On close-to-convex functions

奈良教育大学学術リポジトリNEAR

On close‑to‑convex functions

著者 SAKAGUCHI Koichi, WATANABE Shigeyoshi journal or

publication title

奈良学芸大学紀要. 自然科学

volume 14

page range 7‑12

year 1966‑02‑28

URL http://hdl.handle.net/10105/3354

On close-to-convex functions

Koichi SaKAGUCHI and Shigeyoshi WaTANABE

(Department of Mathematics, Nara Gakugei University) (Kojima High School)

(Received Sept. 25, 1965)

One of the authors C^O has obtained some properties of starlike functions f(z) by studying f(z)-/(-z). In this paper we shall show that close-to-convex functions also have the same properties as obtained there for starlike functions.

§ 1. Preliminaries

Lemma 1. Let <j>{z) be convex in \z\<\. If f(z) is analytic and satisfies 3t C/(zW (z)J>0 for \z\<\, then

(1.1) ^_n At~,\ >o, 4<i, lfi<i-

"l Kz)-KO

PROOF. Take two arbitrary points Zi,Zz in |2J<1, and denote by L the image curve of the line-segment

(<1) which encloses L.

Then there exists a positive number a such that

(zO, ^(z2) under 2=fS '(w). Describe a circle jzj=p

-a

Setting g(z)=f'(z)/<!>'(z)-a, we have

j(/'(z) -af*'(z))fife=j^(z)*5'(2)<fe, \g(z)\<a.

Hence |/GZi)-/(zO-«(*0&WO*i)) '<«! ^(^O-^U)!, that is

/(&) -/(«!)

Therefore

st

(*U2) -(*(2i) -a\<a.

{Kz2) - {KzOå o,

and (1.1) is proved.

L emma 2. If $(z) is convex in 'z ;i, then f5(z)-$K~z) is there starlike

8 ^{Koichi Sakaguchi} and Shigeyoshi Watanabe

with respect to the origin.

PROOF. Since 9ftC(^/(2))7^/(2)D>0, |z,'<l, from Lemma 1 we have z{f{z)+f{-z))

Si ^{>o, :i.}

Kz)-K-z)

Therefore (4(z)-j}(-z) is starlike with respect to the origin in |z|<l in view of the fact that it has an expansion of the form 2<2iZ+ , Oi^fO-

Lemma 3. Let #(?)=«å + be analytic in |c[<l, and let #(c)/r^vO, jf|<l.

If for a positive constant k, ^(c) satisfies

<iargg-(p8^)>-&7r, O<P<1, <Pi<9>2,

icpi

then for q=i?<l we have

(1.2) R(l~R)k ^o_{Y,-M <T} R(l+R)k

=s isv>/| ^

(1.3) (1.4)

\~R2 l-2(k+l)R+R-

£(0

- l-R2

<L «Rcs-'(c) ^ i+2(&+i)#+#2

1-A)2 ^ff(c) 1-R2

all equalities being attained by the function sr0(f) =c(l+?)V(l-c)2+t.

PROOF. As shown by Sakaguchi (2T], #(c) has a representation of the form

(1.5) £(O =s(OXOs,

where s(?) is starlike with respect to the origin in [f|<l, and /»(r) has there a positive real part. From (1.5) we easily obtain all the inequalities of this lemma with equality signs appearing for ^0(f) ; C2j, C3j.

Lemma 4. If k^2 or k~\, then under the same assumptions as in Lemma

3 we have for \<;\--=R<k+l-y'k(_k+2)

«"<»

(1.6) ₁ / 1

+ _£'W ^{2i1 ~u} 5ff(sU, _{I \} _ 2_^1+R2

£(0'Vx kjl-R*

< 4R{k+l+(ks-4k-2)R+(k+l)R2

- ka-R2){i-2(k+i)R+R2} ,

equality being attained by the function #o(c)=c(l +OV(l~f)2+i:

PROOF. From (1.5) we have

(1.7) 9t| s r+a ^'e\ ?+a ^I/* _{å o,} ^\a\- 1, W<1.

l+azJ/ °Vl+ac,

We may assume without loss of generality that (1.7) holds for |cj^l. Setting

S(f) is starlike with respect to the origin in |f|^l, and from (1.7) it satisfies

