A SUPPLEMENT TO THE PAPER

ON EXPONENTIAL REPRESENTATIONS OF

ANALYTIC FUNCTIONS IN THE UPPER HALF-PLANE

WITH POSITIVE IMAGINARY PART(1) BY"

N. ARONSZAJN AND W. F. DONOGHUE in Lawrence, Kansas, U. S. A.

In a paper which appeared a few years ago the authors investigated the

exponential representation of functions analytic in the upper half-plane with

positive imaginary part there [1]. We refer to that paper in the sequel as

A-D. One of the principal results of A-D, there called Theorem A, can be

extended to a considerably more general result, the proof of which is perhaps

simpler than that given in A-D. We give the extended version of Theorem A

here. We will use the notations and results of A-D without further explanations.

Before we present the extension and its proof we would like to add some

information that by oversight was omitted from the list of fundamental pro-

perties of the functions in the class P given in Section 1 of A-D.

In such a comprehensive review one should mention that the classical

theorem on representation of a positive harmonic function in a circle by a

Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of

L. H. Loomis [3] were not given:

XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O

and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0

common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r.

XVIII. I f for two values of 0 in the interval 0 < 0 < ~,

lim Im[~b(~ + re~~ = A < oo r ~ O

-~-4~) exists and equals A / n . then d2 ~

(1) Paper written under Research Contract Nonr 583(04) with the Office of Naval Research.

113

114 N. ARONSZAJN and W. F. DONOGHUE

These properties give rise to a circumstance which was not explicitly mention-

ed in Section 6 of A-D where the standard supports were introduced, namely

the inclusion S O = S ~

Two further remarks should be made, correcting A-D. Assertion IX in the

list of fundamental properties holds only under the additional hypothesis

/~[~] = 0, while the definition on page 376 of the relativized class 9J~a(h, k)

should require that the moment ] # ]a,h-2.~-2 be finite, and not that the moment

[/t [.,h,k be finite.

w For any pair of positive functions t/_(it) and t/+(it) defined for 2 > 0

we define the class 932(t/_,t/+) as the set of all positive Borel measures/1 on the

real axis for which

0 ; / I-t/-( - 2)]-1 d/~(2) + [t/+(2)]-~dp().) < oo. (1)

--oO 0

In a similar way we define class ffi(t/_, t/+) consisting of all Lebesgue mea-

surable functions f(x) with If(x) l ____ 1 for which the measure If(x) ldx belongs to 9~R(t/_, t/+).As in our paper, we make use of such classes as ~ - ( t / ) ,

consisting of all measures # in 9~R(t/, t/) which vanish on the right half-axis.

We are generally interested in the classes defined by comparison functions:

a positive function 11(2) defined for it > 0 is called a comparison function if it

is monotonic and Lipschitzian. Comparison fnnctions may be increasing or

decreasing. When t/(it) is a comparison function it may not be true that

(1 + it)t/(it) or (1 + it)_lt/(it) are comparison functions; we will, however,

make use of the classes of type 9J~ or ffi defined by such functions.

If k is a real number => 1, the function t/k(it ) = (1 + it)2 -kis a comparison

function; if t/h(it) is the function similarly defined with the help of h > 1,

then the class 9~R(t/h, r/k) appears as the class which we have called 9J~(h, k)

in A-D. We will continue to use that notation when the comparison functions

are so chosen.

Thus in the sequel, 9~R(1, 1) will denote the class definedby the comparison

functions t /_( i t )= t /+(it)= it + 1. Since every comparison function is

Lipschitizian, hence bounded by a linear function, it follows that every

9Jr(t/_, t/§ is a subset ofg~R(1, 1) when t/_ and t/+ are comparison functions.

