奈良教育大学学術リポジトリNEAR
A Property of Convex Functions and an Application to Criteria for Univalence
著者 SAKAGUCHI Koichi
journal or
publication title
奈良教育大学紀要. 自然科学
volume 22
number 2
page range 1‑5
year 1973‑11‑15
URL http://hdl.handle.net/10105/2736
Bull. Nara U. Educ, Vol.22, No.2, (Nat.), 1973
A Property of Convex Functions and an Application to Criteria for Univalence
Koichi Sakaguchi
(Department of Mathematics, Nara University of Education, Nara, Japan)
(Received March 9, 1973)
1. Introduction
WedenotebyD the open unit disc. Marx [1] proved that if /(z)=z +S anzn is
re=2
analytic and convex in D, then f'(reie), 0<Ir<l, is contained inthe image domain of the closed disc {2; \z\<Lr) under the function (1-2)~2, and it lies for r=^=0 on the boundary of this image domain when and only when /(z) is of the form z(l-ez)"1,
1*1=1.
If the above convex function f(z) is p-wise symmetric, i.e. /(z) is of the form /(z) = z+ J±ani)+lznP+1, then as is easily seen the function 5(2)= Pc/'C^"2")^2 is also analytic and convex in D. Therefore it follows from the above theorem of
Marx that f'(reie), (h£><l, is contained in the image domain of the closed disc {z; \z\<Lrp) under the function (1-2)~2/p, and it lies for r^O on the boundary of this image domain when and only when /(z) is a function of the form
[\l-tzPy*» dz, |e|=l.
J 0
In this paper we shall show that the same conclusion holds also for convex functions of the form f(z) =z+ J2=J)+1S anzn which are not necessarilyp-wise symmetric, and we shall apply this fact to the study of criteria for univalence.
2. Lemma
Lemma, Let <j>(z)= XI bnzn, bl =^0, be analyticand convex in D. Letf(z)=Jl anzn,
«=1 n=p
p^l, be analytic in D and take there values in theimage domain of D under $(z).
// we set
F(z)= C"£^-dz, 0 (2)=
J O Z
Jr 0
0 (2)
J O Z dz,
then F(reie), (feS><l, is contained in the image domain of the closed disc {z; \z\<^rp}
under 0{z)jp, and it lies for r^O on the boundary of this image domain if and
only if f(z) is a function of the form <j)(ezp) |e|=l.
Proof. The function <f>'1(f(z))=apbr1zp+ is analytic and satisfies I 0"1(/(*)>
1<1 in D. Therefore by Schwarz' Lemma we have |^-1(/(2))l^|z|p for |z|<l, 1
K6ichi Sakaguchi
equality holding for 2=^0 if and only if <j>'l{f(,z))=szp, |e|=l. Hence /W), 0^,t<^l, is contained in the image domain of {2; \z\^tp} under <j>(z), and it lies for t^fO on the boundary of this image domain if and only if f(z)-<j>(zzp), |e|=l.
We now denote by w0 the point in which the image curve of the circle {z; \z\ =r]
under w=<j>(z) cuts the positive real axis on w-plane, and denote by C= {z; z=tpeie{t\
0<^<^T} the image curve of the straight line segment joining 0 and w0 under z=<j>-i(w), where 0(t) is taken so as to be continuous in t. 0(t) is then differentiate with respect to t. We consider another curve Cx= {2; z=tem(t)/p, O^t<Lr} corresponding to C, and put za=reiB{r)lP.Then we have
(2. 1) ReF(20) =Re f ^-^dz=A\ Reå ff^\ !rpJo rf Cte^W)-J(.p+it0'(t)) \dt.
pjo""l
Since f(teie<-t]/p) is contained in the image domain of {2; \z\^Ltp} under <j>(z), wehave Retf(teilxw) <ip+itd'(ty)~)< ma.x Re t<j>(tpeix) (p+it8'(t~))-},
0<x<2it
where equality may be attained for t^fO only by the functions f(z)=<j>(ezp), |e|=l.
