§ 1. Formulation of a certain isoperimetric problemI. Introduction. \z\ > 1 A theorem on distortion for univalent p -symmetrical functions bounded in the circle

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I (1968)

L.

A theorem on distortion for univalent p -symmetrical functions bounded in the circle \z\ > 1

Introduction. Let 27p(0) where p is a fixed non-negative integer denote the class of all holomorphic, univalent and p-symmetric functions of the form

Wp(z)

a.

defined in the domain \z\ >

, with a pole of the first order at the point z =

The symbol 2 p(m) where m is a fixed number from the open interval (

,

) will denote the subclass of 27p(0) consisting of functions satisfying the condition \Wp(z)\ > m. In 1937 J. E. Bazylewicz [2] ob­

tained a sharp estimate from below and from above for the functional

\Wp(z)\ defined on F p(0), for a fixed г Ф

(\z\ >

) and p > 1. Z. J.

Jakubowski [4] obtained a sharp estimate from below and from above for the functionals \W[(z)\ defined in the class 27х(т).

The present paper gives upper and lower bounds for the functional

\W'p{z)\ defined on the class F p(m), under the assumption that z Ф oo (\z\ >

) and \Wp{z)\ are fixed.

We shall also give the method of eliminating the absolute value of the function from the obtained estimates of the functional \W'p{z)\. This method will be presented for p =

; this restriction, however, does not change the basic ideas of the method.

In the first considerations we shall use the theory of K. Lowner [5].

The method of study is similar to that applied by E. M. Eobinson [

].

§ 1. Formulation of a certain isoperimetric problem

I. The basic ideas of the theory of K. Lowner applied to the functions of the class Ep{m) can be expressed as follows (see [

] and [3]). Let Ep{m) denote a subclass of the class Zp(m) consisting of functions which map the domain \z\ >

into the domain obtained from \z\ >

by removing a finite number of analytical arcs. The subclass 27j(m) is dense in Ep{m).

To each function WpeZp(m) we may assing the function F p{z,t), holo-

morphic and univalent in the disc \z\ >

with

< ź < T such that \Fp(z, i)| >

and

■plogm

(

1

1

1

2

(1.3)

Wp(z) = e~TlpF p(z, T ),

„ dFp{ z ,t) ^ ,, Tc(t)F%(z,t) + 1

P си t)

dt m n ( z , t ) - i

F p{z, 0) =

,

where Je(t) is a piecewise continuous function in the interval [0, T], discontinuous in at most a finite number of points, with all discontinuities being of the first kind, and such that \1c(t)\ =

.

In the theory of Lowner the study of functionals \W'p{z)\ in the class F p(m) can be restricted to the study of this functionals the class Ep (m).

The function F p(z, t), in the neighbourhood of the point

=

can be expanded as follows:

F p(z,t) = a{t)z-\-ap_ 1(t)lzp~1 a2p_i{t)lz2p 1+ . . .

where ([2], p. 690) a(t) = et/p, 0 < t < T. Since we have a simple relation m = e~T,p, the assumption that the number

is fixed is equivalent to the assumption that the number Tjp is fixed.

2. To simplify the notations we shall write F p(z,t) = F (z , t), Wp{z) == W{z) and b — \z0\, c = \F(z0, T )|, d — |TP(

o)[ = cm, where z0 Ф

is a fixed point of the domain \z\ > 1 . Then

(

1

6

P dF(z, t)

dt = F ( z ,t ) Jc(t)Fp(z,t) + 1 Jc(t)Fp( z ,t ) ~ 1 '

This equation for г = z0 Ф

is equivalent to the following equations:

(16) d\og\F(z0,t)\ _ \F(z0, t)\2p- l P dt ^ i t j F ^ i z ^ t ) - ! ^ ’

da,rgF(z0, t) = _ J m { k ( t )F p(z0, t)}

P dt \Jc{t)Fp(z0, t ) - l \ 2 '

Since the right-hand side of (

.

) is positive for every £e[

, T], for which the function Tc(t) is continuous, by putting

(

.

