Daróczy and Maksa. Conditions for near-derivations to be derivations are discussed.

1. Introduction. A near-derivation is a self-mapping y of the reals R such

that: y(xy) = xy(y) + yy(x)

y(x +y)>

y(x) + y(y)

for all x,y in R,

(1)

for all x,y > 0,

(2)

y(x) = 0 for all rational x.

(3)

The question immediately arises as to whether there are any near-derivations that are not already derivations. An information function is a mapping/: [0, 1] -» R satisfying the equation

f(x) + (1 - x)f(y/ (1 - *)) = f(y) + (1 - y)f(x/ (1 - y)\ and the normalizing conditions

/(0)=/(l) = 0, The most important function s given by

/G) = i.

example of an information

s(x) = —x log x — (1 - jc)log(l — x)

function is the Shannon for 0 < x < 1.

(Here and throughout we adopt the conventions that log = log2 and 0 log 0 = 0.) The motivation for the above definition of an information function is developed by Aczél and Daróczy in [1], where such functions are studied extensively as part of an analysis of measures of information. The present research was motivated by a problem cited in [1]. Namely, is the Shannon function j the only nonnegative information function? This was recently solved by Daróczy and Maksa [2]. They showed that for any Received by the editors October 2, 1978. AMS (MOS) subject classifications(1970). Primary 39A15; Secondary 94A15. 'This research was carried out partly in the NRC Summer Research workshops held at the University of Waterloo in 1978. 2Research supported by NRC grant A4540 and University of Waterloo research grant

131-7052. 'Research

supported by NRC grant A4655 and University of Waterloo research grant

131-7067. © 1979 American Mathematical

0002-9939/79/0000-0373/S02.SO

117 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Society

118

JOHN LAWRENCE, GEOFF MESS AND FRANK ZORZITTO

derivation 5 on 7?, the formula fix)

= s(x) + 8(x)2/x(\

- x)

provides a nonnegative information function. In our paper we show that/is nonnegative information function if and only if

a

fix) = s(x) - y(x) - y(\ - x), where y is a near-derivation. It follows from this that s is the minimal nonnegative information function. The latter fact is also proved in [2] by a different approach. In addition the Daróczy-Maksa example leads to examples of near-derivations which are not derivations. We conclude with some properties of near-derivations.

2. Connection of near-derivations to information functions. A theorem in [1] leads to the connection between information functions and near-derivations.

Representation

Theorem (Aczél-Daróczy).

A function f: [0, 1] -» R is

an information function if and only if there is a self-mapping 8 of R such that

f(x) = - 8(x) - 8(1 ~ x) for all x in [0, 1], 8(xy) = x8(y) + y8(x)

for all x,y in R,

5(2) = 2.

(4) (5)

(6)

In addition it can be readily checked that / is nonnegative if and only if the representing function 8 satisfies

8(x + y) > 8(x) + 8(y)

for all x,y > 0.

Our first result is a refinement of the representation tive/. Theorem

(7)

theorem for nonnega-

1. A function f: [0, 1] —^/? is a nonnegative information function if

and only if fix) = s(x) - y(x) - y(l - x),

(8)

where y is a near-derivation.

Proof. Let y be a near-derivation. Let a: R -+ R be defined by a(x) = x log|jc|, and let 8 = o + y. Clearly a satisfies conditions (5)-(7). Since y is a near-derivation, 8 satisfies (5)-(7) also. By the representation theorem, the formula fix) = — 8(x) — 8(1 — x), 0 < x < 1, provides a nonnegative information function/. Conversely, let / be a nonnegative information function. By the representation theorem let 8: R -> R satisfy (4)-(7) for/. Let a(x) = x log|x| as before, and let y = 8 — o. We shall prove y is a near-derivation. It will then follow

that fix) = - (o + y)(x) - (a + Y)(l - *) - s(x) - y(x) - Y(l - x) as desired.

