R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )
Zbig n iew Jerzy Jakubow ski (Lodz) (
O n the upper bound oî the functional |/(n,(s)| (w =2,3,...) in some classes of univalent functions
1. Denote by 8 the class of regular and univalent functions of the form
(1) ' f{z) = z + a2z2 + ... + anzn + ...
defined in the circle \z\ < 1.
Let z, z Ф 0 be an arbitrary fixed point of the unit circle.
The function
9 ( 0 = JL± 1
1 +z£
maps conformly the circle \C\ < 1 on to itself and
(2) * 9 ' ( 0 l - и 2
( l + z £ ) 2 '
Let / be an arbitrary function of the class 8. Consider the function
(3) Щ ) = /(g(f))-/w
/'(*)( 1 - И 2) '
Since it is regular and univalent in the circle ICI < 1 and since Л{0) = 0, h'(0) = J, /te S. Tims li"(0)| ■< 2 -2! [1], hence we obtain the estimate [2]:
(4) \ f ’(z)\< 7 7-Ц г > \ г \ = г . (1 - r )
The upper bound is attained for the Koebe function (5)
at the point z = re~lt
/*(*) = z
( 1 - A ) 2
5 — R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I
66 Z. J . J a k u b o w s k i
Making use of the known estimates |Л"(0)| < 2 - 2 !, |&"'(0)| <3*3! [7],
|V4>(0)[ < 4 - 4 ! [4] and of formula (4) Walczak [8] has obtained the fol
lowing acute estimate for n = 2 ,3 , 4
7Ь ~ I 7*
(6) ( l . n „ . , - И = г .
The extremal function is given by (5).
In this paper we shall prove that if Bieberbach hypothesis is true w l for l = 2 ,...., n, then estimation (6) holds for every natural n. We shall also find the upper bound of the functional |/ (и)(г)| (n = 2 , 3 , . . . ) in some subclasses of the family 8.
2. Put w — g(Ç). We shall prove that for every function f e S and an arbitrary n (n = 1 , 2 , . . . ) the following formula
(7) f * ( w )
n —1
( i - и 2r 4 Z l »» /' i l / _л ---(1 +z£)2n~mzm (n — m)l
holds.
In fact from formulas (2) and (3) we obtain f'(w) = f ( z ) h ' (0 (1+zC)2.
Hence differentiating both sides in relation to £ and taking into account formula (2) we obtain
f"(w) = - lf V f 1] - ---- — (1 +2£)4~mzm.
J v 7 1 — \z\2 Zj W 2 - m ) ! V ' m= 0
Thus formula (7) holds for n = 1 , 2 . Assume that it is true for n ^ 2. Then
(n —m) ! / * - 4
\m + 1/
+ + h{n~m)(0
(n — m — 1) !
—1\ *'(£)
(1 - f z£)2№_m_1 zm+l - f . . . - f
+ (n + 1( ( l - Î) ~ j r (1 • Since
(2 » - m ) ("“ M---- — + ( * - ; )
\ m ! (n — m)\ W + 1 /
n -f 1 (n — m)\ \m + 1l (n — m —1)1 (n — m) formula (7) holds for every natural n.
J n )
I \ m + l /
Putting £ = 0 and respectively w = z we obtain from (7) the following relationship
(8) »!/'(* ) у /«-1\ A«"— >(Q) ^
( l - l s l 2) " " 1 Z j \ *» ) (n - m ) \ Z ' ‘ 1 ' m=0
3. Let z Ф 0 be an arbitrary fixed point of the unit circle and / an arbitrary function of the class 8. Then function (3) belongs to the family $, thus there exists such a constant Тсг [1] that |Л(г)(0)| < Тсг-1\ (I = 1 , 2 , . . . ).
Hence by formula (8) we have
\fin)m < »!/'!(*)!
( 1 - 1 * 1 7
.1*1* (n = 2 , 3 , ...)•
, Taking into account estimate (4) we obtain
l/w (*)l < » ! ( l - » - ) - ”- s( l + r) - " + 2 I” ,,,1) h . . mrm,
m= 0 '
1*1 - r {n = 2 , 3 , ...)•
Thus if the Bierberbach hypothesis is true, i.e. if Tct = l (l = 2, 3, ... , n), then
Z ( " « ‘K — ^ ( w + r H i + r ) - » .
m= 0 ' '
Hence the following theorem holds:
I f the function f ( z ) e8 and Тсг = l (l = 1 , 2 , ..., n), then for every natural n and an arbitrary z, \z\ = r, 0 < r < 1 the acute estimate (6) holds.