) Q = \F(z0,t)\

we see that

is an increasing function of t, which, in view of (1.3) and (1.4) increases from b to c as t increases from 0 to T. Thus, each function of tfe[0, T] can be considered as a function of д*[Ь,с]. Put

(1.9) k(t)lF(z0,t) = f v (e).

From the properties of functions 1c(t) and F ( z ,t ) it follows that the function

](

) given by (1.9) is defined in the interval [&,e] and it has at most a finite number of discontinuities of the first kind. Moreover,

\r](g)\ =

. In view of (

.

) and (1.9) the equation (

.

) will now take the form

(

1

10

It follows that

dlogg

2p—1

^ dt \QPy(Q) —

(1.11)

where

(1.12)

dt

dq = P S ( Q ) ~ Q

H ( Q) =

\QPV(Q)-1\

2p-

‘

We see easily that for every qe[b, c] we have the inequality (1.13) 8*4-1 < (e) <

3.

1.1. Let

](

) denote a function satisfying the following properties: (

) r)(g) is a piecewise continuous function, with at most a finite number of discontinuities of the first Jcind in the interval [b, c\ where b = \z0\

and c is an arbitrary fixed real number; (

) \rj(q)\ = 1 . Then for every function rj(q) with properties (

) and (

) there exist functions k(t) and F ( z ,t ) defined for

< £ < T and \z\ >

satisfying equation (

.

) with the condi­

tion F ( z , 0)

and such that (1.9) holds for z = z0.

Proof. Suppose we are given a function r}(q) satisfying the assump­

tions of the lemma. Let us determine function H {

) from (1.9), and put, remembering (

.

)

We see from (1.14) that if

increases from b to c, then t increases from 0 to T. Thus (1.14) allows us to determine

as a function of <e[0, T], hence to determine \F{z0,t)\ and k{t)Fp (zQ, t) from relations (

.

) and (1.9).

If we substitute the function k(t)Fp(z0, t) into (1.7) we find the argument, so that argF(z0,

) = arg^0; thus we find the function F (z 0, t) satisfying the condition F (z 0, 0) = z0. Knowing k(t)Fp(z

,t) and F (z

,t) we find from (

.

) the function k(t) which will satisfy the required conditions, i.e.

it will be piecewise continuous, will have in [

, T] at most a finite number

of discontinuities of the first kind, and \k(t)\ = 1. The functions k(t)

and F(zt, t) satisfy (1.6) and (1.7), which is equivalent to (1.5) for z = z0.

As a consequence, equation (1.5) together with the condition F { z ,

)

defines the required function F ( z , t ) for every \z\ >

and te[

, T ] satisfying (in view of equivalence of (

.

) and conditions

|F(z0, 0)| = |z0| and (1.14)) the condition (1.9). This completes the proof of the lemma.

E e m a r k 1. It follows from the above considerations that there exists a one-to-one correspondence between functions k(t) and

](

) defined by (1.8) and (1.9).

E e m a r k 2. If г](д) is any function satisfying the conditions of the Lemma 1.1, then the function H (

), determined uniquely by (1.12) has at most a finite number of discontinuities of the first kind, and sat­

isfies the conditions (1.13) and (1.14). Conversely, it is easy to see that if 3 {

) is a given function with the mentioned properties, it determines, through relation (

.

), a function rj{Q) with the properties mentioned in Lemma 1.1 (but in general, not uniquely).

4. We easily see that if F (z , t) is continuous, holomorphic in z for every fixed te[0, T ] for which k(t) is continuous, and such that F't(z, t) is continuous, then the derivatives F'Jt(z,t) and F'tz(z,t) exist and are equal. Using this fact, differentiating both sides of (1.5) with respect to

we obtain

9 F '(z, t) W(t)F2t,(z, Ą - Z ^ F ^ z , « ) - 1

P dt ) [k(t)Fp( z , ł ) - 1

where F '(z , t) denotes the derivative F'z{z, t). From this equation, for z ~ z0 Ф

we can separate the real part, which, in view of the relation

d\F(z0,t) 1 P IF{z 0, t

|.F(«0, t)I obtained from (

.

