Internat. J. Math. & Math. Sci.
(1986) 65-70
65
Vol. 9 No.
BOUNDS ON THE CURVATURE FOR FUNCTIONS WITH
BOUNDED BOUNDARY ROTATION OF ORDER 1-b
M. A. NASR
Department of Mathematics
Faculty of Science
Mansoura University, Egypt
(Received April 30, 1985 and in revised form July 2, 1985)
f(z)
analytic functions
f2iRe{
o
< k
zl
a z
n
re
z
r, 0
0 real, denotes the class of locally univalent
b
and
+ X n=2
z
zf’’(z)}Id0
f (z)
+
ture of the image of
Vk(1-b)
k -> 2
Vk(l-b)
Let
ABSTRACT.
i0
n
in
e D
Izl
{z:
D
1}
In this note sharp bounds on the curvaf
under a mapping
r
such that
belonging to the class
have been obtained.
Analytic Function, Univalent Functions, Functions with Bounded
KEY WORDS AND PHRASES.
Bondary otation.
1980 AS SUBJECT CLASSIFICATION CODES.
30C45.
INTRODUCTION
A
Let
D
{z:Iz
I}.
For
G e S(1-b)
G(z)
z
(i-r)
o
D
.
zG’(z)
Re{1 +
and
[g(z)/z]
a z
n
belongs to the class
if and only if there is a function
which are analytic in
S(1-b) (b
I)] 0
g
S(O)
0 complex)
z e D.
It is shown in [1]
such that
b.
(I.I)
-2b
(l_r)-2b
-2b
(l+r)
Vk(1-b)
which satisfy
Y
in
+ X n=2
b z 0 real
(l+r)
Let
G
we say
z
was introduced by Nasr and Aouf in [I ].
S(1-b)
The class
and for
A,
G
G(z)/z x 0
if and only if
that
f(z)
denote the class of functions
2
IRe(l
The class
K -> 2
f(z)
0
and
and
(7}Id0
k
(z)
+-b zf’
f
Vk(l-b
(1.3)
b x 0 complex, denotes the class of functions
D
in
(1.2)
-2b
z
re
iO
gl,g 2
A
D
was introduced by Nasr [?].
if and only if there exist two functions
f
S(0)
It was shown in [2] that
such that
fVk(1-b)
M.A. NASR
66
{g1(z)/z}b(k+2)/4/ {g2 (z)/z}b(k-2)/4
f’(z)
(1.4)
And from (1.1) and (1.4) we deduce immediately that
there exist two functions
f’(z)
{Gl (Z) /z}
The subclasses
(k+2) /4
S(1-b),
tions starlike of order
of order
C(1-b)
e S(1-b)
{G2(z)/z}(k-2)/4
/
V2(1-b
1-b
V
convex of order
f(z)
f
is given by
in
are respectively, classes of func-
l-b
and of bounded boundary rotation
V2(1-b
Vk0
and
respectively by
k.
w
for
Vk(l-b)
and
For a locally univalent function
f
D
in
Re {1
B
Let
inf
K
[z[
r
when
r
[z[
at the point
under the mapping
r
/ ]zf’(z)[
+ zf’’(z)
fv
and sup K
f
Kf(z)
r
the curvature
for the level line, i.e. the image of the circle
Kf(z)
r
if and only if
such that
We shall denote the subclasses
1-b
and
GI,G 2
Vk(1-b)
f c
B
(1.6)
Kf(z)
r
denote respectively, the infimum and supremum of
r
B
belongs to a certain subclass
of locally univalent functions
A, which is normal and compact.
Kf(z)
r
The problem of estimating
considerable attention (see
for various classes of functions has attracted
[3, p.p. 599-601] for
the history of this problem).
V
inf K
The purpose of this paper is to establish
b
o real.
2.
STATEMENT OF RESULTS.
Set
k
(k-2)/4
l-b
A simple calculation shows that H(r) increases strictly with
THEOREM I.
inf K
If
V (l-b)
k
r
f
E
Vk(l-b),
V (l-b)
k
r
0
r
0 < H(r)
and that
r
for
O, then
b
(l-r2)b/r
k
for
(l+r)2b(kl+l)
r(l_r)2bkl
sup K
and
-{log(l+r)/(l-r)} -I
(l+r2)/2r
H(r)
and
k
r
+
2,
0
I,
b
+ (2b-1)r}
{I
bkr
e
(l-r)
(2.1)
(2.2)
otherwise
and
(l_r)r(l-b) +
r(1+r)r(1-b)
sup K
Vk (l-k)
r
r-
+
(l_r)2b(kl
r(l+r)
These bounds are sharp for all
2bk
0
b
{-2"-
(1+r)
(l+r)
2
/2r
(l-r)
22_.$
H(r)
r
log
+ i)
b(k
for
b
bk
+ I)+
r
(l+r.
b(k
(l+r)
+
bkr
+ (2b
+
+
l)r
2
l+bk
(2.2)
+ 1)
bk
(l-r)
otherwise
(2.3)
(2.4)
I.
67
CURVATURE FOR FUNCTIONS WITH BOUNDED BOUNDARY ROTATION
inf K
Vk(l-b),
If
THEOREM 2.
