Typical real functions in the exterior of the unit circle

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PEACE MATEM AT Y CZNE X V II (1974)

S. W

axczak

(Lôdz)

Typical real functions in the exterior of the unit circle

Contents

Introduction...513

Definitions ...514

Chapt er I. Properties of the Class

H r

... 514

1 . The definition of the family

H r .

Relationships between

I I r , Tr , Z® .

. 514 2. Structural and variational fo rm u la s... 515

Chapt er II. Estimation of the modulus of the d erivatives... Properties of the family

H r

...517

1. Estimation of the coefficients...517

2. Estimations of the moduli of the function and the d e riv a tiv e s... 519

3. The properties of the class

H r

: compactness and connectedness . . . . 520

Chapt er III. Boundary Functions with respect to the functionals defined in the class

R r

...521

1. D e fin itio n s... 521

2. The function

F *

( 3 ) and its properties ... 522

3. The extremal functions with respect to real functionals... 527

Chapt er IV. Typical real, symmetrical fu n c t io n s ... 527

1. Definition. The structural formula. The extremal problem s... 527

2. Estimation of the coefficients. Boundary and extremal functions . . . 529

3. The sets of values of some functionals defined in the family

H r

. . . . 530

References...531 Introduction. W. Bogozinski in 1932 [5] introduced the class Tr of typical real functions defined in the circle \z\ <

1

. He has solved in this class several extremal problems. One may define independently of the family Tr the class Hr of typical real functions defined in the exterior of the unit circle.

In this paper we deduce the structural and the variation formulas in the classes Hr and Щ . Basing on these formulas we solve some extremal problems. The upper and the lower bounds of the coefficients, the moduli of the function and the derivative have been found effectively.

In chapter IV we are concerned with typical real symmetrical

functions.

D e f i n i t i o n s .

E = { 3 : l3l > 1 } , Tr — the class of regular functions of the form

f{z) = z + a 2z

2

-f ... + anzn + ...

defined in \z\ <

1

which take real values if and only if z is real. E°r — the class of regular and univalent functions

. . . CO CL,

Е{ъ) = 3+ —- + ••• + З Г + - - -

6 0

defined in

E , F ( 3

) Ф

0

,

F (

00

)

= 00 and an =

a n

for

n

= 1 ,

2

, . . . , M — the class of real non-decreasing functions a(t), te ( — тг, тс] which satisfy the

T C ^

condition fd a (t) = 1 , Hr — the class of functions of the form

■*43) = with a(t)e M ,

3

e E .

3 + e ~U

^ — # da(i)

Chapter I. PROPERTIES OF THE CLASS H r

1 . T h e d e f i n i t i o n o f t h e c l a s s

Hr .

R e l a t i o n s h i p s b e t w e e n t h e c l a s s e s

Hr , Tr , E l. A function of the form

(

1

) -^(

3

) = 3 + aoH--- H ••• H — •••

3 3

holomorphic and different from zero in E is called typical real in the exterior of the unit circle if it has real values if and only if

3

is real.

The family of these functions has been denoted by Hr . It follows from the above definition that im

3

imjF(

3

) >

0

for

3 4 3

and that ak = ak, к =

0

,

1

,

2

, ...

Now we shall prove

T

heorem 1

. I f a function f(z )e Tr , then F(%) — [/(1/з)]_1е Hr . P ro o f. It immediately follows from the definition of the class Tr that the function F (

3

) is real if and only if

3

is real. W. Bogosinski has proved that every function f(z) e Tr may be expressed in the form

7Г

■(2) /(*)= f 1 _J

₁

_—

_{2 0}

_{COS/ +} ,

₀₂

^2da{t),

— TV

where a(t)e M.

The function

g{z, t) = z

1

—

2

^cosif+

^2

is positive for

0

< z <

1

and te ( — n, тс] and is negative for —

1

< z <

0

and te ( — те, тс].

B ec au se of a(t)e M th e fu n ctio n d efined b y fo rm ula (2) is d ifferen t from zero for z —

z

a n d z

Ф

0.

For z

Ф z

we have imz imf(z) > 0, thus f(z)

Ф

0 for z

Ф

0.

T herefore th e fu n ctio n F{%) is r e g u la r in th e set E. I t h as a sim ple pole a t th e p o in t

³

=

^{0 0}

.

T

heorem

2. I f F{^)€Hr , then f(z) = [F (l{ z )Y l a Tr .

The proof of th is th eo rem follow s im m e d ia te ly from th e d efin itio n s of th e classes H r a n d Tr .