), the equality

- Jie{k(t)Fp(z„, t)—1} = j {| F(z„, t ) - l \ 2}

and relations (1.8), (1.9) and (1.12), will take the form

dlog\F'(z0, г)I ( p + 1 )

2

+ (p —

) _

dg S2V—1 P 3(Q) в ' Integrating we get

\F'(z0,T)\

<?-\Ъ2р-

)

(1.15)

where

(1.16) J = V

c

dg

Q '

Formula (1.15) gives the relation between the absolute value of the deriv­

ative of F {z , T) and the value of integral J when the point z0 and |F(z 0, T)\

are fixed.

5. According to (

.

) we have F {z , T) = eT,pW(z) and F ' ( z , T )

= eTlPW'(z). Putting z = z0 Ф oo and using notations (1.4) we get (1.17)

= djm and \F'(z0, T)\ = \W(z0)\lm.

Let us note also, that in view of m = e~T,p we can write instead of (1.14)

C

(1.18) logl/w = j H(g)dg/g.

b

By (1.17), we may transform (1.15) to

\W'(z0)\ (d2p- m 2p)bp~1 _ j трГ - ' ( Ь №- 1

It follows that the problem of finding the extremal values of the functional

|Ж'(з0)[ defined in the class of functions

j(m), for fixed z0 and d = | ТГ(я0)|

reduces to finding the extremal values of the integral J given by (1.16) under the side conditions (1.13) and (1.18). To be more precise, the problem reduces to the following isoperimetric problem with the side condition:

let Л, with notations (1.4) denote the class of piecewise continuous func­

tions, with at most finite number of discontinuities of the first kind in the interval [b ,c ] and satisfying condition (1.13); find in the class H a function H (

) for which the integral (1.16) assumes its maximal value,

€

and the integral f H(Q)dglg is, by (1.18) equal to the constant logl/w.

§ 2. The solution to the isoperimetric problem

1. To solve the isoperimetric problem formulated above we introduce the functions

(2.1)

Р(ж; b, c) a ? - l xp+ l

qp — 1

dg 7 + Г 7 ’

Q{X‘, b, c) xp -j

г dg г gp~f

dg

+ / т з г т ’

defined for x e[b , с]. We easily verify that P(x-,b,c) is strictly increasing

and Q(x',b,c) is strictly decreasing function of xe[b, с]. From the defini-

tion (

.

) we have in particular {< ?+ ! )21Ръ P (

; b, c) = log

(2.2)

P{G) b, c)

c(bp+ l ) 2IP

ć? —

c

^ + T l o g &’

Q(b-,b,e) = l 0 g - ^ _ — ,

л GP + 1 C

Q(c-,b,c) = - t f T Z j loS j -

Since for every x e[b , c] we have P(x-, b, c) < Q(x-, b , c), we have in view of the monotonicity of these functions

(2.3) P (

; b, c) < P{c) b, c) < G(c; &, c) < b, c).

Let H (

) be an arbitrary function from the class H. Then by (1.14), (1.16) and (

.

) we have

(2.4) P (b ',b ,c ) < T/p < ф (5; b, c), P(b-,b,c) < J / p ^ Q { b ‘,b , c ) .

2. Let us now find the maximal value of the integral J for fixed T lpe[P (b : b, c). Q(b; b, c)]. First, we note (see [

], p. 430) that H(o) +

+ 1 I H (

) ^ (

- 1 ) I (

+ 1 ) + (

+ 1 )I(

- 1 ) . Hence we find (2.5) T /P + JIP <P (b-,b,G ) + Q(b-,b,G).

The equality sign in (2.5) is attained for the function b <

< ж,

X < Q < C

s 0 (e)

q

p- i f

+

^ - i

for

for

for any given value of parameter x such that a < x < b. This function belongs to the class H, and its values (

—1)I(

+ 1 ) and (

+ 1 )I{

p—. 1) give respectively the smallest and the largest value of T/p; thus for the appropriate choice of x e [ b ,c ] we can obtain each fixed value of T/p.

The maximal value of J for given T/p we determine from (2.5).

3. To find the lower bound for the integral J , when the value Tjp

= logl/m of (1.18) is fixed, let us consider (see [

], p. 430) the expression

c

(2.6) T / p + m = f [ Н ( е ) + р 1 г1 Щ е ) Ш е ,

b

where Я is a positive constant, and Н(@)еН.