ON EXPONENTIAL REPRESENTATIONS... 115

We can now describe our principal result. The major theorem of A-D

(Theorem A) is stated in terms of the classes ~FJ/(h, k) and (5(h, k) with h and

k __> 1; that is to say, the theorem is stated in terms of the classes defined

by the particular comparison functions 1//, and '/k introduced above. Here

we extend Theorem A to the classes defined by arbitrary comparison functions;

otherwise its content is unchanged. Furthermore we show that the theorem

cannot be extended to classes defined by monotonic and positive functions

q(2) which are not equivalent to comparison functions.

w L e m m a 1. Let q(2) be a comparison function and C its Lipschitz

constant. I f q(2) is non-increasing and 0 < K < C -x then

1 /7(2) =< _ ~ 1 / / ( 2 + Kr/(2)).

I f q(2) is non-decreasing then/7(2) < ~/(0) + CA; for every positive K

1 ,(~0 > = 1 + CKq(2 + Ktl(2 ) ) ,

and further, for 0 < K < C -1 and t > max [0,2 - K~/(2) ],

1 ~/(;t) < ~7(t).

= 1 - C K

Proof . The lemma follows immediately from the definition of the Lip-

schitz constant of /7(2).

If f (t) and g(t) are two positive functions defined for t ~ 0 we shall say

that f and g are equivalent if there exist positive constants m and M such

that m ~_f ( t ) /g ( t ) <= M for all t.

L e m m a 2.

by

I f ~/(2) is a comparison function, then the function ~/(2) defined

oO

1 f dt rl(~) = ( t - :.1~ -+ ,l(t) ~

is equivalent to q(2).

116

P r o o f .

constant.

N. ARONSZAJr~ and w. r. DONOGttUE

(a). Let ~/(~) be non-increasing and C its Lipschitz

~/(2) _ i ~/(2) dt i r/(2) dt ~(2) (t -- ,D 2 + ~(t) ~ -~ (t -- ,l) 2 + ~(;.)a = T

o ,I

~. A + 2 C 2 + 2 C

~/(2) _ + + < (t-- 2)2 + r/(gz) + ~(~) o 2 i i(~) o 2

g + - T ( -

q(2) dt

oo

f ~(2) dt n ~(2) arctan + ( t - , D - - - - - ~ - - < 2 + " ,1(;.)

A + - - 2 C

+2C.

1 Using Lemma 1 with K = .

2C~ we obtain finally

(b). If ~/(2) argue similarly:

is

t/(2) 3 < 2re + 2C. rt(~) -

non-decreasing and C its Lipschitz constant we

t / ( , t ) �9 '1"+ 2C

( t - D z + ~ 2 + -~)- ~/(2) . arctan I-. n(2) 2(~C)) 1

2 ~ - arc tan

Setting to = Max[O, 2 ~(2).] we also have 2C

O N E X P O N E N T I A L R E P R E S E N T A T I O N S . . . 117

t o ,1. t o t o

= + + =< ~/(2) (t:--2) 2 + ( t - - - ~ ~t/(to) 2 0 to ,l. 0 to

+

t o

f r/(2) dt < 3rr ( t _ ; t ) 2 + ~/(;t) 2 = -~- + 2C,

where we have also used Lemma 1 with K = 1/2C.

L e m m a 3. Let r/_(2) and r/+(2) be comparison functions and r a

function in P.

t o t o

r dIt(2) ( Im[r + i~/+(~)) ] d{ < oo then ~ = 0 and J ~/+(2) < 0o. (a) ff j n +(~) o o

0 0

: (b) / f Im[r + i r /_ ( -~) ) ]d~ < oo then ~ = 0 and ,t - ( - r ,1-( - 4)

- - t o - - cO

- - < 0 0 .

(c) Ifo~ = Oand It belongs to ~[l/(t/_, r/+) then It is in ~lJ/(1, 1) and

0 ~O

f Im[r + ir /_(-r ] de ' r Im[r + ir/+(~)) ]

- - t o 0

Moreover, the limits l i m r + ir/+(~)) and lim r162 + it/_( - r both ~-, + to r

exist and equal flto.

