I ntnat. J. Mh. & Math. Sci.
Vol. 6 No. 2 (1983) 327-334
327
ON A GENERALIZATION OF CLOSE-TO-CONVEXITY
K. INAYAT NOOR
Mathematics Department
Science College of Education for Girls
Sitteen Road, Riyadh, Saudi Arabia
(Received July 9, 1979 and in revised form August 30, 1982)
ABSTRACT.
A class T k of analytic functions in the unit disc is defined in which the
A necessary condition for a function f
concept of close-to-convexity is generalized.
to belong to
Tk,
raduis of convexity problem and a coefficient result are solved in
this paper.
KEY WORDS AND PHRASES. Close-to-convex functions, functions of bounded boundary rotation, necessary condon, radius of convexly, generalized Koebe function.
1980 tTHEMATICS SUBJECT CLASSIFICATION CODES.
1
Primary 30A32, Secondary 30A34.
INTRODUCT ION
This paper is directed to mathematical specialists or non-specialists familiar
with multivalent functions [i], and to close-to-convex functions [2].
Let Vk be the class of functions of bounded boundary rotation and K be the class
of close-to-convex functions.
We generalize the concept of close-to-convexity in the
following direction.
Definition.
Let f with f(z)
f’(z)
Then
0.
feTk, k>_2,
Re
It is clear that T 2
cz
+
n
Y.
n=2
a z
n
be analytic in E
{z:Izl<l} Icl=l
and
if there exist a function geV such that, for zeE
k
f’(z)
>0.
g’(z)
(1.I)
K.
E
Using a method by Kaplan [2], we have
THEOREM i.
Let
fgT
Then with z
k.
0
2
f.-l’(z)
iRe I(zf’(z))’
1
re
dO >
i0
and 0
k
2
1
< @
2
(1.2)
K.I. NOOK
328
REMARK i.
From theorem i, we can interpret some geometric meaning for the class T k-
For simplicity, let us suppose that the image domain is bounded by an analytic curve
C.
At a point on C, the outward drawn normal has an angle
arg[ei0f
(e
i0
].
(1.2), it follows that the angle of the outward drawn normal turns back at
This is a necessary condition for a function f to belong to T k.
.
Then from
most
k
It will be inter-
esting to see if this condition is also sufficient.
RE,LARK 2.
Goodman [3] defines the class K(B) of functions as follows.
Let f with f(z)
if for z=re
i6
+
z
and
Y.
a z
n
n=2
61
62
<
n
be analytic in E and f’(z)
0.
Then for B>0, feK(6),
02
iRe(zf’fI(z)(z))’]
d6 > -8
1
We note that
2.
TkCK().
MAIN RESULTS
From remark 2 and results given in [3] for the class K(B), we have at once
THEOREM 2.
(i)
Let feT
k-
Denote by L(r,f) the length fo the image of the circle
A(r,f) the area of
f(Izl=r).
Then for
Izl
r under f and by
0<_r<l,
<_
(a) L(r,f)
L(r,Fk)
_< A(r,Fk)
(b) A(r,f)
where F is defined by, for zeE,
k
Fk(Z)
(-- [\ i_-I
z
+
A (k)z n
I
(2.1)
n
n=2
and clearly Fk e T k.
lanl
(ii)
<
An(k)
n
2,3
where A (k) is defined by (2.1).
(iii)
For z
re
i6
(l-r)
k
>__2
This result is sharp for each n > 2.
0 < r < i,
1/2k
(+/-+r)k+2
<
f’(z)
<
(--+[-)1/2e
(l-r) 1/2k+2
These bounds are sharp, equality being attained for the function F
k
defined by (2.1).
GENERALIZATION OF CLOSE-TO-CONVEXITY
329
We also need the following result.
Lemma 1 [4].
Let ggVk.