We shall show that the integrand is smallest when H (

) — ЯУр.

In fact, let H ^

) be an arbitrary function from the class H such that

H ^

) ф H (

). Then

which shows the truth of the assertion. The equality H (

) — xVp means that for a fixed p > 1 the function H (

) is constant. However, there may be no constant function H (

) in the class H. We avoid this difficulty if, according to (2.3) and (2.4) we consider separately three cases:

(1°) P(ft; b, c) < Tip < P ( c ; Ъ, c), (2.7) ' (2°) P ( c , b , c ) < T / p < Q ( c ; M ) ,

(3°) Q (e ;b ,e ) < T J p < Q (b ;b ,c ).

L

2.1. I f P ( b ’, b , c ) ^ Tjp ^ P(c; b, с), there exists a number xe[b , c]

( 2 .8 )

T/p = P ( x ‘f b, c)

and

(2.9) J > pQ (x; &, c),

where P (x \ b ,c ) and Q(x-,b,c) are defined by formulas (2.1).

Proof. Consider the function

S 2(

) =

xp- l

xv-\-l for b < Q < # for x <

< e

We note easily that the above function satisfies, for every x e [b, c] the condition (1.13), and after multiplying it by 1 I

and integrating from b to c it gives function P (x-,b,c) with domain b and range [P(&; &, c), P(c; b, c)]. It follows from the properties of function P(x', b, e) that in case (1°) of inequality (2.7) there exists unique point xe\b, c]

such that the inequality (2.8) holds. To obtain (2.9) let us take function

C

H(Q)eH, such that Tfp = / H(Q)dQlg, E (

) Ф H 2(

) and let us consider ь

the expression

G {q) = H {q) + K2p I H (q) — [ H 2 {q) + Pp I H 2 (q) ] .

Here Я denotes a positive constant. Put in particular Я = (P’—1)/(P

+ 1) x

X

jVp, where x is the point of [b, c] for which (

.

) holds. It turns out

that for such value of Я we have G (

) >

. In fact, if b <

< x, then

as a product of non-negative terms. Thus, for every

we have G(Q) ^

, which implies that

H (

) - H 2(

) ^ P [ p I H ( 6) - p l H 2(Q)].

Multiplying both sides by 1 fg and integrating from b to c we get

T l p - T l p > » { p f щ de Г

s(e)

- p f Щд)

Using the definition (2.1) of function Q (x ;b ,c ) we obtain

C

dg

6

> pQ (x‘, b, e).

The last inequality is equivalent to (2.9), which completes the proof of Lemma 2.1.

In a similar manner one can prove

Le m m a

2.2. I f T/pe[Q(c-,b, c),Q(b-,b, c)], then there eocists a point xe\b,c] such that T/p ==Q (x}b,c) and J > p P (x ) b, c).

In the proof the essential role plays the function for b < g < x ,

for x < g < c, and the constant Я = {xv-{-l)l{xv—1) ’l/Vp.

Le m m a 2 . 3 .

I f

c), Q(c; ft;

then (2.10) J > (T/^)_

^(log&/c)2.

Proof. We know already that the function H {g) = %Vp, where Я is a certain positive constant, gives the minimum of the right-hand side of the expression (

.

). Thus, if in the case (

0) of inequality (2.7) we put H (g) = iVp for every g e [b ,c ], then for an appropriate choice of the constant Я we get

C

Tip = iVp J dg/g = iVplogc/b.

b H 9(

)

In this case we obtain from (2.6)

C

J > pjXVp j dg/g.

ь

Substituting in the last inequality l^p = (T[p)logc/b we get inequality

(

.

), which completes the proof of the lemma.

The results obtained in Lemmas 2.1, 2.2 and 2.3. can be written together in the following way:

{2.1 1)

where

(

.

) L(Tlp-,b,c)

J > L(T/p- b, c)

(1°) pQ{xj b, c) if Tip = P {x) b, c) (

°) (T[p)~1p(logc/b)2

(3°) pP{x) b, c) if Tjp = Q{x; b, c) .