+ c o

Proof . (a). We may write Im[r f dit(~) '7+(r = ~ + ( , l - ~ ) 2 + n+(~)~

~ 0 0

and if this positive funtion is ~-integrable, it is clear that ot = 0. Also

: .d~ = dit(;t)

o o

CO o0 CO

f f ( ~ _ ~ ) 2 + ,7+(r .d~(a) = j 0-~)" o o '~

118 N. ARONSZAJN and W. F. D O N O G H U E

co

Since 7/+(2) is equivalent to q+(2) it follows that f d#(2) < m . It is evident

o

that the proof for (b) goes through in exactly the same manner. For the proof

of (c) we remark that p in 9J~(r/_, t/+) implies that p is in 9~(1, 1), a fact which we

have pointed out in the introduction as a consequence of the Lipschitzian

character of comparison functions. It is convenient to write the function

~b({) in the form

1 - - 1 cO

[ f " ] f.-, : . . . , ~ ( 0 = l~o+ U-T- + ~ -~ + J ~ - ~ ' - 1 - o o I

and to carry out the proof of (c) for each of the three terms separately.

1

It is obvious that the function q~l(() = fl~ + t ~ - is asymptotically

- 1

C Im[~l(r + iq+(O)] is equivalent to equivalent to floo + Z-~-, and hence that r/+(O

C and therefore is ~-integrable. It is also obvious that ~2 + ,7+(O2

lim (k1(r + it/+(O ) exists and equals fl~, and it is clear that the same argument ~ o o

may be employed when we consider t / _ ( - O instead of r/+(O. Thus (c) is

established for functions of the form (k,(O. Next we turn to

- 1

~ ( 0 = f d~O0 ~ - r - o o

For any function t/(g) we have

and hence

- I

Im[~b2(~ + io(O ) ] = f dp(2) ,1(0 ( a - 0 2 + ~(0 ~

m -I -I oc -I

f f "<" f f ( ; _ 0 2 + ,~+(r = ( ~ _ ~ - d , ( a ) = T / 0 - ~ - o o 0 - o c

ON EXPONENTIAL REPRESENTATIONS.. . 119

since ~l is in 932(1, 1). Moreover,

0 - 1 - I oc - 1

f f +r +:ff < ( ) _ _ ~ ) 2 + F ] _ ( _ _ ~ ) 2 ()+ "~ 4 ) 2 "Jr" q_(r idl~(A) = , J - O - - - - ~ - X ) m

-- 00-- oo --o0 0 --00

since F/_(-2) is equivalent to r /_( -2) and # is in 9-~(q_, r/+).

- 1

f d/~(2) converges to 0 as ~ appro- F i n a l l y , (]~2(~ "~ i ~ + ( ~ ) ) -~- )+-~-iq+(O --CO

aches + oo since 4h(O is regular in the right half-plane and vanishes at infinity.

dl~()t) If we~ write dr(2) for the measure ~ then v is a measure of finite total

mass and

- 1 - 1

~b2(~ + it/-(] ~1)) = f ( '~-~)r/-(- '~))z-dv(2)+i f q - ( I ~ I)t /-(-2)dv(~) �9

-- 30 -- O0

In both of these integrals, the integrands converge pointwise to O; we shall

show that the integrands are uniformly bounded as ~ approaches - m and

therefore will be able to infer from the Lebesgue convergence theorem that

the limit of ck2(~ + iq_( [ ~ ] ) ) exists and equals O. From the Lipschitz con-

dition satisfied by we infer that , / _ ( -2 ) < c1 -r +

2 and ~ being negative. It follows that the integrand in the first integral is

1 bounded by 1 -r + -< c + - . m similar estimate (;, _ r + . _ ( [ r [ )2 - 2

shows that the integrand of the second integral is uniformly bounded by

1 + C / 2 . oo

Lastly we remark that the argument for the function ( J ~ 3 ( ~ ) = 2 - - ~

1

exactly parallels the one just given, and the proof of the lemma is complete.

