a proof of bernstein's theorem on regularly monotonic functions

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a short proof of Bernstein's theorem on the analyticity of such functions. ... theorem [l] on "regu- ... By Rolle's theorem there exists a if on (t, x) such that. (4).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, February

1975

A PROOF OF BERNSTEIN'S THEOREM ON REGULARLY MONOTONICFUNCTIONS J. A. M. McHUGH ABSTRACT. class

(j

order

analyticity

This

paper

of such

a short

tially

case

proof)

theorem,

tion of class

C

if it is of

may depend

of Bernstein's

of Bernstein's

on the

theorem

That

we recall

that

on a real interval

subject

the reader

cited there.

theorem.

theorem

on this

we refer

the proof we present

Bernstein's

(which

proof

For background

original

of Bernstein's

motivated

a short

proof

paper by Boas 12], and the references

special

monotonie"

sign

functions.

functions.

of Bernstein's

"regularly

is of a fixed

We present

presents

monotonie"

tation

is called

derivative

of the derivative).

on the

larly

A function

and each

(and a presen-

to the brief

survey

The book [3] contains

proof for that

here.

[l] on "regu-

Before

a regularly

giving

monotonie

special

a proof of a

case

our proof function

(a, b) for which each derivative

par-

of is a func-

is of fixed sign

on (a, b). Theorem.

// F(x)

is regularly

monotonie

on (a, b), then

F(x)

is ana-

lytic on (a, b). Proof. the even

We assume

for convenience

part of F(x),

f(x),

for the odd part of F(x),

begin

by noting

following

that

a = - b < 0.

at x = 0.

and the analyticity

f(x) is itself

lemma is strategic

Lemma.

that

is analytic

of F(x)

regularly

We will prove

An analogous

thereby

monotonie

proof

follows.

on 0 < x < b.

that holds

We

The

for our proof.

// f{n)(x) < 0, f(n+l)(x) > 0, and /(n+2)W > 0, then for x £[0, b),

\R (x)\ > \R +,(x)|,

where

R (x) is the

mth remainder

in the Taylor

expansion

of f about x = 0. Proof

of Lemma.

Received

Write the remainder

by the editors

November

as

5, 1973-

AMS (MOS)subject classifications (1970). Primary 26A48, 26A90, 26A93; Secondary 26A24, 26A51, 26A84, 26A86Key words

and phrases.

C-infinity

functions,

analyticity,

regularly

monotonie

functions. Copyright © 1975. American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

358

A PROOF OF BERNSTEIN'S THEOREM

(1)

359

RM = —-— fV" >(;)(* - t)n-ldt. "

A sufficient

(n -

condition

1)! J 0

for the Lemma

(2)

to be true is that

-f0 n

for t on (0, x).

Since

ficient

for (2) to be true;

condition

/("+I)(r)

> 0, replacing

~

m by 1 in (2) yields

a suf-

namely

-f("Xt)-(x-t)f(n+1)(t)>o, or, if we write

g for /

,

(3)

-g(t)-(x-t)g'(t)>0.

By Rolle's

theorem

there

(4)

a if on (t, x) such that

-g(f)-(x-f)g'(0

Since

g" = /(n+2)>0, If instead

triples

(analytic

For, in this case,

is a polynomial. triples

or odd.

If /(n)

one has either

of the sign

is immediate.

Only for

+, -, + occur

is even,

/(n+1)U)

(perhaps

for /'*' after

odd.

/-»-/),

/ is even.

/("+1)(0)

> 0 for x > 0.

< 0 for x > 0. Thus /("+1)(x)

proof holds

attainable

when

say, then

> 0 for x > 0. Thus f(n+l)(x)

An analogous

is always

conclusion

can the triple

is even

/("+2)(x)

But also by supposition,

of the Lemma

the Lemma's

functions)

/(n)

0. By supposition,

= -g(x)>O.

(3) follows from (4) (Q.E.D. Lemma).

of the sign sequence

+, +, + or +, +, -,

polynomials

sign

exists

Since

= 0 =» / one of these

it follows

that

for

/ even

(5)

|Rn(x)| > |RB+1M|

for « = 0, 1, 2, ••• ,

for x on [O. b). Frequently = (-1)"

/"

and /"+1

==>/ is polynomial).

(6) then

are both > 0 (or both < 0) (else eventually Rewriting

r (*).,—£— "

for, say,

f*$

(»-«!

(1) as

Cf^Kxùd-ù^-^dt,

J 0

and /"¿ + U > 0 and

x > 0, one has

n ■

(7)

Q