When t^-0, the point <j>(tpeix) (p+it0'(t)) describes a convex curve surrounding the
origin as x varies. Therefore there exists only one value eiX which maximizes
ReC<f>(tpelxXP+tt0'(t))^- If we denote such eix by eioi, then eict is characterized by the following two conditions:
(2. 2) (2. 3)
ReOKW) (P+itO'itmyO, d
Rewriting (2. 3)
(2. 4)
å ^--Re{0(^e'-) (i>+iW(O)}Jjf_ a =O.
we have
whence el<* is determined by (2.2) and (2.4).
On the other hand, since <j>(tpeie{t)) lies on the positive real axis, we have
(2. 5) Im0(Wff)) =O,
(2. 6) Re<KW8W) >0.
From (2.5) and(2.6) we find
(2. 7) Re(.<t>(tpeim) (p+it0'(t))^O.
Furthermore differentiating (2. 5) with respect to t, we have
(2. 8) ImC0'(^e<ew) (/>+«W(O)ci9(/O =O.
Comparing (2.7), (2.8) with (2.2), (2.4), we see that g«»=<?<•E<>(0.
Therefore from (2. 1)
-w. c2*(*)z dz, c2={g;z=tPeiew, O^Lt^r],
=Re$ (z0P) lP^ max Re®(z)lp,
where the equality sign of the first inequality may appear for r^=0 only when /(2)=0(e2*), |e|=l.
Next setting /i(2) =/(e2), |e| =l, the function /x(2) also satisfies the hypotheses
A Property of Convex Functions and an Application to Criteria for Univalence 3 of the lemma, and hence it holds for F1Cz)= f'cfiWl&dz that
*s 0
Re Fx(zo)< max Re 0(z)lp.
In view of the fact that F(s20)=Fi(z0), we thus have max ReF(2)^ max Re<P(z)jp.
Since the hypotheses of the lemma are satisfied also by the pair of functions 7<j>(z}
and Tf(z), \Y\=1, we finally have
(2.9) max Re(YF(z)1< max Re (ri>(z)lp~), \T\=1.
We here note that 0(z) is also convex in D. Since 0(z)lPmapsthecircle {z; \z\=rp}
onto a convex curve, (2.9) deduces that F(rem), 0^d<2it, is contained in the image
domain of the disc {z; \z\^rp] under $0)Ip. Moreover from the above discussion
it is clear that F(reiS) may lie for r^Q on the boundary of this image domain only
when /(2)=0(£2p), |e|=l. On the other hand when f\z)=(j>{szp), |e|=l, we see
that F(reiB), r^O, lies certainly on the boundary of this image domain. Thus the lemma is proved.
3. Main theorem
We denote by C(a,p) the class of functions f(z) of the form f{z)-z+ 2 anzK
which are analytic and convex of order a in D (2~), in other words, which satisfy 1+Re(.zf"(z)If(z)1>oe in D for a constant a such that 0<a<l.
Theorem 1.Let /(2)EC(a, p). Then f'(reie), 0<ir<l, ts contained in the image domain of the closed disc {z; \z\S^rp} under the function (1-2)2<«-h/p, and it lies for
r^Q on the boundary of this image domain if and only if f(z) is a function of
the form f\l-ezpy(x-1>/pdz, |e|=l.
Proof. Since ReCz/"(2)//'(2)3>«-l for |z|<l, the function zf"(z)//'(z)=p(p-h
l)aj,+iZp+ takes values in the image domain of D under the function 2(1-a)zj
(1-2), which is convex in D. Consequently by the lemma it follows that log/'(yeie~), 0s£><1, is contained in the image domain of {2; \z\<,rp) under 2(a-Y)p~x log(l-z), and it lies for r=^=0 on the boundary of this image domain if and only if zf"(z)\f
(2)=2(l-a)s2p/(l-e2^), |s|=1, where log is understood to be that branch which
vanishes at the point 1. Thus the conclusion of the theorem follows.
Remark 1. The function (1-z)^<*-w» which appears in Theorem 1 is univalent and convex in one direction in D. In particular when 2a+p^2, it becomes convex.