R e m a r k 1. Lemma 3 gives a precise relation between the behavior of

~b(() along a curve ~(~) = ~ + i~7(~) for ~ --* +_ m and the behavior of the

120 N . A R O N S Z A J N and W . F. D O N O G H U E

measure at infinity. It is easy to deduce from the lemma a corresponding

assertion relating the behavior of r along a curve approaching a finite point

4o on the real axis to the behavior of d# near that point. Part (c) of the lemma

can also be applied to functions which admit an integral representation,

similar to that of functions in P, but with signed measures; one assumes

dp] in 9J/(q_, r/~), and in the integrals one replaces Im[r by its absolute

value.

L e m m a 4. I f fl > O, the functions Im[fl + w'l and Im[log(fl + w)]

are equivalent in the intersection of a neighborhood of the origin and the

upper half-plane

Proof. If w = u + iv and v is positive we may write

Im[log(fl + w) ] 1 [arctan (~--~---) ] ( ~ - - ~ ) (1) Im[fl + w] - fl + u u "

Since (1/z) arctan z is analytic in a neighborhood of the origin and equals 1

at the origin, the ratio above converges to 1 / fl as w approaches 0 in the upper

half-plane.

L e m m a 5. For r in P , f ( x ) belongs to ffj(q _, q+) if and only i f ot = O,

la is in 9Yl(q_, q+) and floo > O.

P r o o f . If ct = 0 and # is in the asserted class with positive fl~ we infer from

Lemma 3 that

(2)

oo

f [q+(r ] - ~Im[r + iq+(~)) ]de 0

is finite. Since by that same Lemma r162 + iq+(O converges to floo as

increases, the function Im[r + iq+(~))] and Im[log(r + iq+(0)] are

equivalent for large ~ by Lemma 4. Accordingiy the integral

/

(3) _1 [~/+(~) ] - ~Im[log r + ir/q (~)) ] d~ 0

O N E X P O N E N T I A L R E P R E S E N T A T I O N S . . . 121

is finite, whence by Lemma 3 again

oo

(4) f [ q + ( r < oo. 0

We argue in exactly the same fashion using the comparison function q_(2)

to show the finiteness of

(5) J E,r-(I r 1) 3-Y(r de. 0

Thus f ( x ) is in (~i(q_, t/+).

On the other hand, i f f ( x ) is in the class ffi(r/_, r/+) it is surely in the class

ffi(1,1); hence the function log ~b(~) corresponds to the measure f ( x ) dx in

9~(1, 1). Since for functions of the type log~b(~) the coefficient cc in the canonical

representation is always 0, because of the boundedness of the imaginary part

of log ~, log ~(() converges to a finite real limit ao~ as ff converges to infinity

in angle. Since the integral (4) is finite, so also is (3) by Lemma 3, and

log~b(~ + iq + ( 0 ) converges to ~roo as ~ increases. Thus ~b(~ + iq + ( 0 ) approaches

a real limit, which must be positive since the limit of the logarithm is real.

Now Lemma 4 makes the integral (2)finite and Lemma 3 implies that for 2 > 0

it isin 9Jr + (r/+). The same argument carried out with q_(2) then puts # in

9T~(r/_,r/+) and shows that c~ = 0 and floo > 0.

As an immediate corollary of the preceeding lemma we obtain the following

generalization of Lemma 5 in A-D, w :(2)

L e m m a 5'. (a) Let c~" eP" and ~1+ be a comparison function; then

f " ~ + ( r / + ) i f and only i f #"~9J~+(r/+) and fl~ > O.

(b) Let (D' e P' and q_ be a comparison function; then f ' e f f j-(q_) i f and

only if #' E~R-(q_), ~' = 0 and fl'oo > O.

We now state our main theorem.

(2) We remind the reader of the notation adopted in A-D especially of the classes P' and P" (dcfincJ by f(x) = 0 for x > 0 or x < 0 respectively) and oftho facts that 1 ~ for 4 ' e P', d#'(3.) = 0 for ;. > O, and that 2 ~ for q~'" ~ P", d#"(2) = 0 for 2 < 0 and a = O.