Then there are two starlike functions s and s 2 such that
I
for zgE
.gk4
g’(z)
(Sl(Z)/Z)
(s2 (z)/z)
THEOREM 3.
fgT
k-1/2
k if and only if
(k (z))
f’(z)
i
(k (z)) k-1/2
kl k2ek
From definition i, we have
PROOF:
f’(z)
g’(z)h(z),
geV k and Re h(z)>0.
Using lemma I, we know that there are two starlike functions s
(s(z)/z)
g’(z)
(s2(z)/z)
Thus
(s (z)/z)
f’(z)
(s2(z)/z)
and k
I
Lemma 2.
2
and s
2
kk-1/2
1/2k
((Sl (z)h(z))/z)
k_. h (z)
((s 2 (z)h(z))/z)
1/2k
1/2k-1/2
are two suitable selected close-to-convex functions.
Let H be analytic and be defined as
H(z)g’(z)
(zg’(z))’, ggVk and H(z)
7+
hl(Z)
h2(z)’
Re ho(Z) > 0, i=1,2, h (0)= 1
i
Then
2
[H(z)
2
dO
<_
IH’(z) Id0
<
1
+
(k2-1)r
l_r2-
2
(z
re
i0
0
and
2i
1
2
l-r 2
0
PROOF:
such that zgE,
k
(k (z) k+
(k (z))k-1/2
where k
I
By the representation formula due to Paatero [5], we can write
K.I. NOOR
330
+ ze it d(t),
+/-
1
H(z)
+/-
-ze
2-0
where
27
2, and
da(t)
Id(t)
0
I + I
Let H(z)
-
Th en
c
i
n
ICn]
<_
k
0
<
c z
n
n=l
n
27
[
-in
e
td.t,
and so for
n>_l,
0
I
Id(t)
< k
Thus
2
i
2
IH(z)l
2
E
n=0
de
0
Aso
ICnl
2 2n
r
2
2n
(k2_l )r 2
i+
Y. r
(l+k
<
l_r2
n=l
2
i
H’ (z)
e
(l_zeit
2
d(t)
Thus
27
i
2
iH,(z)id e
THEORF21 4:
<
ll-rei(e+t) 12
Let feT k.
Then for
deld(t)
<
l,r’2
Id(t)
<
l_r 2
n>_l,
2
< c (k)n
where c(k) is a constant and depends only on k.
PROOF:
Since
feTk,
f’(z)
we have for zgE,
geVk
g’(z)h(z),
and Re h(z)>0
Set
F(z)
z(zf’ (z))’
where Re h(z) > 0 and
H(z)g’ (z)
i
Thus, for
E and n>l;
zg’ (z)[H(z)h(z) + zh’ (z)],
(zg’ (z))’,
2(z)’
with
Re h i(z) > 0, i=1,2, h i(0)=l
(2,2)
GENERALIZATION OF CLOSE-TO-CONVEXITY
331
2
(n+l) 2
an+l-n
I-1
anl< 2rn+l
and by using lemma i and (2.2), we obtain
k-
2g
{n+t)
2 an+l-n 2 anl<
f
iz_
0
where
Sl,
s
is i (z)
IH(z)h(z)
Is2(z) lk-1/2
+ zh’(z)
Id0
(2.3)
are starlike functions.
2
It is well-known [i] that for starlike function seS,
(l+r)
(l-r)
(2.4)
2
Then by a result of Golusin [6,p162], there exists a z I with
Let 0<r<l.
Zll
2
z[
r such that for all z,
IZ-Zl] ISl(Z)
r,
2r
<
l_r
2
(2.5)
2
From (2.3)-(2.5), we have
hk-1/2
(n+l)
2
an+l-n2 an I<-- n+ll (i<k-1/2(i!r’22)(
)r
(2
2r
NOw as in [7], we have with z
re
lh(z)12d0
l+3r
2
l_r
<
0
I
]H(z)h(z)+zh
(z)
Id0
(2.6)
i0
.2
1
2n
2
2
(2.7)
and
2i
2
flzh’(z) Id8
0
<
where Re h(z) > 0.