T

heorem

3. The fam ily 27° is a subclass of the fam ily IIr i. e. 27° cr R r . P ro o f. Let F{3)e27°. The function F(%) is regular and different from zero in E and it is real for

3

=

3

. Thus it suffices to prove that m

3

im F(

3

)

> 0 for

3 Ф 3

. Suppose that contrary is the case, i. e. there exists a point

3

o such that im

30

> 0 and im F(

3

) < 0 or ^Ш

^{1 3}

< 0 and im F(

3

) > 0.

For

3

->

0 0

on the semistraight line гез = 0 and Ш

1 3

> 0, im_F(

3

) ->

0 0

. Thus in the upper semiplane there exists a point зх such that im F(

3

1) >

0

, Therefore there exists in the interior of the segment

3

— 3

0

+ ^(

3

i —

3

o)-

0

< t <

1

, a point

3

a such that W 2 = F ( з2) is a real number. Simulta­

neously the point W2 is the immage of the point з

3

= F~] (W 2) which lies on the real axis. This contradicts the univalence of the function F (

3

).

The argument is analogous if im

3

i <

0

and im F(

31

) >

0

. We have proved that im

3

im F(

3

)

> 0

for

3

^

3

. Thus the function F (

3

) belongs to the family Hr .

2 . T h e s t r u c t u r a l a n d t h e v a r i a t i o n f o r m u l a s .

Making use of the struc­

tural formula deduced by Gelfer [2] one may easily prove that every func­

tion of the class Hr may be represented in the form (3)

H — Roczniki PTM — Prace Matematyczne ХУП.

where a(t)e M and conversely that every function of form (3) belongs to E r . Let

F(b) = ³⁺⁶

3 - 6

— it

— it

da(t) be an arbitrary function of the class E r .

Putting

3

=

1 / 0

we obtain

(4) m eu + z

ei l - z

e i l -\-z

e~u —

0

da(t).

We denote by G the set of functions of form (4) with a(t)e M. In this class one may apply the variational formula of Goluzin (comp. [3], p. 606).

Employing this formula we find that together with the function f(z) the functions

<2

(5) h (z) -/ (« ) + A /

with Ae ( —

1

,

1

); G, — тс], a(t)e M, с = lima(<), с = lim a(0, belong

to the class G. tMt

If a(t) is a step function, the function

( 6 ) h i * ) *=/(*)+ ^ ^e**1 _e

_{" 1}

_—

^{+ 0} ₀ ¹_{1 — 0}

^”f"

⁰

_{tfh _ ^} ^e

^,72

⁺

⁰

e -a* + z\l

e~i{z —

0

) J ^{+ о(Д),}

where tj, G are discontinuity points of the function a(t) and A is a real sufficiently small number, also belongs to the class G.

Putting in equalities (5) and (

6

)

0

- =

1 / 3

we get the variation formulas in the class E r .

(7) ^л(З) = ^ (

3

) + A 2це{ 2i%e

,it\2

(3 —

6

) {b_ e-ü )2 l\a (t) - C\dt’

(

8

) Ел(з )= Е (з ) + Д Г/ 3 + ^ i

Из-

3 +

6

ih

3

— б

_г<1

3 +б

г'<2

3 — S 5 ) ] + « i .

variation formulas in the class E r .

It follows from equality (3) and from the definition of the class Hr that every function of the family E r may be represented in the form

(») ^ (

3

) = i l - i # (

3

),

23

where F (^ )e E r .

Substituting in (9) the function F x(%) defined by formula (7) for F{%) we hare

(

1 0

) I ' M = F ( Z) + W 1(b),

where

<2

Г ieü ie~ü

1

V (3) = / 9À i,t ) W ) - o \ d t , g ,(3,0 = (3

2

- i ) [

( 3 _ ea)2

- (

8

_ e-« ).J- If a(t) is a step function we obtain another variational formula is the class Hr from (

8

) and (9)

(11) -^л(З) = F(%)-{-ÀU2(%)-\-

o

(

â

), where

„ , , , , ,

^з

2- 1

г з + « й

, 3 + e -“ i p«(3) = g(5,< i)-ff(3, h ) ,9 ( 3, t] Л- + - — >

Яe (e, e),

^e

>

0

and is sufficiently small, t2 are the discontinuity points.

Chapter II. ESTIMATION OF THE MODULUS OF THE DERIVATIVES PROPERTIES OF THE FAMILY Hr

1 . E s t i m a t i o n o f t h e c o e f f i c i e n t s .

We shall prove

00

T

heorem

4. I f F(%) = 3 + J- belongs to Hr , then n

= 0

3

—

2

^ a

0

^

2

, 3 < ax < 1

— 4 max sinwtfsmi < an < —4 min sinwisintf.

tfe(—те, те] t e ( —те, те]

The above estimations are sharp.

P ro o f. It is known that F(%) may be represented in the form 32- i r h + eit , з + е -г7\ л

•F(3) = oW I ( -— УГ+-:— y^ H «(z).

