799
Internat. J. Math. & Math. Sci.
VOL. 18 NO. 4 (1995) 799-812
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
V. SRINIVAS, O.P. JUNEJA and G.P. KAPOOR
Department of Mathematics
Indian Institute of Technology
Kanpur-208016
India
(Received May 31, 1991 and in revised form September 7, 1993)
ABSTRACT.
Given 0 s R 1 s R 2 s =, CVG(RI,R2) denotes the class of
normalized convex functions f in the unit disc U, for which af(U)
satisfies a Blaschke Rolling Circles Criterion with radii R 1 and R 2.
Necessary and sufficient conditions
for R
R2, growth and
1
distortion theorems for CVG(RI,R2)
rotation theorem for the
and
class of convex functions of bounded type, are found.
KEY WORDS AND PHRASES.
univalent functions, Convex functions,
Curvature, Subordination, Distortion theorems, Growth theorems.
1991 uMS SUBJECT CLASSIFICATION CODES.
30C45, 30C55.
1,
INTRODUCTION.
Let S be the class of functions f(z) which are analytic and
and have the
univalent in the unit disc U
< 1
z:
z
0
S
normalization f(O)
and
r
(O,l),the
f’ (0)-I. For f
radius of curvature, p(z) of the curve f(Izl
r) at the point f(z),
is given by [6],
z’ (z)
zf"
Re(l + 7 (z)
p(z)
(z))
of
re i8
Goodman [2] introduced the class CV(R I,R2)
z
functions f(z) having p(z) restricted as Izl tends to i. Thus, let
where
p,(r)
min
R,
lim
r 1
,
p(z),
max
p (r)
p(z)
and
(I)
p,(r)
R
,
lim
r-- 1
DEFINITION I.
p (r)
-
Let R] and R be fixed in [0,=}. A function f
S_
2
be in the class
if R
and R
R
where
R,
1
2
R, and R are as in (i). For 0 < R I
a function f
is called a convex _function of bounde_d_type.
is said
to,
CV(R1,R2)
R2<
A
function,
and R
f(z) is said to be in
where R,
are as in (1).
For functions f(z) in the class
,
-Q(RI,R2)
CV(RI,R2)
,
CV(RI,R2
if,
RI=
R,
and
R2=R
,
Goodman [2] obtained
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
800
(i)
first
the
approximation
moduli
the
for
of
the
Taylor
coefficients, (ii) covering theorem and (iii) bounds for d, where d
is the distance of af(U) from the origin, in terms of R
and R 2.
1
Goodman [3], Wirths [8] and Mejia and Minda [4] extended this study
by finding certain other interesting properties of functions in the
class
CV(RI,R2).
Styer and Wright [7] introduced the following class of functions
based on Blaschke’s Rolling Circles Criterion:
DEFINITION 2for each
R2,
0 s R
n
1
and
2
R 2 z i,
let
with the property that
and D2(n of radius R 1
in
af(U) there are open discs D l(n)
respectively, such that, n e aDl(n N aD2(n) and
D I()
If R
R
1
be the class of functions f(z)
CVG(RI,R2)
and
Given
0 or R
2
,
DI(W)
f(U)
_
and
D2(n
-
D 2(D).
are to be interpreted as the
empty set and an open half-plane, respectively.
It follows that [7]
CV(RI,R 2)
K CV
CVG(RI,R 2)
where, CV is the subclass of functions f(z) in the class S,
f(U) is convex.
for which
Mejia and Minda [4] showed that, in fact, CVG(0,R2)
CV(O,R2).
for
R 1 > 0, whether CVG(RI,R2)
CV(RI,R2) still holds,
remains an open problem. The difficulty to settle this problem lies
in the fact that, for
f
CVG(RI,R2), R 1 > 0, the radius of
curvature p(z) of the curve f(Izl
r) at the point f(z) may not
be a continuous function on U
z
Izl 1 ), (see [7]).
Let g(z) be analytic and univalent in U.
A function f(z)
g(z))
analytic in U, is said to be subordinate to g(z) in U (f(z)
if f(O)
g(O) and f(U)
g(U).
