On a conjecture about zeros

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On a conjecture about zeros

Citation for published version (APA):

Overdijk, D. A. (1983). On a conjecture about zeros. (Memorandum COSOR; Vol. 8304). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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Department of Mathematics and Computing Science

Memorandum COSOR 83 - 04

ON A CONJECTURE ABOUT ZEROIS

by

D.A. Overdijk

Eindhoven, the Netherlands March 1983

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ON A CONJECTURE ABOUT ZEROS

by

D.A. Overdijk

Introduction and summary

Consider the set W of meromorphic functions

A(z) N

I

j=1 A. J 2 (z-a. ) J such that N E 1N, A

j > 0 (j=1 ,2, ••. ,N) and al < a2 < ••• < aN. The (2N-2)

zeros of a function A E Ware denoted by zl,zl,z2,z2, ••• ,zN_I,zN_1 with

Define the subset peW

and 1m z. > 0 (j=1,2 , ••• ,N - I). J Re z. J m ~

I

j=1 a. J for m 1,2, •.•

,N-I} .

It is a conjecture of F.W. Steutel [IJ that P

=

W.

In this note we neither prove nor disprove this conjecture but report on the way we struggled with the problem.

In Section 1 we introduce a function ~ and give a formulation of the above

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In Section 2 an integral representation for ~ is given on which an algorithm

can be based that decides whether a function A E W belongs to the set P or

not. This algorithm avoids calculation of the zeros of A.

I. The function ~

In the first lemma we collect some simple properties of the zeros of functions

~n W.

Lemma 1.1. Every function A(z)

N

I

j=1 A. J E W satisfies (z-a.)2 J a l < Re zl < aN A. J A.a. J J := N

I

j=1 1,2, ••. ,N-I) • N

I

j=1 where 13 0 (j Re z. J N-I

I

j=1 b. a. c • Re z1 - a1 > 0 N-I

I

j=1 Re z. -J N-I

I

j =1 a. J

Proof.a. Let X

_o

+ iyO be a zero of A(z). We have

N

I

j=1 A. J N

_I

j=1 2 2 A. [(xO-a.) -

Yo ]

J J 2' 2 2 2 - ~yo [(xO-a_j ) + yO ] N

I

j=1 A.(xo-a.) J J

Since

Yo

#

0 we conclude that

N

I

j=1

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3 -N N 2

L

A. II (z-a. ) j=1 J i=l, ilj J b. A(z) N ₂ II (z-a. ) j=1 J N _2N-2 N N _2N-3

L

A. z - 2

_L

A.

_L

a. z +

...

j =1 J j=1 J i=I,ifj ~ N 2 II (z-a.) j =1 J Hence N N

N-I _j=I

L

A.J _i=I,i~j

L

a.~ N

L

Re z.

_L

a.

-

8₀

.

j =1 J N j =1 J

L

A.

j=I J

c. Immediately follows from a. and b.

o

Definition 1.1. For every function A E Wwe define the function W lR+lR

L

Rez.<8 J (Re z. - 13) -J _a.<f3

L

J (a.-f3), J (13 E lR) •

Some obvious properties of the function Ware collected in the following lemma.

Lemma 1.2. Every function A E Whas the properties

a. ~ is continuous.

b. ~ ~s linear between consecutive points of the set

{Re zl ' Re z2 , .•• , Re zN_1 ,a I ,a2,··· ,aN} .

c. At the points Rez. (j=I,2, •.. ,N-I) the derivative ~'

J

has a jump of -I.

At the points a. (j=I,2, •.. ,N) the derivative ~' has a jump

J

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< 0

w'

(S) 0 ~

#

{j

I

Re z. < S} > 0 J > <

#

{j

I

a. < S} • J Proposition 1.1. A function A(z) N

L

j=1 A. _---=-J_-::- E W (z-a.) 2 J

belongs to the set W \ P iff there exists an integer 3 :s; nO :s; N - I such that

Proof. Suppose A E W \ P. Then we have to show that there exists an integer

such that w(a

n -I) < 0:

o

w(a) < O. We may

3 :s; nO :s; N - 1 such that w(a I) < 0 and w(a ) < O. I t is sufficient to show

n O- nO

that there exists one number an -I E {a

2,a3,··· ,aN_I}

o

Suppose a number a E {a

2,a3, ••. ,aN_1} exists such that

assume that nO is the smallest index such that 1jJ(a_{n -I)}

0

easily from Lemma 1.2. that also w(a ) < O.

nO Define the set

< O. Then it follows k

I

j=1 Re z. < J k

I

j=1 Since A E W \ P we have K

#

0 .

Let k

O := max K then in view of Lemma 1.1.c we have 2 :s; kO:s; N - 2 and because

k

O is maximal in K

Define the set M

Re zk +1

o

> ak

o

+1 • M :=

{m

E K

I

m

L

j=1 Re z. J m <

I

j=1 a. and Re z J m

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5

-First suppose M

_0.