23 3 - e 3 - e

Expanding the under-integral function into the series of Maclaurin we have

02 it —2г<

fj

1

) - +

¹

>

F(%) = /Г8 + (вв -в - " ) + (в2<|

— 7T L

OO - J

+ ^ ( { e i{n+1)t + e~i{n+1)t) - (e ^ -1)*+ e-<(n"

1

)<))g-n da{t).

n =±2 J

Intergrating the above series we get (

1 2

) a 0 =

2

J costda(t),

— Я

(12') cq = J (

2

cos

2

Z — l ) d a ( t ),

— 71 Я

(12") an — — 4 J sinnt-smtda(t), n = 2 ,3 ,...

—-Я

From the equality

J da(t) =

1

and (

1 2

), (

1 2

'), (

1 2

") the following estimations of the coefficients result (13) - 2 < a

0

< 2, - 3 < cq < 1 and

— 4 max sinwZsinZ < an < —4 min sinnt sin/, n =

2

,

3

, ...

<e(—7i,n] te(— гг, Tt]

The above estimations are sharp.

The equalities a 0 = ± 2, cq = — 3, cq = 1 are satisfied by the functions

^ (3 ) (32 ± 1 ) 2

---f

3

-^

2

(

3

) 32- i ( d ± i г - t \

2 3

\ 3-+ b + i !

Л ( 3 )

(3 + 1 ) 2 --- J

3 respectively.

The function <p(t) — sinwZsinZ attains in the interval ( — и, тс] its bounds. Suppose that it attains its greatest value at the point Z^ and that its smallest value is attained at the point Z™ . By what has been said above the equality an = — 4 min sin nt sin Z shall be realized by the function

■*43) = 32- l

2 3

3 + f Ъ-е*

3+ 6

^—it

da(t)

³

2- l

/ 3

+ e

23 З -б

+ 3 +

6

-«I

a(Z) where

23

_{— Я}

0

for

Ze ( — 7C,

C ],

1

for Ze (/" , Tü] .

Accepting for

^a(t)

the function

we obtain

°(t) =

⁰

1

for

foi* te [t^j, 7г],

F ( 3) = 3+ e

M

This function realizes the equality an — (

1 2

") implies immediately that

3+ e

^—it

;in M

—it^,;П M

4 max sin nt sin t.

te(—n, re]

Inequality

\an\ ^ n — 2 , 3 , . . . This estimation is sharp for n = 2& + 1 because

|sin(

2

ü-|-l)tfsint| =

1

for t = тг/

2

.

Another proof of this theorem has been given by Zamorski [

6

].

2 . E s t i m a t i o n o f t h e m o d u l i o f t h e f u n c t i o n a n d o f t h e d e r i v a t i v e s .

Basing on inequalities (13) we shall prove T

heorem

5. I f F (з)е Hr , then

l ^ ( 3 ) K l 3 l + 2

l ^ (

3

) K i

and

\F{k){%)I < Ш -

(I

3

^{I -}

1

^{)2 ’}

^

1

з

1 + 1

(

1

з

1

—l

)fc+2

’ Tc = 2, 3, P ro o f. Making use of (13) we have

l-P(

5

)l < I

3

l + У K l

1

з Г “ < 131+2 +

4

У |зГ” =

131

+

2

+ —— — ,

« = П « . = 1

131

| - P ' ( 3 ) | < l + 4 ÿ » | 3 | - ! ” + 1' = l +

И =

1 з !- 1 ) 2 ’

l ^ t4,(

3

) l < i > ( « + i ) - - - ( « + f c - i ) K I

1

^зГ <“+1)> » = 2 , з , . . . , 1

п = 1

|^*>(3)|<4fc!

(!з 1 - !);

к+± j k —

2

, 3, ...

The above estimations are not sharp. The sharp estimations may be found from formula (3). We have

ПЛ r dk [

1

b

<1 4 >

_{— TC}

J ^ [ 3 + “ + 7 + 7 ^ i with a = ea + e~u,

6 = 1

— e2lt, c =

1

— e~2it.

From equality (

2 2

) it follows that

c

бГг<- da(t),

l^(3)l < max

te(—71, it]

1

3

+

2

cosJ + —---

¹

- в " 2й е~и -Ъ 1

\ F 'm < max te(— n,Tc]

1 1

---- ~ +

3

2

1

- e2it

₁ 1

— e~2it (е~и - Ъ ? Г

<e(— max тг.тт]

( -

1

)" . Tcl( l - e 2it) h \ { l- e - 2it) -Jfe

+1 1

* (еи- ъ )к+1 + {e~u - i ) k+l

3 . P r o p e r t i e s o f t h e c l a s s 7 / r : c o m p a c t n e s s a n d c o n n e c t e d n e s s .