Moreover, we have
(9) ! f n](z) _ _ _ _ _ /fy ! ---n + r
I / #(*) " ’ ( l - r r - ^ l + r) (n = 2 , 3 , . . . ) .
4. Let Ê denote a subclass of the family 8 of functions of form (1) which map the circle \z\ < 1 onto convex regions. l i f e 8, then he В (comp.
(3)) thus |й/г,(0)| (I = 1 , 2 , . . . ) [6]. We also have estimation [6]
\ f ( z )I < (1 - r ) 2, \z\ = r the upper bound being attained for the function
f ( z ) = z { l - e itz)~1
t the point z = re~%t. Thus formula (8) implies the following theorem:
68 Z. J . J a k u b o w s k i
I f a function f(z) e j§, then for every natural n and an arbitrary z, \z\
= r, 0 < r < 1 the acute estimate
(10) \f(n){ z) \ < п\ {1- г )~п~1 holds. Moreover, we have
/ (n,(*)
/'(*) < nl ( l — r) n+I,
Besult (10) has also been obtained by Walczak [9] in another way.
5. Let 8* denote a subclass of the family 8 of functions which map the circle |^| < 1 onto star-like regions in relation to the origin.
It is known ([5], p. 204) that if /e §, then the function F(z) = zf(z)
belongs to the class 8*. The converse is also true. Hence we obtain the formula
F {n) (z) = nf{n) (z) -f 0/ (n+1> (z).
Thus the following theorem follows from estimate (10) [9]:
I f a function F(z)e 8*, then for every natural n and an arbitrary z,\z\
= r, 0 < r < 1 the acute estimate (6) holds.
6. Let 8R denote a subclass of the family 8 of functions f(z) of form (1) with real coefficients. Let z = r, 0 < r < 1. Then the function h e S R (comp. (3)) thus \h(l){0)\ ^ l - l \ (I = 1 , 2 , . . . ) [3].
By formula (8) the following theorem follows:
I f a function f(z) e 8R, then for every natural n and an arbitrary r, 0 < r < 1 the following acute estimate holds
}/(n)(r)| < n\ n-\-r ( l - r ) n+2’
f (n){r) n + r
--- --- ^ T _____
f { r ) ' ( l - r f - ^ l - f r ) ' The upper bound is attained for the function
at the point z = r.
/*(*> = z (1 - z ?
References
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[21 — Über einiqe Extrem alproblem e im Gebiete der konform en A bbildunq, Math. Ann.
77 (1916), p. 153-172.
[3] J. D ie u d o n n é , S u r les fonctions univalentes, С. E . Acad. Soi. Paris 192 (1931), p. 1148-1150.
[4] P . E . G a r a b e d ia n and M. S c h if f e r , A p ro o f o f the Bieberbach conjecture fo r the coefficient, J. Eation. Mech. Anal. 3 (1955), p. 4 2 7 -4 6 5 .
[5] Г. M. Г олузин, Геометрическая теория функций комплексного переменного, Москва 1966.
[6] К . L ô w n e r , U ntersuchungen über die Verzerrung bei konform en Abbildungen des E inheitskreises \z\ < 1, die durch E u n ktio n en m it nicht verschwindender A bleitung geliefert werden, Leipzig Ber. 69 (1917), p. 89-106.
[7] — U ntersuchungen über schlichte konform e Abbildungen des Einheitskreises, Math.
Ann. 89 (1923), p. 103-121.
[8] S. W a lc z a k , Oszacowanie \f"{z)\, \ f"' {e)\, \ f ^ ( z ) \ fu n k c ji klasy 8 , Zesz. N auk.
ILL. Ser. II, 39 (1970), p. 6 9 -7 3 .
[9] — E xtrem al problems in the class o f close-to convex functions, Ann. Polon. Math.
25 (1971), p. 2 3 -3 9 .
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