Th e o r e m

2.1. In the class Ev(m) for a fixed z0

|s0| = b >

, for a fixed |W(#0)| = d, we have the following sharp bound

(2.13) df+1(b2p- l )

bp+1(d2p- m 2p) <|ТГ(г

<

bV-l{d2p_ m2P)

^ d F 4 t g - i ) e ł p [ - - b № ; ł ’ a>]’

where the expression L(T/p-, b, c) is given by (2.12), p > 1 is the fixed degree of symmetricity of function, m is a fixed number in (

,

), Tjp

= log

jm and c = djm.

Proof. By (1.15) we have

Г)| (c

p—l)bp_I c°~l (blp- l ) e ’ and since by (1.17), lP'(z0f T)\ = |W' (z0)\/m, we get

|W'(*0)I m{c2p—l)b p

_ j cp~1(b№- l ) 6 ’

Inequality (2.5), after substituting in place of P{b-,b,c) and Q { b ;b ,c ) the corresponding values (

.

) can be written as

(2.15) J < log (c2p- l f b 2pmp {b2p- l ) 2c2p Using (2.15) and (

.

) in (2.14) we obtain

cp+

(

*p- 1)

w2,- 1&p+1(c2?,- l ) < \ W'(z0)\ < m(c2p—l)b p 1

<?-\b2p-

) e x p [ - X ( T / p ; b, c)].

If, in addition, we put according to (1.17) c = djm, then dp+l(b2p-

)

bV + l(d2p_ m2P)

bp~l {d2p- m 2p)

mpdp- 1(b2p~ l ) e x p [ - L ( T / p ; b, c)].

which gives the assertion of Theorem 2.1.

§3. The estimate of |TT (0o)l as a function of b = |я0| and m

1. In this section we shall show how one can eliminate from estimate (2.13) the value d = \ Wv{zQ)\, thus expressing the lower and upper bound for the functional \W'{z0)\ only as a function of b and m.

At first, we shall determine the upper bound for this functional. In the considerations we shall restrict our attention to the case p —

. In this particular case we have instead of (1.15), (2.11) and (2.12) respectively

(3.1) l ^ o ^ ) !

(3.2)

(3.3) L(Tib, c)

J

( 1

°)Q (x -,b ,c ) if T = P(x',bj c)j (

°) l/T(logb/c)2,

(3 ° ) P ( x ; b , c ) if T = Q ( x ‘,b , c ) . In view of (3.2), equality (3.1) yields

(3.4) log\F'(z0,T)\ ^ N ( T - , b , c ) where

(3.5) S { T - ,b ,c ) = \ o g ^ - - - L ( T - , b , c ) . b

It follows from (3.4) that the problem of finding the upper bound for functional |Ж'(г0)1 = e~T \F'(zQ, T)\ for fixed b and m = e~T reduces to the problem of finding the maximal values of N (T -,b ,c) for all possible values of c. The value of N (T -,b ,e), according to (3.5) depends on the value of log(c

—

)/(&

—

) and on the value of L(T-, b , c). The last value, in turn, by (2.12) depends on the fact whether

(1°) P(&; b, c) < T < P ( c ;

,c), (3.6) (

°) P ( c ; & , c ) < P < Q ( c ; b , c ) , (3°) Q (o ;b ,c) < T < Q ( & ; & , c ) .

The cases (3.6) were distinguished under the assumption that b and c

are fixed. Kow we assume that only b is fixed. We see, that in order to

find the maximal values of N (T ',b ,e ) for varying c, we have first to

determine in which intervals should lie the value of c so that the cases

(l°)-(3 °) hold. To answer this question, we show first that the expressions (see (2.2))

( c + l ) 2ft {c—l f b

P ( M ’ c) = log^ W ’ (3.7)

^ ^ ^ С-]-

c

P{c\ Ъ, c) =

Q(c; b, c) = ---

c-f

b c

b

treated as functions of

c

b

b

P'c( b ;b ,c ) = > 0 , Qe(b-,b,c) = > 0 , (3.8)

2 Г e е * - П

We shall show the tru th of inequality

Q'c(c; b , c) > 0,

,

Г c c2—l l

(3.9) <?efe; b , e) --- --- _ _ J .