(1.8) ^{r_ _ .} / fJi-m \~Vk

dt \ S(rt(a+rt(iL +art/cel -±-!-Z-\

JV1+cKr+ac ^j ^{0, |f^l.}

The expression in the brackets has the following Taylor expansion in |c|f£l:

c{ l+]-(»,-«,+A +S) c+i[*,-«,-|(l-|>l+i(l+|-)«S

where c= 'is{a)/g{ayj/k and

S(O=S^(c+kc2+&sC3+ ), g(4^-) =g(a)(l+flif^aaf+ )•E

i~r«-* /

a

Therefore by Caratheodory's theorem we have

(1.9) b2-<?!+-+aa <2k,

(1-10)

6a-«a ~(1- )^«+4-(V

-i) al-\aA

_l/i IVIx^^h _A +af+±{b2-a1~){±+a)+^ <2k.

On the other hand from (1.8) we have

The expression in the brackets has the following Laurent expansion in 0<C|fS^l:

A^,r,

!+1^-*+<*-i>(i+s)}Hf*-fc-!(i-i>;

n^UiL +-!(i+!)«

1-i)(^)(v+«)+^-1#

-i-'^-m-^)

where c1=a\ig{a)/s{a)J'k.

Therefore by Robertson's Lemma T4J we have

(1.ll)

«.-h-J-( ^-

i>-

+(1_^)(ai_ftl)(I.+5)+ (*_!)!

<k+2 a1~b2+(k-l)[~L+a 1 -\

a

Combining (1.ll) with (1.10) multiplied by k-\, we have

k(a2- ^- «+l(i- ^{- l+kjbr: ^}

Ki-1-*)^' i-|)^+2(i-^;

(«!-&)+y(-|-3+^ _.<x

2\k

From (1.9) we may write

(1.12)

^*(2*-l)+2

-I \

ai-62+ (A-l)(-i-+a) a

-+a=Iks+d-b2,^{å <l.}

10 ^{Koichi Sakaguchi} and Shigeyoshi Watanabe

Substituting (1.12) for a 1+a in the last inequality, we find a2-b3+b22-a1b2+2(k-l)ea1-2(k-l)eb2+2(k-l')(k-2)e2

<.2\a1-b2\ +6&-5.

Therefore under the assumption that k~2i2 or k=l we have

\a2-aA+2(&-1)eal\<2\a1\+2k2+Ak,

since \b2'\<2, b22~bj{<l because of the starlikeness of S(f).

Substituting {a~l-\- a-a1+b2)/2k for e in this inequality, we find

a2 1

-(1-4- Ha,-

"1

1+^\a\

a

<2(l+-^)+2&(&+2)K1 \

/ j"ll

Since

«=*$-a-*>. *- r$ «d<)L|g>a(1-K), 0<k<i.

it follows that

1+*g0(A)_9{, iyg'W ( 2\i+K

s k"-j \ «å / gyvcj kj l-\a\

<A\l+l+k(k+2)

JL -I* K

\a\ #(«) ^}å

k 1-I«P I «*'(«)

which holds also for a=0. Using the left-hand inequality of (1.3) and replacing

<* by c, we have (1.6) for !c|=R<k+\-/k(k+Z).

The statement concerning equality can easily be verified.

§ 2. Properties of close-to-convex functions

Theorem 1. Let f(z)=z+ be analytic in 'z,<ll. A necessary and suff-

icient condition for f(z) to bs close-to-convex in \z\<l is that there exists a convex function <j>(z) defined in |z<i such that

(2.1) SR

/(z) -/(c)

*0z)-*(O

PROOF. From Lemma1 the necessity is clear. The sufficiency can be shown by making c-^z in (2.1).

Theorem 2. Let f(z)=z+ be close-to-convex in 'z\<\. Then the func- tion F(z)= (/(z)-/(-2)) is close-to-star in jzi<l, and satisfies for.z=r

(2.2) (2.3)

(1+r2)2 ='rw'= (l-r2)2 ,

1-6^+r4^,=S i-i- K*J\..,^£S 1+6^+/

(i+r>y (1-r2)3

(2.4) (2.5)

1-f1 = F(2) =

1+6^+y4

1-r4

1+6^+r4

1-r4 ;

all equalities being attained by the function fo(z) =z/(l -z)2.