For a function f(z) in S, the unit exterior normal to the curve
where
n(z)
r) at the point f(z) is
zf’(z)/Izf’ (z)
f(Izl
univalent
normalized
a
that
r
(0,i). Styer and Wright [7] found
U
function f
CVG(RI,R2), if and only if, f CV, and for every
However,
-
for which f()
is finite,
(2)
and, in the case
(3)
where
f(U)
D(f()
R2n(),R2)
R 1 > 0,
D(f()
Rln(< ,RI)
f(U).
D(a,R) is the open disc of radius R cenetred at a.
For a function f(z) in the class
CV(RI, R2) Goodman [2]
obtained bounds for d and d* where d and d * are respectively the
distances of the nearmost and the farthermost points on af(U)
from
the origin. Thus he proved that
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
R2
_]
R2
R
2
and
(5)
R1
d s R
2
-
1
801
RI2_R1
2
2d
1
-<
R2
where the right hand side inequality in (4)
-+ f"
.v,.-’’-’2-,- (5/’ ,2f, f
rurther’
and the left hand side
i
d*
(6)
-
R2 +
J
2
R2
R2
Styer and Wright [7] observed that inequalities (4) and (6) continue
to hold for the class CVG(RI,R2). The method of proof of inequality
(5) in [2] shows that this inequality also holds for the class
CVG(RI,R2)
and is sharp. These inequalities are necessary conditions
and R
in terms of d
d(f) for a function f(z) to be in the
1
2
class CVG(RI,R2). However an analogue of these conditions in terms of
d
is not known.
distortion
lower bound on
Further,
If( z
on R
,
,
properties involving d or bound on larg f’ (z)
for functions f(z) in
the class CVG(RI,R2) have not been investigated so far.
Section 2 is aimed at the determination of necessary and
sufficient conditions for R to be equal to R2, if the function f(z)
1
is in the class CVG(RI,R2). In this section analogues of conditions
(4)
and
(5)
involving d
,
in place of d, for the functions in the
are
also
found.
Section 3 consists of theorems on
CVG(RI,R2)
growth of If(z)
for functions f(z) in the class CVG(RI,R2).
class
the
Finally,
Section 4 consists of a distortion theorem for the class
CVG(R1,R2)
and a rotation theorem for the class
2.
CVG(RI,R2).
PRELIMINARIES.
we first find some relations
f e CVG(R 1 ,R2)
the largest distances of the image curve
f(U) from the origin. We first prove the following lemma
function
For a
between
the
[[44A
I.
smallest
Let
fe
and
CVG(R1,R2).
(ii)
f(U)
D((R2-R) e iu,
(iii)
f(z)
e iu
(iv)
d*
FR(Z e-iU),
sup
eOf(U)
II
R +
If
RI
R2
R < m, then
R), for some real
where
FR(Z
z e U
R2-R
PROOF.
(i)
(ii)
Follows by (4)
By the definition of
CVG(RI,R2)
if R
1
R2
R "< m, f(U)
is a
802
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
disc of radius R
real, then
If the center of the disc is at
ro
R
ro e i
o
R2-R
d
Or, equivalently,
D((R2-R)
f(U)
(iii)
i
R)
ei
Since f(U) is a disc, d+d
d*
[4AK.
The function
,
FR(Ze
-i
2R. Consequently, by (i),
R +
R2-R
R 2 (denoted as
[2] as an extremal
of Lemma 1 with R
FR(Z
function for a number of problems concerning
PROPOSITION 1.
zf
f
(7)
CVG (R1, R 2)
1
inequalities
are
-
Thus,
R).
).
in the sequel) was first used by Goodman
FR2(Z
The
D(R2-R,
maps U conformally onto the disc
FR(Z
f(z)
(iv)
e
CV(RI,R2).
then
* 2
(d)
R2
2d -i
sharp for the
function
FR2(Z),
R 2 z I, of
Lemma l(iii).
PROOF.
Let
is increasing in x if
Thus inequality (7)
d =-, inequality (7)
The function
,
with
,
d
1 s x <
follows
41
It
is
clear
that
(x)
and is decreasing in x if 1/2 s x <i.
from inequalities
,If
(6) and (5).
follows from Definition 2.
FR2(Z
i/(i
x2/(2x-l).