L et us write this derivative in the form

2 Г _q "i

Qe(o;b,e) =

and put logc— (c

—l)/2c =

o(c). We see that cp'{c) — — (c —l )

/2c

< 0.

Thus, for b < c we have

(c) <

(

), and since ft

we get

(c) < logft, which is equivalent to the inequality logc

—

—

(c

—l)/2c

This, together with (3.9) implies Q'e{c\

c)

I t is easy to see th a t as c increases from ft to + ^{0 0}, then each of the expressions (3.7), treated as a function of c increases from 0 to + 0 0. Thus, for given ft and

m,

T = logl/m

€ P

C

,C

, C

P(ft;

, CP) == P ( £ P ;

, CP) = Q(Pq -,

, CQ) = Q(

;

, CQ) = T . It follows from inequality (2.3) that these numbers satisfy the condition

< C Q < 6 P < CP. The inequality P (

;

, с) < T < ^(ft; ft, c) (see (2.4))

will now take the form C

<

< ( 7p . The cases (3.6) are equivalent respectively to the three following cases:

(

°) C p ^ c ^ C p ,

(3.10) (2°) C

^

^ C

,

(3°) C

^

^ G

.

Let us now find the derivative of L{T\ b, c) treated as a function of e in each of the cases (3.10).

By (3.3) we have L'c (T'j b, c)

(

°) Q’c{x\ b, c) + Q’x{x-,

, c)[

(

°) -^ -lo g ^ ,

Tc b

(3°) P'c{%-,b, c)+P*(a?'; b, c)

P'c(x-, b ,c) 1 P'x(V) b, c)J

Q'c{x- b, c)l Qx{fflj bj c)J

if T = P ( x ; b, c),

if T = Q(x; b, c).

Differentiating functions (2.1) we get (3.11) K ( T - ,b ,c )

(1°)

c

c { c -

)

2

c

-To10gJ

ж+

c

— 1 / c(c + l )

c

/ x

c

c(c + l ) " \ а ?

-

/ c(c—

)

if T — P ( x ; b, c),

if T — Q{x) b, c)..

We shall show that the derivative L'c( T ; b , c ) exists and is continuous in the whole interval [CQ, <7P]. Indeed, it is seen directly from (3.11) that that derivative is continuous in every interval (3.10). To find its values at points CP and CQ it suffices to put x = b in cases (1°) and (3°).

The values L'c{ T m , b, CP) and L'C{ T ; b, CQ) are the right- and left-side values of L'C(T-, b, c) at the points CP and CQ. It is easy to check that these limits exists and are equal respectively to

L'c(T-,b,Cp)

(CP+1)

C p { C p —1 )

L'c(T -,b ,C Q) 2(G

- 1 )

Gq( Gq+ 1 )

Thus the function L'c(T-,b,c) is defined and continuous in the whole

interval [C

, <7p]. Moreover, we see from (3.11) that L ’C{T-, b, c) > 0. I t

means that the function X(T; b, c) is increasing in each of the cases (3.10).

We shall now investigate the derivative of the expression N ( T ; b , c ) treated as a function of c. By (3.5) we have

N ’e( T ; Ъ, c) = 2c](c2- l ) - L ' c( T ; b, c).

Thus, in view of (3.11) we get Ж'с( Т ; Ь , с )

2

o - l

--- if T = P{x\ b, c), (

°) c

+

c

/ OB+1

c(c

- l )

1

(

°)

c c c

c2- l

T logv

(3°) c

+

c

1 c(c

—

) W + i

o + l

if T = Q(x-, b, c).

We shall study the sign of N'C(T; b, c) separately in each of these three cases.

(1°) In this case, after some simple transformations N'c{ T ) b , c ) takes the form

с

Г /ж +

, с - 1 Г /л?+1\П

с(с

-

)

It follows that Ж'С(Т-, b, с) < 0, hence the function N{T\ b, c) is strictly decreasing in the whole interval [CP , CP]. At the points CP and CP the function N'c( T ’,b , c ) assumes the values

, CP — 2CP—1 / b + 1

CP —1 N‘ { T ’ b ’ C*> = ' - ( w ) С Ж + Т )

if T = P (b -,b ,C P) , (3.12)

N '(T -,b ,C P) = - Bp{^p _ 1} if T = P{GP-,b,GP).