122 N. ARONSZAJN and W. F. DONOGHUE

T h e o r e m A. Let c~(() be a function in P and in our standard notation p,

f , etc., the corresponding elements of the representations of dp. Let r/_(2)

and q+(2) be any two comparison functions; then

(a) f ( x ) - "A-(x) belongs to (~(q_, q+) if and only if o~ > 0 and It is in

~(2r/_, ;s/+).

(b) f ( x ) belongs to ~(rl_, rl+) if and only if ~ = 0 and # is in 9J~(rl_, q+)

and floo > O.

(c) f ( x ) - 1 belongs to (~(q_,rl+) i f and only if ct = O, It is in 9J'ffrl_,q+)

and fl~ < O.

( d ) f ( x ) - Z+(x) belongs to ffi(q_,q+) if and only if c~ = O, It is in

gJ~ --s -j~+ and [3oo = 0.

P r o o f . The proof is completely analogous to the proof of Theorem A

in A-D. Instead of the classes A(h, k), ..., O(h, k) and Al(h, k), ..., Dx(h,k)

introduced there we will have now A(q_,~/+), . . . ,D(r/_,q+) and

Al(q_,q+), ..., D~(q_, ~/+) respectively. The latter coincide with the former

when q_(2) = (1 + ,~)2-h and q+(2) = (1 + ).)2-k The main tools in the first part of the proof are: 1 ~ the factorization theorem

(Theorem I of A-D) and 2 ~ Lemma 5' which replaces Lemma 5 of A-D. We

notice that the case (b) is contained in our present Lemma 5.

The second part of the proof is exactly the same as in A-D.

R e m a r k 2. The simplicity of our present proofs is due essentially to the

replacement of the rather involved function-theoretic arguments in Theorem

III and Lemma 5 of A-D by the more elementary ones leading to the present

Lemma 5. The greater generality is achieved essentially by the use of the paths

+ r/• + ~), ~ ~ + oo in Lemma 3; they replace the more special tangential

paths of order z(a) of Theorem Ili, A-D.

R e m a r k 3. By using comparison functions one can also extend Theorem

of A-D.

(3) These correspond to the comparison functions rl(2) = (1 + ;t) 1-L

ON EXPONENTIAL REPRESENTATIONS... 123

w In the rest of the paper we investigate the necessity of the hypothesis

that a comparison function be Lipschitzian for theorems of the type of Theorem

A to hold.

It is obvious that if q_(2) and r/+(2) are positive monotone functions on

the positive real axis which are not Lipschitzian but which are equivalent

to comparison functions ~/'_(2) and r/+(2) respectively, then the class 9J/(q_,r/+)

coincides with the class 9J~(rl'_, ~/+). Thus in this trivial sense Theorem A

will hold for functions r/equivalent to comparison functions. We will show,

however, that no further improvement can be expected.

T h e o r e m C. Let r/(2) be a positive, monotone function on 2 > 0 with

the property that for every c~ in P corresponding to f in ~(1, 1), /t is in

ffJ~(r/, r/) if and only i f f is in ~(~l, ~7). Then if rl is monotone non-increasing,

it is equivalent to a comparison function, and i f rl is non-decreasing, the

function r/'(2) = Min[r/().), 2 + l] is equivalent to a comparison function.

R e m a r k 4. The reason for introducing ~1'(2)in the non-decreasing case

is that for r/(2) > 1 + 2 we have automatically f e (501, r/) and /t ~J~(r/, r/).

The proof of this theorem will consist in the construction of counter-

examples. For this construction, certain preliminary lemmas are

required.

We consider the following properties of positive monotone functions.

(6a) q(2) is non-increasing, and there exist positive constants K and M

with M=> 1 and r/(2) =< Mq(2+Kr/ (2 ) ) .

(6b) t/(2) is non-decreasing, and there exist positive constants K and M

with M < 1 and q(2)> mr/(2+Kr/(2)) .

L e m m a 6. A monotone and positive function q(2) on 2 > 0 is equivalent

to a comparison function if and only if there esixt positive constants K and M

and an infinite interval 2 > a > 0 such that q(2) satisfies (6)for 2 > a.