@(x)
of Lemma l(iii)
I/R 2
is in the class
CVG(RI,R2)
and gives sharpness for inequality (7).
REMARK. For f CVG(RI,R2), inequality (7) sometimes gives a
better lower bound on R than that of
inequality (5).
In fact,
2
(d*) 2/ (2d*-l) >
if and only if d(2d-l) < d*
There does
exist a function in the class CVG(R R
satisfying d/(2d-l) < d*
1 2)
consider for example, f(z)
21og(l-z/2) -I
CV(I, 2/-3)
d2/(2d-l),
PROPOSITION 2.
zf
f
CVG(RI,R 2)
(d, * 2
2d -i
z
with
R
1
R1
-
I, then
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
803
and
z R
The inequalities are sharp when
for some real 8
R -R
R1
R 2.
I
i8
and f(e o)
f(e
0f(U) be such that
By making a suitable rotation of
f(z}, we
d* <
Let
R00
I
o
d*=
i8
i8) I,
may
-d*
assume that f(e o)
Then the unit exterior normal to af(U) at
ie
ie
f(e o) is n(e o)= -i. And, by the containment relation (3), we have
D(RI-d
,
f(U)
RI)
equivalently,
z
where
implies
B
B
-
(2Rl-d*)d*/R 1
f(z)
Az
1
and
* * /R
(Rl-d)d
1
A
i, or,
for
R 1 > 0. This
,
2d -i
which is inequality (8). The case
inequality (8) follows directly.
R1
0
is trivial
When
d*
Inequality (9) follows from inequality (8) and
Definition 2. The
sharpness of inequalities (8) and (9)
follows by considering the
function
of Lemma l(iii).
FR2(Z)
COROLLARY.
If
f
(i0)
CVG (RI, R 2), then
-
R1
PROOF.
(d)2
s
2d -i
R2
Proposition 1 and inequality
(8),
corollary.
together, give the
RE4ARKS, (i) For f
CVG(RI,R2) with R 1 z I, it is easily seen
inequality (8) sometimes gives better upper
bound for R than
1
that given by inequality (5). In
fact, (d* 2/ (2d * -i) < d2/(2d-l), if
and only if,
d
< d/(2d-l).
There does exist a function in the
class CVG(R R 2),
with
R 1 a i,
satisfying
1
d* < d/(2d-l)
consider, for example, f(z)
eZ-I
(ii) For the function f
C(RI,R 2) with R 1 < i, inequality
that
(8) is not sharp
.
and
R
1
(d* 2/ (24
< 1
-i)
GROWTH OF IF(Z) I.
For
()
because
f
CV(RI,R2),
Goodman ([2],[3]) found that
If(z)
-
2R
2
d
804
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
(12)
zl
in the disc
are sharp.
the
for
-
R 2 (l-r) + rd
where
1
II-
inf
d
f(u)
Both the inequalities
His proof shows that inequality (ii) continues to hold
CVG(R I,R2) also. However, analogues of inequalities
d
sup I<I, are not known. In this
involving
(12)
class
and
(II)
r
rd (2R2-d)
If(z)
af(u)
section these analogues are derived.
Goodman [3] also showed that, if f
then
CV(RI,R2)
R
2
2
rR 2
R2
R
2
Izl r [r ,I) where r
+ d
and the
inequality is sharp. In this section an analogous inequality for
the
functions in the class
CVG(RI,R2) is found wherein the number r is
for
2)
2R2(R2-d)/(2R2(R2-d)
independent of d.
In the following proposition, an analogue of
inequality (ii)
,
involving d in place of d is found. In Theorem i, an
improvement of
this proposition will be obtained.
P0POSII0N 4.
If
(13)
f
CVG(RI,R 2)
-
f(z
in the disc
P00F.
z
r s i.
with
then
d*
IR2
r(R 2 +
-,
R2 <
The inequality is sharp for
From the definition of d
-
If(z)
R
1
R 2.
we have that
d
,
in the disc
zl r s i. The triangle inequality and Schwarz lemma
together with the above inequality completes the proof of (13).