In this section we shall prove two theorems. They are of basic importance in the study of extremal problems in the class R r .

We shall prove the following

T

heorem

6 . The fam ily Hr is compact.

P ro o f. Consider an arbitrary sequence {-Рп(з)} of functions belonging to the family Hr . By Theorem 2 we have [,Fn( l /г

)]- 1

=/„(»), where f n(z)e Tr . The family Tr is compact. Thus from the sequence {/„(«)} a sub­

sequence {fnjc{z)} may be chosen which converges to a function f 0{z) e Tr . The subsequence F n (1 /z) = 1 lfnk(z) converges to the function F 0(l/z)

= l/ fQ(z). Put z =

1

/

3

. By Theorem 1 we conclude that the function F 0(

3) =/<Г1(

1

^/з)

belongs to Hr . Thus the family Hr is compact.

A family В is called connected if for arbitrary functions /

1

(

2

), /

2

(г) of this family there exists a class W cz R of functions f(z, t), ze E ,te [a , b]

which satisfy the following conditions :

1

° f(z, t) is close to uniformly continuous in F with respect to t,

2

° f(z, t) converges close to uniformly to f x(z) as t-+ a, 3° f(z, t) converges close to uniformly to f 2(z) as t -> b.

T

heorem

7. The fam ily Hr is connected.

P ro o f. L et F x(

3

) and F z(

3

) be arbitrary functions of the family H r . The class W mentioned in the definition shall be defined as follows:

(15) F (b t) =

( l ~ t ) F 1

( t ~ l ) F 2 1 — t

t - 1

for

0

< t <

1

,

for t — 1 ,

for

1

< t <

2

.

For every > e [0, 2 ] the function

F (% ,

t) belongs to

H r .

We estimate the difference

h = \F(

3

, tx) F (

3

, ^

2

)I L ? ^

2

e [

0

?

1

]•

For an arbitrary

3

, |з| >

q

> 1 we have

h =

|(1

tx) F x{z, tx)

( 1

t2) F x(z, t2)I

•where B (

q

) = é(n-\-l)Q n.

n = 0

From the estimation h < \tt —12\ - B (

q

) it follows that the functions F (

3

, t) are close to uniformly continuous in E.

Putting respectively t2 = 0 and t2 =

1

we find that F (

3

, tf) converges 4ose to uniformly to F x(

3

) as i ->

0

and ^ (

3

, i) converges close to uniformly to

3

as t ->■

1

.

In a similar way we may prove that

\F{%, t i ) —F(fo h)\ < \h — 4\’ B(Q) for [

1

,

2

].

Hence it follows, that F(%,t) converges close to uniformly to

3

as t tends to

1

and that it converges close to uniformly to F 2(

3

) as t tends to

2

.

We have proved that the class W defined by equality (15) satisfies conditions 1-3. Thus the family Hr is compact.

Chapter III. BOUNDARY FUNCTIONS WITH RESPECT TO THE FUNCTIONALS DEFINED IN THE CLASS IIr 1

1 . D e f i n i t i o n s .

Let the function

(16)* 0 ( F ) --

be given and differentiable

\F\<A,

(16') \F'\< B,

|F(fc) 1 < <7, 1 where

A = 13o

1+ 2

+ Ш

° ~ (13.1-1У

l3ol—1 ’ В = 1 +

(!3 o l-D

²

>

* + l » 3o

— is a fixed point of the set E.

(И) Ф(Щъ о)) = 0 ( F ( t o) ,F ( 3о) ,..., *#»>(з0), & n)(boï)

defined in the family Hr . з

0

is a fixed point of the set E. Denote by A the set of values of the functional Ф(Е). It follows from Theorems

6

and 7 that A is a closed set. Denote by Г the boundary of region A. It can be proved, that (comp. [4]) for every point of Ф 4 A there exists а Ф0е Г such that for every Фе A and Ф sufficiently close Ф

0

the inequality

(18) |Ф

0

-Ф |<|Ф -Ф |

is satisfied. The point Ф

0

is called a regular point of the boundary Г. Denote the set of these points by Г '. One can prove (comp. [4]) that Г ' is a dense set in Г. A function F ($ )e H r is called a boundary function with respect to a functional Ф{Е) if Ф(Т(з))еГ. A boundary function which satisfies the condition Ф{Е(%0))е Г ', shall be denoted by F*(%). The functions F(%) and a(t) which correspond to the function F * (

3

) are denoted by F*(%) and a* (t).

2. The function F*(%) and its properties. Preserving denotations (16) and (17) we accept

0Ф - ^дФ

a* + /?â p r ' for h = 0 >1 > "■>“ > m = 1 - Let

9

o(t), te ( — тс, тс] be a function defined by the formula

П

(19) <p(t) = 2 ' a*9?)(3»> + *)•

*=o We shall prove

L

emma

1 . The function <p(t) satisfies the equation h

(

2 0

) f <p{t)\a(t)-e\dt = 0,

h where tx, t2e ( —тг, тс].