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NEAR-DERIVATIONS AND INFORMATION FUNCTIONS

119

Now, property (1) of near-derivations holds for y because 8 and a have it. Property (3) for y follows from [3, Theorem 2], where it is shown that 8 = a on the rationals. Thus property (2) remains, namely that

8(x + y) - (x + y)\og(x + y)

> (8(x) - x log x) + (8(y) - y logy) for all x,y > 0. This inequality will follow from the subsequent lemmas. Lemma 2. Suppose 8: R—> R satisfies conditions (5)-(7). Then

8(1 + x) > (1 + x)log(l + x) + 8(x) - x log x for any x > 0. Proof. For any positive integer « and x > 0 we apply (5) and (7) to get

«(1 + x)"-x8(l + x) - 8((l + x)") = §( îo{"ky)

> îoS(("k)xk).

Thus

In [3] it is proved that 8(r) = r log r for any positive rational r. Hence

S(l+x)>

l

Î (nk)kxk-x8(x)

«(1 + x)

+

k=0^K/

1

£**GWï)

n(\ + x)"~x k~o

xk

= 8(X)+i±ï ± (D—^—u « k%\k)(l hJcy (1 + X?

= 8(x) + an~

bn + c„,

where

^ =^Jo(^7rfVlog(^) (l + x) cn= -4^ 2 (Dtt^ « k=0XK/ (1 +

x)

=xlog;c'

M(i + *)")= 0 + ^)iog(i+ x).

Since


(9)

120

JOHN LAWRENCE, GEOFF MESS AND FRANK ZORZITTO

it follows from [1, Theorem 1.3.7] that 1 + x,

Kl < -!og("

,

,.

+ 0-

Thus a„ —>0 as « —»oo. We therefore get in the limit

0(1 + x) > 8(x) - x log x + (1 + x)log(l + x). Lemma 3. For 8 satisfying (5)-Çl) and any x,y > 0, inequality (9) must hold.

Proof. By using property (5) and Lemma 2 we get

8(x+y)

= 8(x(l +y/x)) >x((l+

= x8(l + y/x)

y/x)\og(l

- (y/x)\og(y/x) Use of (5) to expand 8(y /x) latter expression to

+ y/x))

+ (1 + y/x)8(x)

+ 8(y/x)

+ (1 +y/x)8(x).

and some logarithmic properties reduces the

(x + >-)log(x + y) + 8(x) - x log x + 8(y) - y logy, whence (9) follows. Theorem 4. Iff is any nonnegative information function and s is the Shannon

function, then f > s. Proof.

Let 8 represent / according to the representation

theorem. By

Lemma 3, inequality (9) holds for 8. With 0 < x < 1 and y = 1 - x, (9) specializes to

ô(x) + 8(1 - x) < x log x + (1 - x)log(l - x),

which says that/ Theorem

> s.

5. Not all near-derivations

must be derivations.

Proof. Formula (8) yields an example of such near-derivations nonnegative information function/different from s.

for every

3. Some properties of near-derivations. We are interested in conditions for the additivity of near-derivations. For example, it can be seen from [3] that if F, K are fields of real numbers and K is quadratic over F and y is a near-derivation vanishing on F, then y is additive on K. By Theorem 5 this need not happen if K is not algebraic over F. If, for a transcendental x, a near-derivation y vanishing on F were linear on the field F(x), then y would be constant on the coset x + F. We are therefore led to examine y restricted to x + F. Theorem 6. Let F be a real field, x a real number and y a near-derivation vanishing on F. Then for a,b in F: (a)0 0, yx(x + a) increases to a finite value ßx(x) as a —>+ oo through the rationals. By the definition of y, it follows that ßx(x) = 0. Thus yx(x) < 0 for all x > 0. However, one can check that any function satisfying (1) and (3) must have a dense graph in R 2 or else vanish everywhere. We must conclude

that y, = 0, or ß = y. A corollary of Theorem 7 seems worth noting. Theorem 8. A near-derivation y is a derivation if and only if, for all x, a(y(x — a) — y(x + a)) —>0 as a -» + oo through the rationals. Proof. By formula (10), ß(x2) = 2xß(x). This makes ß a derivation, and hence y.

References 1. J. Aczél and Z. Daróczy, On measures of information and their characterizations,

Academic

Press, New York, 1975. 2. Z. Daróczy and Gy. Maksa, Nonnegative information functions (to appear). 3. J. Lawrence, The uniqueness of the nonnegative information function on algebraic extensions, (to appear).

Department of Pure Mathematics, Canada N2L 3G1

University

of Waterloo,


Waterloo,

Ontario,