For the function
of Lamina l(iii),
and
R 2,
R1
FR2(Z)
IFR2(1)
41-1/R 2
i/(i
R
2+IR2-d*l.
inequality (13) follows.
C00LLARY.
If
f
CVG(RI,R2)
If(z)
in the disc
zl
r-
i.
-
wi.th
d
,
,
r (2R2-d)
Thus,
the sharpness of
-
then
R2,
PROOF. The inequality in the corollary is straightforward in
view of inequality (13).
RARKS. (i) The corollary improves Goodman’s result [2] given
by inequality (ii).
(ii) The functions f(z) in the class CVG(RI,R2) satisfying
,
d < d <
R2<
integer
k a 2 and 0
do exist as can be seen from the following example. For
< a < I/k
the binomial pk(z)=
CVG(R I, R2)
2,
z+azk
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
with R
2
so that for
,
d
(l-ka)2/(l-k2a).
,
k
Further, for
< R
d
2,
1/8 < a < 1/4
for
2
d*
l-a <
d
pk(z),
805
and for
l+a,
k z 3,
for 0 < a < I/k 2.
2
An analogue of inequality (13) involving R can also be
(iii)
1
found. Thus, if f
then
CVG(R1,R2) with R 2 <
< R
,
s
f(z)
Rl-d*l)
r(R 1 +
rd*
s
r(R 2 +
IR2-d * I)
in the disc zl
r s i. The above inequality is sharp for R
R 2.
1
Next, a growth theorem is derived for the class CVG(RI,R2) with
the help of the following lemma:
LEMMA 2 [5].
univalent in U, then
F (z)
If
f(z)
is in CV
and f (z)
F(z) in U implies that
-
Iz:l
in the disc
I.I
< R, where
_R
IF(z)
0.543
If
rd
(14)
R1
f
with
CVG(RI,R2)
-
and
is the least positive root of
arc sin x + 2 arc tan x
THEOREM I.
convex
is
0 < R
,
12Rl-d ,
(l-r) +rd
2
rd
R 2-
1
s R
2
<
-,
then
12R2-d
R2-dl r
r,
hand side inequality holds in the disc
Izl
< _R, B is as in Lemma 2 and the right hand side inequality holds
in the disc zl s i. Both the inequalities are sharp.
where
the
left
zl
P00[.
ie
f(e
o)
By making a suitable rotation of
-d*
sup
af(u)
II,
for some 8
f(z) we may obtain that
o real. We have n(e
o)=
-i.
Now, the by containment relation (2), we get
f(U)
=
D
,
(R2-d ,R2)
or
I-Az
where
d*(2R2-d*
(R2-d*)/R2.
B
)/R 2 and A
The inverse of the function g(z)
and the function
n(z)
(hof) (z)
Schwarz lemma. So,
l(z)
in the
disc
zl
r s I.
Bz/(l-Az) is h(z)
z/(Az +B)
satisfies the conditions of
r(IAf(z)
+ B)
This implies that
rlA
By substituting the values of
A and B in this, the right
hand side
inequality of (14) is obtained.
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
806
Now, to prove the left hand side inequality in (14), we apply
the containment relation (3) and obtain
,
where
B
A z
1
.
d (2R
A
and
d )/R
1
1
d )/R
1.
(R 1
Further,
,
A z
I
Rl+(d * -Rl)r
in the disc
zl r < i.
Hence, by Lemma 2, we have that
. ’2Rl-d
.
Rl(l-r)+rd
B Z
I=(z)
A*z
1
in the disc zl < R where _R is as in Lemma 2.
hand side inequality of (14).
of Lemma l(iii)
The function
FR2(Z
For
CVG(R2,R2).
,
rd
this
(2R2-d)/(R2-1R2-d
r/(l+ar)
=IFR2(-r)
,
d
function,
i
This gives the left
in
is
a)
/ (i
the
z R
2
class
so that
and rd
I/(R l(1-r)+rd
r/(l-ar)
is attained
equality
now
and
41-1/R 2
12RI-d
r)
where a
in inequality (14).
i the
r
and
(i) For f
CVG(RI,R2) with R 2 <
by
given
that
than
is
larger
upper bound of l.f(z) in inequality (14)
the
both
of Lemma l(iii),
inequality (13). For the function
R[MARKS.
