The Radius of α-starlikeness in the Family of Close-to-convex Functions

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ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN-POLOMA

VOL .XXXIX .20_____________ SECTIOA_________________________I960

Instytut Matematyki. Fizyki i Chemii Politechnika Lubelska

B.Ś WITONI AK, J.Z DERKI EWICZ

The Radlus ofa-starłikeness

ln the Family ofCloee-to-convox Functlons

Promień e-gwiażdzistośei funkcji prawie wypukłych

Радиус а-звездообразности семейства функций близких к выпуклым

1 Introduction. Let denote the ciassof all functions holomorphic inthe unit disk A = {z G C : jzj < 1}. The function / G ff(A) is said to be close-to- convex S’ if/(0) = /'(0)- 1 = 0 and Re /'(*)

/<-’) >0. z G A,fora certain univalent function g G H(A) that maps the disk A onto a convex domain. The class of close-to-convex functions will be denotedby A', According to the results of papers (1,4—6j the class K is identicalwith the classL of linearly accessible functions |2(, that is of such univalent functions f G A), /(0) = /*(0) -1=0 that C - /(A) is the union of closed half-lines suchthat the corresponding open half-lines have no points in common. Moreover. Z. Lewandowski in jot has proved that for

=/'(*) jzl < 4\z2 - 5 we have Re

/(*) > 0for any / G L and that this result is sharp.

This means that 4v'2 -5 is the radius of starlikeness of the class L. Theresult of ZXewandowski was establishedby makiXg use of the following theorem of M.Bie rnacki :2i:

zl ] =

^(I ^(SI< r.itj < r (1)

.2>

where r = :z>.

I I /(*)

(2)

z

1*56 B.Switoniak, J.Zderkiewici

In this paper we shall determine min | Re :/ £ L | whence we shall obtain the radius of o-starlikeness of the class L — K. 2. Main result. Making useof (1)weshallprove:

Theorem. If z£ A and jxj = r, then for any f & L the following estimation holds true:

I - r

for 0 < r < 2v'3- 3 1 +r

—___ ---2 ' for 2V3 - 3 < r < I . V(l-r) 1-r 2

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The result is sharp.

Proof. By (1) we have:

.min“K,)SR'W:'e£} =

( Re((l + .l2(2 + J))-j.||l+.|2 ,, )

= 1 ---

^{i(l -} ^{Re .)}

--- = {.I = r| =

= min{w(x) :-1 < x < 1} , where

/ \ 1 “ p2 , « (1 + r)(l ~p2) w(x) = —- ---h 2rx +

rx— Re s, -I < x < 1.

If

2(1+rx)

=('4)^-h-.

then w'(x) > 0 for x £< -1,1 >, that is w(x) > w(-l) = (1 - r)2. For any r £ (0,1) we havex„ < 0. Ifx, > -1 then

/ x / x (1 - r)(4v1 ~ p +r — 3) w(x) > w(x„) = --- Itsufficesnow to solve the inequalities x„ < -1 and x0 > -1.

Definition. The number:

r(a) = inf sup $r: /ei I

Re > ° for ***< r}

is called the radius of o-starlikeness ofthe class L.

From (2) we directly obtain:

Corollary.

fora< vi - 1 r(O’) = (1 +v i

1 - a

• 1-r a for —--- < or < 1 . 2

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The Radius of ft-starlikeness in the Family of Close-to-convex Function» 167

REFERENCES

|li В ielee к 1, A . , Le wan dowsk i.Z . , Set •» théorème concernent lee jonction» en,voient et linétirement erccaUce de M.Biemach, Ann. Poion. Math. 13, (1963) 61-63.

,3; Biernaeki. M. , S er le repruentetion conforme dea domamet linéairement acceaibiet, Praee Mat.Fix.

44 (1936), 393-314.

(8( Kaplan,W., Cio ее-to-convex tchlicnl fenctian», Michigan Math.J. 1 (1953), 169-185.

]4( Lewandowski,Z. , Ser l’identité de certain! ciattet de fond,ont enivatenlct I, Ann. Univ. Mariae Curie-Sklodowska, Sectio A 13 (1958), 131-146.

[6] Lcv/undov/ib,,!. , Ser l’identité de certeint dattet de fonction! eruvelcnlct U, Ana. Univ. Mariae Curit-Sklodowska, Sectio A 14 (I960), 19-46.

|6| Pommerenke, Ch. , Univalent f'enctiona, Vandenhôeek and Ruprecht, Gottingen 1975.

STRESZCZENIE

W pracy xnaietiono dokładne ostacowanie od dołu funkcjonału Re prawie wypukłych i wyznaczono promień a-gwiazdristcśei tej klasy.

/(-’) w klasie L funkcji

РЕЗЮМЕ

В работе получено точную нижнюю оценку функционала Rc

-гм

/м

^{в классе}

функций близких к выпуклым и определено радиуса-эпеэдообразности этого класса.

!

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