2r
l-r
2
Aso
2
1
2
IH(z)h(z) + zh’ (z)Id8
<
<
2
2
]H(z)h(z)[dO +
Izh’ (z)Id6)
(l+(k2-1) r2)1/2 (i+3r2) 1/2
l-r
2
+
2r
l-r
(2.8)
2
by using Schwarz’s inequality, lemma 2 and (2.7).
Hence from (2.6) and (2.8), we have
(n+l)
2an+I
1 21/2k
nl<_ rnfl
2
-n a
[(l+(k2 -i) r2 1/2+i]
1
(l-r) 1/2k+l
K.I. NOOR
332
2
and so choosing
II
fan+ll-lanl
n
we obtain for
r
l+(k2-1)r
<
+
n>_l
1
n
\3!
Thus
1/2k-I
<_ c(k)n
lan+ll-lanll
(}- I)
The function F defined by (2.1) shows that the index
k
We now evaluate the radius of convexity for the class T
THEOREM 5:
Let feT
k.
is best possible.
k.
Then the radius R of the circle which f maps onto a convex
domain is given by
1/2k+2>-X/k)],
R
The function F
k
defined by (2.1) shows that this result is best possible.
2, R
ticular, when k
2aV--,which
In par=
This result also follows from
is well known.
the remark in [3,p.23].
PROOF:
By definition
ag’ (z)h(z)
zf’(z)
geVk;
Re h(z)>O.
Thus
(zf’ (z))’
f’(z)
(z)
+ zh’
h(z)
(zg’ (z))’
g’(z)
and so
(zf’ (z))’
f’(z)
For
geVk,
it is well known
Re
(z))
Re (zg
>
izh’(z)
hz)
g’ (z)
[9] that, for z
(zg’ (z))
g’ (z)
>
re
i8
0<r<l,
r2-kr+l
l_r
2
Hence
Re
(zf’ (z))
f’(z)
>
r2-kr+l
l-r
2
2r
l-r
2
2
r -(k+2)r+l
2
l-r
This gives the required result.
(i).
We also note that the extremal function
function as
FB(z)
Fk(Z)
defined by equation (2.6) in [3].
defined by (2.1) is the same
As A. W. Goodman has pointed out
that this function is sometime referred to as the generalized Koebe function.
GENERALIZATION OF CLOSE-TO-CONVEXITY
(li).
333
We conjecture that the class T is a proper subclass of the class K(B) as
k
defined in [3], since in the definition of Tk, gV and we know that
k
gVk, 2<_k<__4,
is
convex in one direction and all the functions in one direction form a proper subclass
of the class of close-to-convex functions.
(ii).
It remains open whether T
ACKNOWDGEMENT.
k
is a linear in variant family.
The author wishes to thank Prof. A. W. Goodman and the referee for
their interesting and helpful comments, which influenced the final version of this
paper.
REFERENCES
i.
HAYMAN, W. K. Multlvalent Functions, Cambridge University press, 1967.
2.
KAPLAN, W. Close-to-convex Schilcht Functions, Mich. Math. J. 1 (1952), 169-185.
3.
GOODMAN, Ao W. On Close-to-convex Functions of Higher Order,
E6tvos. Sect. Mathematica, 15 (1972), 17-30.
4.
BRANNAN, D. A. On Functions of Bounded Boundary Rotation, Proc. Edln. Math. Soc.
2 (1968/69), 330-347.
5.
PAATERO, V. Uber Geblete von beschrankter Randdrehung Annal/Acad. Scl. Fenn.
Set A 37 (1933), 20pp.
6.
GOLUSIN, G. M.
7.
POMMERENKE, Ch. On Starlike and Close-to-convex Functions, Proc. Lond. Math. Soc.
3 (1963), 290-304.
Univ. Sci.
Budapest.
Geometrische Functionstheorie, Berlin, 1957.