FR2(Z
r < i, the upper bound given by inequality
bounds are equal.
(14) is better than that given by inequality (13).
(ii) From the proof of Theorem l,it can be observed that
replaced by d everywhere, continues to remain
inequality (14) with
true and sharp; i.e., if f
CVG(RI,R2) with 0 R 1 R 2 < m, then
For
--
d*
(16)
Rl+iRl_dlr
If(z)
R2(l_r)+rd
r, the left-hand side inequality holds in the disc
Izl
as in Lemma i, and the right hand side inequality holds
is
R
<
R,
Izl
of Lemma l(iii)
The same function
i.
in the disc
where
Izl
-
FR2(Z)
gives the sharpness in this inequality also.
(iii) Let Q(r,R2,x)
x(2R2-x)/(R2-1R2-xlr)
-
r
It can be seen that for
R2
decreasing in x for x
function
Q(r,R2,x)
and hence the upper bound of If(z)
[r ,i),
the
is
in
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
inequality (14) is better than that in inequality
r
where
2
807
(16)
for R 2 z d
,
R2-R2/(2R 2-I)
It can be seen
Let p(r,Rl,X
function
the
that for r
[0,R), R is as in Lemma 2,
P(r,Rl,X) is
decreasing in x for x
[RI,2RI] and hence the lower bound of If(z)
in inequality (16) is better than that in inequality (14) for
d -* d
R1
2RI; the last inequality does hold for the function
3
where 0 s a s 1/15.
z+az
p3(z)
For f
(v)
C--Q(RI,R2) with R 1 < R2, strict inequality holds
(14), because, when
in the right hand side of the inequality
Cz/(l-Dz) where
f(z)
equality holds, inequality (15) gives that
i#
i#
e
# real, so that
and
D
e
C
xl2Rl-Xl/(Rl+IRl-Xlr).
(iv)
- ,
CVG((I+3a)2/(I+9a),R2),
d*(R2-d*)/R 2
d*(2R2-d*)/R 2
R 2.
f(z) has R 1
For f
in inequality
CVG(RI,R2), the upper bound of If(z)
(14) (or (16)) is dependent on d* (or d). The following theorem gives
an upper bound of If(z)
that is independent of both d and d*.
TH[0[4 2.
If
f
CVG(R,R2)
R
where
Izl
[r * ,I] and
r
r*
2
R <.,
with
JR2-R
then
2
2
2
2
The inequality
IR2-R2/(2R2-1)"
is sharp.
PROOF. Set Q(r,R2,d
d(2R2-d)/(R 2 (l-r)+rd)
is the upper bound of
in inequality (16)
f(z)
r*
2JR 22-R2/(2R2-1 ).
decreasing in d.
For
r
,
(4),
Hence, for r
[r ,i], we may replace d by R 2
and obtain the assertion from inequality (16).
The function
z
of Lemma 1 iii
inequality (17) for
REARKS.
d
we have
rQ(r,R2,d
rQ(r,R2,d
[r ,i], the function
By inequality
FR2
z
Then,
Let
z
R2
is
R22 _R2
rQ(r,R2,d)
R
in
gives
sharpness
in
r.
inequality (17)
in
the upper bound of f (z)
f e CVG (R2,R2)
is better than that in inequality (13). Indeed, for
the function
(r,R2)
Q * (r,R 2)
(i) For
Q
.- rR2/(R 2
R
2-R2)=
2
r
rR2/(R 2
r(R2+
J
R2-R 2)
2-R2)
R2
we have
for r e
[2J R2-R2/(2R2-1),I]2
If f
(ii)
C--q(RI,R2) and equality holds in (17), then as in
R 2.
Remark (v) following the proof of Theorem I, we obtain that
Hence strict inequality holds in (17) when R 1 < R 2.
involving both
In the following result an upper bound on If(z)
R 1 and R 2 is obtained.
RI=
808
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
THEOREM 3.
If
f
CVG(RI,R2)
a
I
R1
RI-R
2
PROOF.
is
r **
r
the
2R
set Q(r,R2,d
upper bound of
inequality (4), we have that
RI-R
2
R1
by
1
inequality (16).