P ro o f. Employing the first variation formula (10) we get Ф [-P*(3o) + , F (3 » ) + А»,, • • •, ^ ( 3 o ) + M n|]

n

= 0 [ i" ( 3 , ) . . . ? W ( 3 1) ] + i +<&<>) + <> ( A

k= 0

дФ дФ

Vk = J W ’ lk = â p r - Hence АФ — Я ( P k ^ + °(1)'-

k= 0

Consider the functional

Inequality (18) can be written in the form

| Ф - Ф о 1 2 + | Ф о - Ф | 2

+

2

ге(Ф -Ф

0

)(Ф

0

-Ф ) > |Ф

0

-Ф|2- Because of Ф -Ф

0

= ^Ф we have

Are/? • JT {PkuW + qku{k)) + ° ( * ) > 0, k

= 0

where /? = в targ(0°

Since A can have any value from the interval ( — 1 ,1 ) we find

re/? + =

0

.

ft = 0

By some transformations we have

П П

P 2 (PkU? + <iMc)) + p £ (ркй(к) + qku[k)) =

0

,

fc

= 0 к—0

or

X K ^ + a r f ) =

0

.

&=o

By (10) and by the theorem on the sum of integrals we obtain

<9 П

f [< * k ffi* 4 fo , t ) + <*к91Л)(Зо> = 0 .

1

1

&=o

Taking into account (19) we get the assertion of the lemma.

L

emma

2. I f at every point w = {F, F , F^n\ F (n)) of the region D defined by (16) and for every /?1? |/?| =

1

the condition

t ■ Л дФ - дФ

(

2 1

) >, \ak\ >

0

, where ak = P —^ r + £ -= = -, Jfc =

0

,

1

, »,

&=o

is satisfied, then the function <p(t) has no more than 4w +

8

foots m the interval ( —

7

t, тс], if go 4= 3

oj

n°l wore than 2n + 4, if д

0

= з0.

P ro o f. It can be easily proved that

(

22

⁾

№ e A io ,t) d/'

w "(3S-

1

) te-e (3 S -l) — г<

/_2 (e“ - 3 o ) 2 ( « - “ - S o ) 2

kU a Wk(eil) k l i r “ W,.{e~il) («“ —3.)-

^{* + 2} \ & + 2

3o)

for Jc = 0, for & =

1

,

2

, ..

with 17fc(^) = г[(/с-1)е2гЧ2зег*-(& + !)].

After substituting the above equalities into (19) we get (23) <p(t)

~- l ) ^~г

7

(Зо~

1 ) 1

- r - i e - « f â - l ) tew( g -

1

)

Зо)2 ^{(е~и - Ъ ?} J L '*"-*-'*

йЖй(в<4) е^Ж^е")

Г ^ ^(Зо

4 (eü - n Г elt

ak k ! —jT-

* L(^ ■

+ 2 _7c

₌₁

- 3 o

)A+2

(«"* 3o)

^{fe-f- 2}

Consider the function

2- l ) ^ (3 o --l)

Y " ' j c ' \ k ÿ 3o) - ü Wh{é

(^и-Зо) + eu Wk{e~ü)

(24) G(v) Г Щ и -

“"I ( « - ;

n

^Г

fc=x L

Зо)2 (1-®Зо)а vWk(v) v7c+1Wk(l/v) (®-3o)' k+2

_{( 1}

—®3o> ^fc

^{+ 2}

Mi

-(е -и-Ъ »)м (e"-3o )*+2_

L ( l-^ 3 o ) 2 (^-3o )2 J '

vk+lWk{v) vWk(l/v)-\

(l-® 3 o ) + («-3o> ^fc

^{+ 2}

Comparing (23) and (24) we find that

G(eu) = <p(t).

By assumption there exists a & < w such that ak Ф 0.

To fix our attention assume that an is the non-vanishing coefficient.

By taking l[(v — з

„)” +2

before brackets G(v) becomes (23) (25) G (v)-(v- do)n+2 = a0|^(32 o - l) ( ^ - 3 o )

' 2

i) w(i

8

- i )

= « .[ * ^ ( з ? -

1

)

Ч-

(l-^ 3 o )2 p x g - i ) ^ ( g - D l .

° L ( i

- ^ . ) 2

( * - 3 .) s J ( 3o) +

( » - 3 o f +2 +

п

Г

+ y ^ a kJ c ll

7c=l *-

vWk{v)(v-% 0)

<1 — »3o)

- 7., Г

“ * f t ' L ( i - ^ . ) * +a ( ® - 3 o ) ‘ + s J ®

+

3o)

^{И + 2}

<

t

(^) is a regular function for v ф g0, v Ф з

0

and

0

#

1

/з„, v ^ l /Зо- From equality

( 2 5 )

it follows that

И т б г ( 'У ) = oo,

thus G(v)

=é 0 .