Proof . If r/(2) is a comparison function Lemma 1 shows that there exist

appropriate K and M such that (6) is satisfied. It is easy to see that any mono-

tone function equivalent to one satisfying (6) also satisfies that condition.

124 N . AKONSZAJN and W . F. DONOGHUE

with perhaps a different choice of the constants K and M which nevertheless

will satisfy the requirement M > 1 or M < 1 corresponding to cases a) and b).

Suppose now that ~/(4) satisfies (6) for 4 > a > 0. Define by induction:

4-1 = 0, 4o = a, 4k+ 1 = 4tr -t- Krl(4k) for k > 0. We have 4k~ O0 (otherwise,

in case a) 7(4) would vanish for a finite 2; in case b) the statement is obvious).

We construct a function 7*(4) as follows: q*(4k) = ~/(4k) for k > - 1 and,

in the interval 4k < 4 < 4k+1, k = - 1,0,1 . . . . . r/*(4) is extended by linearity.

Obviously ~/*(4) is monotone of the same type as 7(4). It is also clear that for

4k <=4 <=4k+l, k > - 1 ,

r/(4k+l)/r/(4k) < r/(4)/r/*(4) < tl(4k) / rl(4k+l) for non-increasing type,

r/(4k)/~/(4k+ 1) < ~/(4)/~/*(4) < r/(4k+ 1)/~/(4~) for non-decreasing type.

Thus, by using (6) we get for 4 > 0, M1-1< r/(4)/r/*(4) < MI, where

M1 = max(M, ~/(0)/r/(a) ) in case a) and = max(M - 1, rl(a)/~/(0) ) in case b).

Finally to prove that ~/*(4) is Lipschitzian we notice that in each interval

4 k < 4 < 4k+ ~ the Lipschitz constant oD/*(2) (by linearity) is ] t/(4k+ 1) -- r/(4k) I = 4 k + i - 4~

For k > 0, by using (6), we get a uniform bound for this constant in the form

n(4~) - n (4k+ ~)< - -

1 in case a) and n(4k+ 1) -- r/(4~) < K Kt/(4k) = MK

in case b).

We can now prove Theorem C by the construction of appropriate counter-

examples. We will suppose that the monotone function ~/(2) is not equivalent

to a comparison function, and shall exhibit a ~ (0 in P with f E (5(1, 1),

# e~l/(1, 1), ~ = 0, floo = 1, such that either p is in 9J/(r/, 7) a n d f not in tf(r/, 7)

or f is in (5(7, 7) but # not in 9J/(q, ~/). These functions will be of the form

4,(0 = 1 + #.

1 ~ with /L. > 0 and 2. > 1 increasing monotonically

to infinity. We will have ~ 2~-1p. < oo and therefore # will be in 9J/(1, 1). 1

The masses gn and the points 4. will be chosen inductively with the help of

the given function ~/(4). Let to. be the zeros of ~b(O; we have 4. < to. < 4.+1;

for our construction we will require

(7) / ~ . < o 9 . - 2 . < 2 # , , , for n > 1.

O N E X P O N E N T I A L R E P R E S E N T A T I O N S . . . 125

This can be guaranteed in the following fashion. We write

k

~ k ( 0 = 1 + ]~ #" 1 2 . = ~

and denote its zeros by o~ k), j = 1, 2 , . . . , k . If we require that

/~t < w~ k ) - ht < 2/tt for 1 < 1 < k, then, in the passage from Ok(0 to r

the term #k+X is added to Ck(0, and this term is positive for real ( < 2~+~. 2k% 1 - - ~

Thus o~k+X)< co[~)for I < k. On the other hand, Ak+ 1 being chosen so large

that ~k(O is sufficiently close to 1 in the neighborhood of 2k+ t, it is easy to see

.(k+l) 2k+ < 2/~k+1. that/~k+l <Wk+l - t If in addition ~k+1/~+1 is small

enough, the inequality /~l < CO~k + l ) -- 21 for 1 < I ~ k will be preserved.