For
RI=
R2,
The function
(18) for
in
21)
+
where
d s R
and
R 2.
1
increasing
d.
By
I. Thus, we may replace d
obtain
and
in
For
rQ(r,R2,l)
the
equals
assertion
from
rR2/(R2-r R2-R2).
l(iii) gives sharpness in inequality
of Lemma
(i) The number r
RI-R
I R12-RI
R1
i
is
2
rQ(r,R2,d)
r.
RMARK$.
r ** defined in
(ii) For
holds in (18)
R1
21)
+ rd). Then, rQ(r,R2,d)
Let
inequality (16)
rQ(r,R2,d)
the upper bound
FR2(Z
z
in
f(z)
function
the
< m, then
2
d(2R2-d)/(R2(l-r
2(R2-I)/(2R 2(R2-I)
[O,r ** ],
-< R
2R 2(R2-I)/(2R 2(R2-I) +
The inequality is sharp for
1
1
2R2-I
r -< r **
zl
in the disc
1 -< R
r i
R 2 (l-r) +r 1
l(z)
(s)
with
,
defined in Theorem 2 is larger than
if and only if, R
R 2.
1
-<
f
with 1
R 1 < R 2 < m, strict inequality
for, when equality holds, it can be seen as in Remark
(v) following the proof of Theorem i, that R
R2, a contradiction.
1
4.
Theorem 3.
Both are equal,
C-V(RI,R2)
DISTORTION AND ROTATION THEOREMS.
For
CV(R I,R 2), Goodman [3] found that
f
I’(z)
()
-"
R2
l-r
in the disc
Izl
r < i.
The function
2)
FR2(Z),
of Lemma
l(iii), for
R2
shows that inequality (19) is sharp for each r e (0,i).
I/(l-r
From the proof of inequality (19), we observe that inequality (19)
continues to hold for the class CVG(RI,R2). However, an analogue of
,
inequality (19) in terms of d
is not known. In this
sup
8f (U)
section a result in this direction is found for the class CVG(RI,R2).
Finally, in this section, a rotation theorem is derived for the
class CV(RI,R2).
Its validity for the class CVG(RI,R2) remains open
for investigation.
in the sequel:
The following lemma is needed
z O,
f(z) in U and g’ (0)
g(z)
LEMMA 3 [7]. If f S with
4-8 m 0.171.
in the disc Izl -< 3
then Ig’ (z) <- f’ (z)
II
e
<
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
THEOREM 4.
zf f
,
(20)
12
+ rd
(Rl(l-r)
zl
in the disc
d
r
,
4-8.
R
< m, then
2
d (
If’(z)
3
-
with 0 < R
1
CVG(RI,R2)
809
d
,
-IR2-d *
(R 2
The inequalities are sharp for R
1
R 2.
As in the proof of Theorem i, we obtain
P00F.
,
d
B
together give
where
(2R2-d)/R 2
Bz
<
(z)
-z
A
and
,
This and Lemma 3
(R2-d)/R 2.
B
I’ (z)
-
4-8. By substituting the values of A and B
3
r
in the disc zl
in this inequality, the right hand side inequality of (20) is
obtained.
To prove the left-hand side of the inequality (20), we have, as
in
the proof of Theorem I,
,
B z
,
f(z)
I-A z
B
where
,
,
d
(2Rl-d)/R 1
A
and
Therefore, by Lemma 3,
(Rl-d)/R I.
,
(I-A z)
,
in the disc zl
inequality (20)
r
For the function
so that
Rld
(2R 1
a
41
R2d
d*
1/It 2
(2R 2
-
Rld
(Rl(l-r)
3
FR2(Z)
*
d )/(R
2
I/(R I(I
+ rd
2
48, which is the left-hand side of the
of Lemma l(iii), R I
R2
2
r) + rd *
d*
Ir) 2
R 2 and d
ar)
i/(i
2
i/(i + ar)
so that equality is attained in
2
FR2
inequality
,
I/(l-a)
(r)
FR2
(-r)
and
where
(20).
CVG(RI,R2), the upper bound of If’ (z) in
The sharp
that in inequality (19).
than
is
better
inequality (20)
function given in the proof of Theorem 4 is independent of the point
under consideration whereas the sharp function used for inequality
4AKRS.