«->00

The function G(v) as a regular function must have a finite number of roots on the circumference v — eu.

By (23) we have

<p(t) = L n+,{eu) (е“ - г „ Г

+2

Ln+Ле “) Ln+Ле a) (е-г'-З о

)“ +2

(е~й-З о

)“ +2

Zw (ea )

(e“ -b o )n+2 ’

where Ln+2(eu) denotes some polynomial which is not identically zero

.and whose degree &<w +

2

.

If

3

o #

3

o> we have

<p{t) = ____________ -Ljn+si®1 )_____________

l(eü - 3o) (e“" - bo)(e-u - So) («" - So

)]n+2

*

It is seen that the function <p(t) has not more than I n + 8 roots in the interval ( — тс,

тс

].

If

3

o —

3

o ? we have

cp{t) = L 2n + M l)

[(ей -Ь о )(е -й-Ь о )Т +2'

In this case the nnmber of roots of the function <p(t) does not exceed 2w + 4.

Denote in succession the roots of q>{t) by tx, t%, ..., tm+s.

We shall prove:

L

emma

3. In every interval (ti,ti+1), i = 0 ,1 , . . . , I n + 8, t0 — —тс, tm+9 = тс, the function |a*(i) — c| ~ 0 or a (t) = const.

P ro o f. Suppose |a*(i) — c\ ф 0. There exists a point t\ in the interval (ti,t{+

1

) such that \а*($)

^—

с\ > 0 .

To fix our attention assume that <p(t) > 0 for te (t{, ti+1) and c —

= lima* (

2

).

t-+tj

Since a*(t) is a non-decreasing function in the whole interval [tf°, ti+1), we have \a*(t) — c\ >

0

.

Thus 4+i

J <p(t)\a* (t) — c\dt >

0

.

The above inequality contradicts equation (

2 0

). Therefore a*(t) = const in every interval ( f , ti+1).

Prom formula (3) it follows that

N

(26) F(b) =

23 fc

= 1

3 3 + e'

3 — eük З -б ' where tk are the discontinuity points of a*(t)

4

= a* (h+i +

0

) — a* (ti —

0

), N ^ l n + 8 .

<p(t) is an odd function (comp. (10')). Therefore ( —tk) is a root of this function together with tk. Grouping in pairs the terms of sum (26) we obtain

^*(3) 32- i у . ( ъ + f t

23

2 j AkU - e « *

l+ e~ itk\

3

- e~itk)

with tke [

0

, тс] and W < 2n-\-l.

L

emma

4. The function a* (t) has in the interval [0, тс] not more than n -\-2 step points.

P ro o f. We have found that a*(t) is a step function. In this case the second variation formula may be applied in the class Hr .

By an analogous argument as in the proof of Lemma 1 one may conclude that the boundary function satisfies the equation

y~r [ dk df I

(27) 2

j

у*к-^с[9(Ъо,%)-д(Ъо, [д(ъо,Ч)-д{го, b+i)]j = °-

Consider the function V>(*) = ^

k= 0

dk

W la kg{bo, t) + akg{%0, *)]•

From equation (27) it follows, that Ж ) = vik+i)-

Thus in the interior of the interval (t{, ti+1) there exists a point £,•

such that ip'(ty = 0. It may be easily observed that ip'(t) = <p(t). Thus

=

0

. a* (t) is a step function and the number of its steps in the interval [0, тс] does not exceed n-\- 2. In fact, supposing that a*(t) has more n-\-2 step points, "we find that (p(t) has more than

2

w + 4 roots.

This is impossible by Lemma

2

.

In an analogous way it may be proved that in the case 3o = 3o the number of steps of the function a* (t) is no greater than

The lemmas proved above and formula (3) yield.

T

heorem

8 . I f at every point w = (F, F , ..., F^n\ F (и)) of the region D defined by (16') and for every (5, |/?| =

1

, the condition

n + 2 "

2

П

2

^{k i}

> °>

fc

=0

ЗФ - ЗФ

ak — P dF(k) f or = ° ? 1 ’ , П is satisfied, then the boundary function F* (%) with respect to functional (17) is of the form

(28) F * M = 32- l VW / 3 + e%

2 3

Z

j

k U - e uk

3 + e itk\

where tke [

0

, n], Àk ^

0

, if 3o = -3o*

JV

2 К =

1

, N < n + 2 7/3o ф 3o and N < [> +

2

/

2

]

A

=1

3 . T h e e x t r e m a l f u n c t i o n s w i t h r e s p e c t t o r e a l f u n c t i o n a l s .