Accordingly, the inductive procedure defining ~b(O must be such that at the

n-th step it is possible to choose/~.+x arbitrarily small and 2.+x arbitrarily

large. This will also permit the selection of /~. and 2. so that the series

/~./2,, converges, thereby putting # in ~lJ~(1, 1).

We suppose first that ~/(2) is non-increasing; since it is not equivalent to a

comparison function it converges to 0 with increasing 4, and for arbitrary

choice of K. and M. => 1 there exist infinitely many points 2 in any interval

of the form a < 2 < oo for which ~/(2) > M.q(2 + K.~/(2)). We choose

K. = 2-", M. = 2" and select 2.+~ so that this inequality is satisfied for

2 = 2.+~ and so that /a.+~ = 2K.~/(2.+0 is so small and 2.+, is so large

that the required inequalities for the zeros of r are satisfied. This is

clearly possible, and the function r can therefore be constructed. It is

clear that the corresponding measure /s has finite total mass, and in fact,

f d # ( 2 ) _ ~ #. _ #x - . - - x , 1 ( 2 . 1 (2,1

the other hand co.

f ;:----2 ,l + K - 1 ~ ( 2 . )

- - + ~ 2K._1 < 0% thus # is in ~0/(q,q). On n = 2

,TCx---j-

= . ( .k + I,;._ x.(2.))

co

> ~ Kn-l-~/~n-l= +o0, n----2

126 N . A R O N S Z A J N and W . F . D O N O G H U E

Hence f ( x ) is not in (5(tb ~/).

We have next to establish the counterexample for the case that r/(2) is non-

decreasing, and the function r/'(2) = Min [-t/(2). 2 + 1] is not equivalent to a

comparison function. Thus r/'(2) will be unbounded, and for any choice of

the positive numbers M, < 1 and K,, the inequality

(8) t/'(2) < Mnt/'(2 + Kn~/'(2 ))

holds for infinitely many arbitrarily large 2.

We will set K n = 2-", define Mn and In+ 1 inductively and put/a n = t/'(4n).

Suppose that q~n(~) is already defined. We find then a n > 2n and a positive en < 2 -n so that for any 4n+l > an and Un+ 1 < en4n+l, q~n+l(~) will satisfy

our requirements. We put then M n = e,,/~4 and choose 4n+ t > a n satisfying

(8)'. Since t/'(2) < 1 + 4, we get by (8)

r/'(4n+l) < --~--(1 + 2n+ 1 + 2-51'(4,,+t)),

n+l = n'(4n+l) < A+I, n'(4n+l) = n(in+l).

Hence ~bn+ l (0 does satisfy our requirements and thus the construction of ~b ( 0

Y is completed. We have d#(2) < #1 + < o% therefore #egX(1, 1); 4 = 2-1 =2 en

f d#(2) ~. /A _ ~1 ~, t/'(2n)

On the other h a n d , f is in (5(1, 1) by Theorem A, and we shall presently show

that it is in (501', ~/'); this will p u t f in (5(r/, t/) and complete the counterexample.

We have

a~ n cot 2 . + K . r/'(2~) 2 + 2 / t

I. . ~ l ~... t . . + K tlt(2n)

<

= / , ( ( 4 3

ON EXPONENTIAL REPRESENTATIONS... 127

REFERENCES

1. N. Aronszajn and W. F. Donoghue, "On exponential representations of analytic functions in the upper half-plane with positive imaginary part". Journ. d'Analyse Math., Vol. V (1956-57), pp. 321-388.

2. G. Herglotz, "~3ber Potenzreihen mit positivem, reellem Teil im Einheitskreis", Leipz. Ber. 63 (1911).

3. L. H. Loomis, "The converse of the Fatou theorem for positive harmonic functions". Trans. Amer. Math. Soc. Vol. 53 (1943) pp. 239-250.

DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF KANSAS LAWRENCE, KANSAS, U. S. A.

(Received August 6,1963).