(3°) In this case the derivative A

'(T; b, c) can be written in the form

,

(c +

) Г 2x

]

Жс( Т ;Ь ,с ) = c(c_ 1)

“ (c- l.)2J

and it follows that its sign depends on the sign of the expression

= g(c).

2x We see easily that

(ж+ l

( c + l ) ‘

g (c Q) 2b

( b + 1

” {CQ+ i y > 0

and

Thus, at the points C

and CQ we have

if T = Q(bj Ъ, CQ), (3.13)

From the last formulas and from the continuity of N'C(T; b, c) it follows that this derivative is positive in a certain neighborhood of the point C

. On the other hand, the derivative N'c (T \ b , c) is equal to the difference of the derivatives of two functions which are continuous and strictly increasing, namely functions log(c

—1)/(6

—1) and L ( T ; b, c). Inequal­

ity (3.13) implies that the first of these functions increases faster than the second one. Thus, for every C€[CQ,C Q] we have N'c( T m , b , c ) > 0, and it follows that N(T\ b, c) is strictly increasing in the interval [CQ, C

(2°) In this case we have

We have already established that (see (3.12) and (3.13)) N'c(T-,b,C Q) >

and N'c(T ',b ,C P) <

.

We shall show now that in the open interval (C

CP) there exists exactly one point С0 such that N'c(T-,b,C 0) = 0 . Consider the functions r(c)

— c2l(c2—

) and s(c)

l/Tlogc

defined in the interval [C

,C p]. In the whole interval

CQ,C

the function r(c) is strictly decreasing, and the function s(c) is strictly increasing. Moreover,

Thus, in the interval (C0 ,C P) there exists exactly one point C0 such that r (CQ)- 8 (C0) =

, or

(T-,b,c)

c2- 1 T

c

i°g{]

t(Cq) - s(Cq) - CqI2N'c(T-, b ,C Q) > 0

and

r (C p )- 8 ( C P) = CPI2NC( T ; b ,C P) < 0.

(3.H)

The last formula and the form of the derivative N'c( T ‘,b , c) imply that N'c( T ; b , c ) >

for CQ^ c < C 0, N ’e{T\Ъ, C0) =

and N'C( T ; Ъ, c) <

for (7

< c < Ć p . At the point G0 the function A(T;&,c) assumes its unique maximum. By (3.3) (2°) and (3.5) we get

-шггг Ъ

L Coy m a x jy (T ;

,c)|c, Co = — ^log— j . Using inequality (3.4) we get

lo g ic s ., T)\ < log A A -

(

)'-

Substituting here, by (1.17), the values \W'(z0)\lm and d0jm instead of

\F'(z0, T) I and C0 we obtain

log |ТГ («о) I < log do —w

m(b2—

) Equation (3.14) takes the form

(3.15) dl

d20 — m

do mb

The last inequality, using (3.15) can be written in the form

log I W' (ą>) I <log which yields finally

|Ж'(^0)1 <expjlog

dl — m

dl , bm

—--- h-^5—55— log--- m(b2 —

) d l - m 2 d0

dl — m2

+ dl

m(b2—

) d\ — m

bm\

logl

i

where d0 is the root of the equation {d\ — m2)d0

log&w/d

= logm. This equation was obtained from equation (3.15) by substituting T = logl/m.

2

. Now we shall find the lower bound for the functional | W'(

0)|

as a function of b and m. First, let us note that the inequality (2.15) can be written in the form

(3.16) J < log ( x - l ) 2{ c + l ) 2b

( x + l ) 2( b - l ) 2e = A (x; b, c), where the expression {x —l )

/(<r+l

is defined by (3.17) m { х + 1 ) Ц с - 1 ) 2Ъ

° ё ( x - l ) 2{b-\-l)2c B(x-,b, с).