If 0 (F )

i s

a real functional the assumptions of Theorem

8

may be weaker. By an analogous argument as in Section

2

one may prove

T

heorem

9. I f the functional 0 (F ) is of form (17) and if for every point w = (F, F , F (n\ F (n)) of the region D defined by (16') the con­

dition

V

k=0 3 0 3]?(k)

^{> 0}

is satisfied, then every extremal function is of form (28).

The assumption of this theorem is weaker than that of Theorem

8

. This results from the following implication

(29)

П k 1 =0

3 0 - 3 0

dF(k) 3F (k) > o » . y 3 0 Z

j

3F[k) k—0

>

0

^,

P roo f. Suppose y 3 0

(30)

k= о 3 0

~3F{k) c)F{k)

=

0

0. Hence follows that

for h =

0

,

1

, ..., n.

(31)

Since the functional 0 (F ) is real we have 3 0 3 0 dF(k) dF(k) From inequalities (30) and (31) it follows that

n.

---

3 0

к У

—0

P <jF{k) F P 3 0

o.

This equality contradicts the assumption of (29). Thus implication (29) is true.

Chapter IV. TYPICAL REAL SYMMETRICAL FUNCTIONS

1 . D e f i n i t i o n . T h e s t r u c t u r a l f o r m u l a . T h e e x t r e m a l p r o b l e m s .

Typical real functions of the form

(32) Л (3) . <*i . «s

3H---

1

— r

3

^з3

. . .

+ —

1

+

^{. . .}

Rre called typical real symmetrical functions.

Denote by H\ the class of these functions. It is obvions that FL\ c Hr.

Now we shall prove

T

heorem

10. The function _Р

2

(з) belongs to the fam ily ll\ if and only if (33) F 2(

3

) =

^{( 1 - з 2}

/ ((в " — +

(34)

23

— TC

^j \(«“ -3 ) (« - i‘ - 3 ) (e“ + 3)(e-" + 3)

P ro o f. From the inclusion Щ c= R r it follows that d -

32)2

Г

1

da(t),

•^

2

(

3

)

_{3 — 71}

/ (ей j («й- з ) ( « - а - з ) Since — F 2( —

3

) — F 2(%) we have

(35) Л (3) =

TC

f —---^ ⁷ -

J (ег1 + г)(е tl d - 3

2)2

________

3 l (e" + 3)(*~" + 3) da(t).

da(t).

Adding side-wise (34) and (35) we obtain

^

2

(

3

) ( 1 - 3

2)2

Г/

¹

2 3 _{— 7Г}

J \(ей- -3)(«“fl- 3 )

or

(36) г/с2й + 3

2

j \ e“il — 3

2

+

_it

da(t).

0—2 it

e~2it- f da(t).

Let

9

^(

3

) e l? }. and

9

o2{^)eHr . The function (37) 9=>o(3) =

2

l>i(3) + ^

2

(

3

)]

also belongs to Hr , because

im3-imç?0(3) = i-img-im^^j)-f ^ т з Т т у ^ з ) > 0 . Similarly the function

(38) 0 i (3) = - F ( - b ) , F ( t ) e H r , is typically real because

й п з-н п ^ з ) = im

3

- i m [ - F ( -

3

)] = im( —

3

)-im F (-

3

) >

0

. Putting in (34)

we find

a(t) =

F(b) =

0

, .

1

,

( 1 - 3 2)2

■

tu

<C t ■'C ? t'Q ^ t ^ TU j

(ег‘° -з )(е ° —

3

)

Therefore with arbitrary t, te ( — т:, iz] the function

(39) F (

3

) = —

belongs to H r .

It follows from (38), that

1 (еи -Ъ)(е~и -%)

(40) 0Л г)

( f - з 2)2

3

1 ( ^ + з ) ( ^ Ч з ) is also a typically real function.

It can be easily seen the function under the integral symbol in (33) is the arithmetical mean of (39) and (40).

By (37) it belongs to the family Hr . Since a(t) is a non-decreasing function (33) also belongs to the family Hr .

It is easily observed that formula (36) the function F 2(

3

) may be expanded into the Laurent series of the form

-^

2

(

3

) — 3 + — + ••• + - f£ rr +

0 6

2 . E s t i m a t i o n o f t h e c o e f f i c i e n t s . B o u n d a r y a n d e x t r e m a l f u n c t i o n s .

By an analogous argument as in Chapters II and III one may prove the following theorems:

T

h eo rem

11. I f

^*(3) a ² n-l

~2ll

— 1

6 + .• 1 belongs to H 2 r , then — 3 < a x < 1 and |а2гь_г| < 4.

The above estimations are sharp.