Corollary. Let f(z) EC(a, p). Then
(3. 1) (arg/'Cre1*) |^-(1. -^sin-'r^ 0<r<l,
where arg/'(O)=O, sirr1 0=0. Equality is attained for r^O by and only by the functions of the form f\l-szpyia-1)/p dz, |e| =l.
This is an immediate consequence of Theorem 1.
•E4 Kbichi Sakaguchi
When p=l, Theorem 1 and Corollary reduce to results of Pinchuk £6].
Remark 2. As is easily seen the two conditions /(z) £ C(a, p) and
J [/'(2)1/(1~"W2 £ C(0, ^) are equivalent. Therefore Theorem 1 is equivalent to the following:
Theorem V. Let /(2)£C(0, p). Then fire16), 0<r<l, is contained in the image domain of the closed disc {z; \z\<rp] under the function (1-z)~v, and it lies for r^O on the boundary of this image domain if and only iff(z) is afunction å of the form (\l-szp)^vPdz. [el=1-
J0
4. Application
Let /(2)=2+ 2 anzn be analytic in {z; |z|<i?}. Ozaki C3] proved that both (4. 1)
and
(4. 2)
Re(1+2TOr)>-i' lzl<R'
Re(1+zTw)<i' |2|<*'
are sufficient conditions for f(z) to be univalent in {2; |2|<i?}, and both
(4.3) R e( l+2/"(2) >
- T \z\<R.
and
(4. 4)
/'(«)
H1+zT§)<P-2-2' \ z l<R>
are sufficient conditions for f(z) to be at most ^-valent there. Afterwards Umezawa [4] showed that (4.1) and (4.2) are sufficient conditions for f(z) to be convex in one direction in {z; |z|<7?}.
By using (3.1) we shall show that for functions of the form f(z)=z+ 2 anzn
both (4.3) and (4.4) bscome sufficient conditions for the univalency and the close- to-convexity.
Theorem2. Let/(z)=z+ f] anzn be analytic in {z; |z|<7?}. // f(z) satisfies one
n=p+l
of the conditions (4.3) and (4.4), then f(z) is univalent and close-to-convex there.
Proof. Since we can consider the function f{Rz)jR instead of /(z), we may
assume without loss of generality that R=\. Suppose that f(z) satisfies (4.3) for R=l. Then f'(z) has no zeros in D, and so the function
<4i00= Vf'{zynp+2)dz=z+2(p+2)-ia!1+lzp+l+ is analytic in D, and satisfies
J 0
Therefore (j>i(z) is a member of C(0, p), and by Corollary of Theorem 1 we have
Hence
A Property of Convex Functions and an Application to Criteria for Univalence
|arg0'1(z) |^j,sin-i|2r, |*|<1.
/'(2) !_
arg l'(2)
=|arg^1'(2)p/2 ^sin'M2!"<-£-, TV \z\<l
so that f(z) is univalent and close-to-convex in D {S}.
Next suppose that f(z) satisfies (4.4) for R=l. The function
^2(2)= I /'(2)^2/prz dz=z-2p"1aJ>+1zp+1+ is similarly analytic and convex in D.
*/ 0
Hence
|arg^'(«) |^-|-8in-1|2|1>, |«I<1,
sothat
arg/'^ l ^ larg^'^-^l^sin-ilsl^y, |z|<l.
Therefore f(z) is univalent and close-to-convex in D. We thus complete the proof.
References
C U A. Marx, Untersuchungen iiber schlichte Abbildungen, Math. Ann. , 107 (1932), 40-67.
C2} M. S. Robertson, On the theory of univalent functions, Ann. of Math., 37 (1936),
374-408.
C33 S. Ozaki, On the theory of multivalent functions, II, Sci. Rep. Tokyo Eunrika Daigaku, 4 (1941), 45-87.
C4] T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan, 4 (1952), 194-202.
£53 W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), 169-
185.
C6] B. Pinchuk, On the starlike and convex functions of order a. DukeMath. J., 35
(1968), 721-734.