Then we conclude that Re z < a

k and

k

O 0

k

O kO

1jJ(ak +1)

L

Re z. _{- koak +1}

-

_L

_{a. + kOa}

k +1 < 0

.

0 j== 1 J 0 j == 1 J 0

Now suppose M

#

0

and define m

O :== min M. Since Re zm O we have m -1

o

L

j==1 Re z. < J m -I

o

L

j ==1 a. J

and we conclude that Re z 1

m O -< a • We now have mO-1 m

_o

-I

L

j=1 Re z. - (mO-I) a -J m O m -I

o

L

j=1 a. + J (mo-I )am_O < 0 •

Conversely, suppose that 1jJ(a. ) < 0 (i

OE{I,2, ••• ,N-I} then we have to show

~O that A E W \ P. Since N-l

L

j == 1 Re z. -J N-I

L

j=1 a. J (lennna 1.1.6)

and 1jJ is continuous we may consider the smallest zero 13] of 1jJ larger than a . • We have a. < 13

1 < aN'

~O ~O

Let nO and m

Obe the largest indices such that Re zn

o

< 13

1 and a_mo < 131,

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Furthermore

o

Re z. J - n

o

S -

1 Re z. -J a. + J m

o

Since

I

j=no+1 (SI - a j) > 0 we conclude that Re z.J - a. < O. J Hence A E: W \ P .

o

Because of Proposition 1.1 we can give an equivalent formulation of the

conjecture of F.W. Steute1: For all A E: W we have ~ is nonnegative.

2. An integral representation for the function ~

Proposition 2.1. For every function

we have A(z) N

I

j=1 A. J 2 (z-a. ) J E: W co S - So ₁

J

1

[~A(S+it)Lld

~(S) ₂ ₂_'IT og N

J

t, 0

I

A. j=1 J N

I

A.a. j=1 J J So := _N

I

A. j =1 J (S E: JR.) •

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Proof. For J 7 -1,2, ••• ,N-l put z. = r. + iw .• J J J since we have A(z) N

I

A. j=1 J N

I

A. j=1 J N-l II _ j=1 - N 11 j=1 (z-z . ) (z-i' . ) J J 2 (z-a.) J N 11 j =1

From an elementary calculation we get

I

10g[~~(S+it)

\] dt

I

A. j=1 J N-l

I

j=1

~

t log { 2 «S-r. ) J 2 2 + (t-w.) )«S-r.) J J 2 2 2 «S-a.) + t ) J {

(S~r.

}2+ (t+w.)2 } +

~

wj log ---",,:.I'--:::2--...:::.J---"-2 (S-r.) + (t-w.) J J { t-w. t+w. } t ]

+ \S-r·1 arctan ~ + arctan ~ - 2 I s-a. I arctan ~::::-r J IS-r~1 Is-r~1 J IS-aJ~·1

J J

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Hence

00

J

log

[£J

_NA(I3+i t)

L

ld

_J

_t = 'IT N-I

I

_{IS-r./ -} 'IT N

I

IS--a.

I

.

j=l J j=l J 0

I

_A. j=1 J Using N-I

I

j=1 r. J

completes the proof.

We now can formulate a stronger version of Proposition 1.1.

N A.

Proposition 2.2. A function A(z)

L

J 2 belongs to the set W \ P iff

j=1 (z-a.)

J

there exists an integer 3 :S nO :S N - 1 such that

o

i

ii 1jJ(a ) < 0 •

nO

Proof. Since 0 <_-

£J

A(I3+it)_N

L

I

A.

j=1 J

< 1 it follows from Proposition 2.1. that

1jJ(S) > 0 if IJ

On the basis of Proposition 2.2. an algorithm can be constructed which

decides whether or not a function A(z) E W belongs to the set ? This

algorithm avoids the calculation of zeros of the function A(z).

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9 -Example 2.1. A(z) _ _1--;:- + I + I (z+3)2 (z+2)2 (z+I)2 2 3 +-2+--~2+ z (z-6) 3 2 • (z-IO) 1jJ(-2) 0.4 1jJ(-l) 0.8 1jJ(0) 1.2

Because of Proposition 2.2. we conclude A E W.

Example 2.2. A(z) 0.5

o.

I O. I 3 + 2 + 2 + 2 + 2 (z-3) (z-4) (z-5) (z-12) 4 O. I 6 10 + ₂ + ₂ + ₂ + ₂ (z-20) (z-25) (z-40) (z-90) 13 0 52.98 1jJ(4)

₌

0.8 1jJ(5) 0.9 1jJ (12) 0.9 1jJ (20) 4.2 1jJ(25 ) 9.2 1jJ(40)

=

13.5 •

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[IJ F.W. Steutel, Infinite divisibility of mixtures of gamma distributions, Colloquia mathematica societatis Janos Bolyai, 21 (1977), 345-357.