T

h eo rem

12. I f a functional 0 (F ) defined in the fam ily H i is of form (17) and if at every point w = (F, F , ..., F <и)) of the region D defined by (16') and for every /?, Щ —

1

, the condition

Л дФ дФ

У, lafcl >

^{a k}

~ ft Qpi(k) + @

Q p ( k )

’ h — ^{0, 1 , n ,}

is satisfied, then the boundary function F* (

3

) is of the form -^

2

(

3

)

1 — 3 2 V . /

e2itk - f f

в ~ 2г^ + 3 2 \ 23

k \ e2itk —

3 2

e~2itk —

32

/ ’

N

where tke [0, те], Xk > 0, £ Xk = 1, N < n + 2, if з

0

Ф з

0

<шй N <

k= l

if 3o — 3o*

3 . T h e s e t s o f v a l u e s o f t h e f u n c t i o n a l s d e f i n e d i n t h e f a m i l y

Щ

.

Basing on structural formula (33) we shall prove:

T

heorem

13. The set of values of the functional 0 (F )

=

F(%0), |з0| > l,

30 3

0, is the segment of the circle determined by the points A = 0 , В = ,

3o bounded by the arc with the equation

(32o ~ l)2 3o(l + 3o)

(41) w(t) ( з ; - 1 ) 2(1+ з 2о) i

Зо 1 — + 3o

and by the chord

h(X) + (1 -X ) ( 3 o - l) 2 3o(l + 3o) ’

- 1 < 1

?

< A < 1 .

I f

3

=

3

the set of values of the functional 0 (F ) — F ( з0) is a segment of the real axes

and

~ l + 4 (3 o ~ l)2 ] 3o 3o(l ⁺ 3o) J

Г

(Зо- l ) 2

^{1 + зП}

L 3o(l + 3o) ^’ 3o J

4 3o< - i

if 3o > 1 -

P ro o f. It is known (comp. [

1

]) that the set of values of the functional 0 (F ) — F (

3

o) is a convex hull of the curve

(42) W(t) ( l- 3 o ) 2

23o [(«e - Зо)(е й -З о ) ‘ (eM + 3o)(e M + 3 j]

It can he easily noticed that the set of points of the curve (42) coincides with the corresponding set of the curve

(43) W (t) = ⁽³

^q

^{~ l)2(l} + 3o) 3o

1

1 - 2 з 2о*+зГ ^{te [ -}

¹

^,

¹

^].

Making use of the general properties of homography we find, that equation (43) represents an arc of the circumference passing through the points

A = 0, В =

°3o

( 4 - D 2

3o(l+.3o)

The convex hull of curve (43) is the region bounded by arc (43) and the segment

(44) h{À) = Я 1 + 3° + (1 -Л ) ^3o ^ 2 \ ’ le [0,1].

3o 3o(l + 3o)

H 3o = 3o the assertion of the theorem immediately follows from the structural formula.

(45) where

T

^heorem

14 . For every function F (f )e H r and for arbitrary

3

Ф з

0

^(3o) a < arg <Ъ,

• 1A 1 + 3*0 л (3o-l)2

a = mm ( A rg 2— , Arg - 2 -- 1 3o

1 + 3

2

Ъ = max ( Ar g— Ar g 2 33(1 + 32)

(32o - l ) 2

3o 3o(l + 3o)

We mean by F{f)fe this single-valued branche L (

3

) of a multi-valued function

<r(3) = arg^ (

3)/3

for which L(oo) = 0.

P ro o f. Estimation (45) follows from the fact that the set of values of the functional 0 (F ) = F ( з„) is a segment of a circle whose circum­

ference passes through the origin.

References

[1] И, Я. Аш невич и Г. В. Ч лина,

Об о б л а с т я х значений ан али ти чески х функций п р едстави м ы х и н те гр а л о м С т и л т ъ е с а ,

Вестник Л. Г. У. 11 (1955), р. 31-42.

[2] С. А. Гельф ер, О

коэф иц иентах типично-вещ ественны х функций

, Д. А. Н.

G.G.G.P. 94 (1954), р. 62-70.

[3] Г. М. Го Лузин,

Г ео м етр и ч еск ая т е о р и я функций комплексного леременного,

Москва (1966).

f4] W. Ja n o w sk i,

S u r un e certaine fa m ille de fonctions un iv alen tes,

Ann. Polon.

Math. 23 (1966), p. 171-203

[5] W. Rogosin.sk i,

Tiber positive harm onische E ntw icM ungen un d typisch-reelle Potenzreichen,

Math. Z. 35 (1932), p. 93-121.

[ 6 ] J. Z am orski,

O istotnie rzeczyw istych meromorficznyclh fu n k cjach ,

Prace Mat.

6 (1961), p. 41-49.

15 — Roczniki PTM — Prace Matematyczne XVII.