PROOF. From Theorem 1 there exists a convex function jiS(z) defined in 2<1 such that

By Lemma 2, Kz)-^(-2) is starlike with respect to the origin in 2<1. There- fore F{z) is close-to-star in j2<l T5J, and odd. Thus if we set g(<;)=F(Kz)'1,

c=22, then g(c)/?^O, Cj<l, and

«>2

r2rfargg-(pO>-2<~, O<P<1, <P!<<Pz.

Jn

Hence g(c) satisfies (1.2), (1.3), and (1.4) for k=2- These inequalities yield (2.2), (2.4), (2.5), and (2.3) with aid of the relations g(?)=F(z)2, r#'(c)/

g(<;) ~zF(z)/F(z), f=22. The statement concerning equalities can be easily ve- rified.

Theorem 3. Letf^z)=z+a2z2+a3zi+ be close-to-convex in ,2<1. Then

the function F(z)=z+a3z3+ which consists of the odd terms of f(z) is

univalent and starlike with respect to the origin for

(2.6) |2i<-/2-l,

and convex for

(2-7) j2 <-,/6--i/5.

Moreover all the partial sums of F(z) are univalent and starlike with respect to the origin for

(2.8) ]*<l/3,

and convex for

(2.9) 2<l/3/3.

These bounds are all sharp.

PROOF. The bound (2.6) follows at once from (2-5). The bound (2-8) can

be proved by the same argument as used in [[1] by making use of the estimates in Theorem 2 and 'a2»+i!^2«+l; [&}.

Next by Lemma 4 the function g(r) defined in the proof of Theorem 2 satisfies

1+ ^c£"(0_*f(0 ^c£'(c)_g(.e)

-2SO^K+3^)_ R -

W ith aid of the relations 2Cl+r£//(O/£/(O-c£XO/£(c)>=l+2F//(z)/F'(z)-

12 K6ichi Sakaguchi and Shigeyoshi Watanabe

zP(z)/F(z), <g'(c)/g(<)=zFi(zyF(z), <;=z\ and the estimates (2.4), (2.5), this inequality yields for \z\~r<.-i/2-1

(2 10) ^(1+^F%z)\ 1-24^+46^-24^+^

and

(2.ll) ^z F"(z) ^<r ^'_s2r'(9-21r?+ 3/+r5)

F {z) (l-O(l-6r3+/) •E

From (2-10) the bound (2.7) follows. The bound (2.9) can be proved by the

same argument as used in £lj bymaking use of (2.10), (2-ll). The sharpness of these four bounds can be easily shown by considering the function fo(z) =z/

(I-*)2.

We here remark that the bounds (2-6) and (2-7) can be extended to more

general functions, in other words the following theorem holds:

Theorem 4. Let a p-fold symmetric function F(z)=z+ap+1zp+1+ be

analytic in \z\<\, and let F(z)/z^0, ]z\<l. Ifforapositive constant k, F(z) satisfies \ JargF(re'9)>-kn, 0<Xl, (9i<^2, th?.i F(z) is univalent and

J«i

starlike with respect to the origin for

and if further pk^2 or pk~1, then FQz) is convex for

!2:<{M-1/M2-1}1^, M=-£{(p+2XPk+l)+Pi/ Kp+4XPk+2)+l ).

These bounds are sharp.

A proof can be given just as in the proof of Theorem 3 by applying Lemmas 3, 4 to the function ^(c)=F(z)p, c=2p which satisfies #(0/f^O, ic,<l, and

[ ^d&rggW^-pknt 0O<l, <Pi<<p2.

The details therefore will be omitted.

References

C1D K. Sakaguchi, On functions starlike in one direction, J. Math. Soc. Japan, 10 (1958), 260- 271.

C2} K. Sakaguchi, A representation theorem for a certain class of regular functions, J. Math. Soc.

Japan, 1j (1963), 202-209.

C3'J K. Sakaguchi, The radius of convexity for a certain class of regular functions, J. Nara Gakugei Univ., 12 (1964), 5-8.

CC) M. O. Reade, On close-to-convcx univa'.ent functions, Michigan Math. J. , 3 (1955-56), 59-62.