(i) For f
(19) is dependent on the point.
From the proof of Theorem 4, it can be seen that
ii
,
replaced by d
inequality (20) continues to remain true with d
everywhere, i.e., for f
CVG(RI,R2) with 0 R 1 R 2 < m, we have
--
810
V. SRINIVAS, O. P. JUNEJA AND G. P. KAPOOR
that
(RI+ IRl-dl r
zl
r
For
f
in the disc
(iii)
R2-R
-
E
(R 2 (l-r) +rd)
-
4-8.
3
R
CVG(R
d* R
If’ (z)
with
s
i/(12-2
16) and
r s 3
4-8, the upper bound of
in inequality (20)
is better than that of the inequality (21).
For
(iv)
f E CVG(RI,R2)
R 2 < m, the lower bounds of
with
s
/R
If’ (z)
in inequalities
(21) are equal by Proposition 3.
Similarly, the upper bounds of If’ (z) are also equal.
,
For
f
(v)
CVG(RI,R2) with R 1 s d s d s 2RI, the lower
bound of If’ (z)
in inequality (21) is better than that in inequality
(zo).
Finally, we prove a rotation theorem for the class CV(RI,R2).
Its validity for the class f
remains open for
CVG(R I,R2)
investigation.
TH[0R[h 5.
larg
f’ (z)
in the disc
R00[.
Izl
If
2 in
f
(20)
cv
and
R 2(l+r) +
-r
-
(RI,R2)
C(r,R2)
+
r < 1 where C(r,R)
+ __2
,
then
(4R2_I
R(l+r)
C(r,R 2
l+r
(l-r).
For each fixed A in U, the function
f[z+l
g(z)
l+Xzlz +
is
R2 <
with
f(A)
c2(A)z2+...
-
If’ (I) (1-1112). It is known
CV(RI/A(A ),Rz/A(A)) where A(A)
then
Therefore,
g
Ig"(O)/2!
CV(R[,R)
41-1/R
[8] that if
R2
2f’ (A)
which, by using the distortion property
for the
function f(z) in CV, gives
2f’ (A)
R (i+
2
CONVEX FUNCTIONS AND THE ROLLING CIRCLE CRITERION
21ll/(l-Ill 2,
Multiplying the above inequality by
x
Replacing
2112
f ,f((x)
()
ll
_ll
I_IX
811
we obtain
i-
2
by p in the above inequality, we get
1_p2
"- (X
R (l+p)
2
1_p2
-<
1_p2
1
l-p,
R 2 (l+p)
Thus,
2
(22)
1-p
2
1
..,
Im (X f"(X)
since,
ca arg f’ (X)
ap
1-p
R2 (l+p)
2p
(X)
2
l_p2
p
a
_<
2
l-p
1-p
R 2 (l+p)
arg f’ (X)
Now, integrating the terms in inequality (22) along the straight
linepath from X
0 to X
re
the required inequality follows.
i8,
REFERENCES
I.
BIERNACKI, M.
(1936)
2.
Sur
les
fonctions
univalentes,
Mathematica
1--2
49-64.
GOODMAN, A.W. convex
functions o__f bounded type, Proc. Amer.
541-546.
(1984)
More o_n convex functions o__f bounded type, Proc. Amer.
Math. Soc. 9--7
(1986), 303-306.
MEJIA, D. and MINDA, D. Hyperbolic ggometry in k-convex regions,
Pacific J. Math. 141 (1990), 333-354.
Math. Soc. 9__2,
3.
4.
5.
TAO-SHING, SHAH.
On the radius o_f uperiority i__n subordination,
! (1957), 53-57.
Science Record
6.
7.
8.
STUOY, E.
Konforme Abbildung Einfachzusammenhanqender Beeiche,
Teubner, Leipzig add Berlin, 1913.
Convex functions with restricted
STYER, D. and WRIGHT, D.J.
curvature, Proc. Amer. Math. Soc. 10__9,(1990), 981-990.
WlRTHS, K.O. coefficient bounds for convex functions of bounded
type, Proc. Amer. Math. Soc. 103 (1988), 525-530.