Roczniki р т м — Prace Matematyczne X I I 4

The equality sign in (3.16) is attained for the function for b <

< x ,

for x <

< c

for an arbitrary value of the parameter x such that b < x < c. For p = 1, inequality (2.15) takes the form

(c2—l ) b 2m m (3.18) J < l o g {b2-_ _ ^ cT = K { T - , b , c ) .

#o(

) =

Q - 1

£?+l

e + i

The functions A ( x , b , c ) , B (x-,b,c) and K (T -,b ,c ) satisfy the equation (3.19) iC(T; b, с) = А (я; b, c) for T = B ( x , b , c ) .

Consider now K { T ; b , c) as a function of the variable c in the interval

[Cq, Op] .

Applying the theorem on differentiating composite functions

we get, for fixed b: K'C(T', b, с) =

(c

+ l ) e -

(c

—l

>

. It follows that the function K{T\ b, c) is strictly increasing in the interval [CQ, Op], From (3.1) and (3.18) we get

(3.20) log|F'(

0, T) 1 > Jf(T; b, c) where

c

(3.21) M { T : b , c ) = log--- K( T- , b, c ) with Т = Б(ж;&,с).

b2—1

Let us consider M(T-, b, c) as a function of c where CQ < c < Op. It is easy to check that M'C( T ; b, c) — — 2c~1(c2—l

< 0. Thus the function M { T ; b, c) is strictly decreasing in the interval [C

, Op], It assumes its lower bound for c = Op, but then x = b. For simplicity of notations let us write О instead of Op. Thus, according to (3.19) and (3.21) we have

c2- l (

+ l

(3.22) inf M( T , b, c) = log — - l o g ■

cącq

,C] b2—1 (

+ l

where О is the root of the equation

(3.23) T = log ( C - l ) 2b

(b—l ) 2c

In view of (3.22), the inequality (3.20) takes the form logl-F'C^o, T )I > l o g ( H - l ) ( 0 - l ) 0 - l o g ( f t - l ) ( 0 + l ) b .

Substituting here [W'{zQ)\fm and dfm instead of \F'(z0, T)\ and O we

get I TF'(

0) j > d2 (b2 —

)/b2 (d

— m2), where d is the root of equation

dj(d2 — m2) — b(b— l ) 2. This equation is obtained from equation (3.23) by

substituting T — logl/w and C = djm. Thus, as a result of the above considerations we obtained

Th e o r e m

3.1. In the class S x(m) we have the following sharp esti­

mate

d2(b2-

)

b2(d2 — m2) < |W’/(«o)l < exp {log

d0 — m

m(b2—

)

d20 bm

d\ — m2 d0

where b — \z0\ (z0 Ф

\zQ\ >

and d and d0 are respectively the roots of equations d /(d ~ m )2 = bj(b— l

and {dl — m2)jdl\og(mbjd0) = l o g m .

This estimate was obtained by different methods by Z. Jaku­

bowski in 1960 (see [4], p. 38).

R eferen ces

[1] I. E . B a s ile v ic h , Zum Koeffizientenproblem der schlichte Funhtionen, Mat.

Sbornik 1 (43) (1936), pp. 211-228.

[2] — Complement to the paper „Zum Koeffizientenproblem der sehliehten Funic- tionen” and „Sur les theoremes des Koebe-Bieberbaeh”, Mat. Sbornik 2 (44) (1937), pp. 681-697 (in Russian).

[3] — On distortion theorems in the theory of univalent functions, Mat. Sbornik 28 (70) (1951), pp. 283-292 (in Russian).

[4] Z. Ja k u b o w s k i, Theoremes sm la deformation dans la familie de fonctions univalentes a un pole simple a V inf ini et bornees inferieurement dans le cercie К (oo, 1), Bull. Soc. Sci. Lettres, Łódź, Sectio III Nr 76 (1960), pp. 6-40.

[5] K. L o w n e r, Untersuchungen uber schlichte Tcomforme Abbildungen des EinheitsTcreises, Mat. Ann. 89 (1923), pp. 103-121.

[6] R. R o b in so n , Bounded univalent functions, Trans. Amer. Math. Soc. 52 (1942), pp. 426-449.