Vk(l-b)
r
O, then
b <
(l_r)2b(kl
r(l+r)2bkl
+ I)+
(l_r)r(1
r(l+r)r(1
b) + b
{1 + bkr + (2b
1)r
2
(2.5)
and
sup K
b)
(l+r)
for
r
sup K
c(l-b)
Keogh [7] for
Noonan
inf K
inf K
sup K
r
Vk(1-b)
r
c(O)
sup K
inf K
V
and
sup K r
for
b
inf K r
c(l-b)
c(O)
r
(2.7)
(2.8)
otherwise.
by Zmorovic [6
sup K r
sup K
vk
c(1-b)
r
b < 0
for
r
0 < b
-<
our results
by Zederkiewicz [4] and
k
2, b
l, coincide
and those given by
our results agree with those reached by
and
and
r
1-b)
k
vk
c(l-b)
Moreover, for
b
Finally, if
for
2}
2,
k
sup K
and
r
and
r
c(0)
8] and Singh [9
1-b)
k
c(l-b)
5] for inf Kc(1-b)
r
inf K
ledge the values of
V
(2.6)
r
inf K
are reduced to those give[: for
with those given for
bkr + (2b- l)r
{I
by Singh [7], also for
r
those given by Eenigenburg
b(kl+i)-l
our results coincide with the results given for
b > 0
2,
k
and
2/
l+r.
log (--)
2r
bk
+ I)-I
+
0
These bounds are sharp for all
c(l-b)
(l-r)
(l+r) < H(r) -< r + b(k + I)-1 (l-r)
b(k + I)-I
r
(l+r)2b(kl
r (l-r) 2bkl
inf K
(l+r)2/2r
bk
V (l-b)
k
Indeed, if
(l-r)
e
{-
b
But to the best of our know-
and also the values of
V 1-b
k
and
and aiso the values of tnf K r
for
b < 0
are not known as yet.
3. PROOFS.
We need the following:
LEINA 119]:
If
S(0),
g
z
D, then
r e
(1-r
2) log{ (l+r) / (l-r)
Both sides of the above inequality are sharp.
COROLLARY I.
B(r, G(z)/z)
If
G
S
A(r,G(z)/z)
-< Re zG’ (Z)
G(z)
A(r,C(z)/z)
(l-b),
z
re
i0
D, then
for
b
0
(3.2)
for
b
0
(3.3)
<_
B(r,C(z)/z)
M.A. NASR
68
where
+ (2b
A(c,x)
(l-r)2b xl
+ _2_r log
l)r
(l-r2)log[ (l+r)/(l-r)
r
and
(l-b) + b(l-r
B(r,x)
2) lxl I/b
The proof will follows immediately from (I.I) and (3.1)
PROOF.
PROOF OF THEOREM i.
[Gl(Z)/Z
Set
[Gm(Z)/Z
and
u
Then from (2.2), we find that
[i/(1+r)
In view
2b
and
u
lie in the interval
v
2b
1/(I-r)
of (1.5) and (1.6) we need to find the
k
Kf(z)r
(3.4)
v
+ I)
[(k
V
zG’(Z)l
Gl(Z)
zG(z)
G2(z /
-k
(3.5)
extreme values of
kl+l
ru
(3.6)
In view of (3.2) and (3.6) we need to obtain the minimum of
k
k
F(u,v)
+ 1)B(r,u)
[(k
V
A(r,v)]
-k
/
+1
(3.7)
ru
and the maximum of
k
H(u,v)
when
u
k
[(k + l)A(r,u)
V
and
klBir,v)] /
lie in the interval given by
v
+I
(3.8)
ru
(3.5).
This reduces the problem to finding extreme values of functions of two variables.
It is easily seen that for
0 < b N
and
k
2 (k
O)
the minimum is attained
for
u
i/(l-r2) b
(3.9)
and because the value of
minimum.
u
lies within the interval given by (3.5) this gives the
We thus obtain (2.1).
2, b >
k
If
or
k
2, b > 0
it is readily
confirmed that the roots of
3F
3F
u
v
do not give the minimum.
u
I/(l-r)
2b
and
v
Hence, the minimum is attained on the boundary for
2b
This yields (2.2)
I/(l-r)
In order to maximize
H
H
u
v
H(u,v),
it is found that the equations
give
v
{2r/(l-r2) 2
log [(1+r)/(l-r)]}
b
(3.1o)
and
log
(l-r)
The value of
2r/(1+r)
v
2
2b
U]
+ bk
+ k
[1
+ bk
+
k- - - -
2br (1-r 2)
+ l--2r
-2r log
l-r
(l-r)
(3.)
given by (3.10) lies in the interval given by (3.5) because
-< log[(l+r)/(1-r)]
_<
2r/(l-r) 2
(3.12)
69
CURVATURE FOR FUNCTIONS WITH BOUNDED BOUNDARY ROTATION
but the value of
given by (3.11) lies in the interval given by (3.5) if
u
[b(l+kl)(l+r)l(l+bkl)]
r
< r
+
[b(l+kl)(1-r)l(l+bk I)].
(3.13)
When (3.13) does not hold, the maximum values of (3.8) is obtained
(2.3).
This gives
-< H(r)
on the boundary for
1/(l-r)
u
2b
and
I/(I+r)
v
2b
This yields (2.4)
The above inequalities are sharp and the extremal functions are given below where
equality in each case, is attained at
r
z
For equality in (2.1)
(i)
fl’(Z)
cos t
i/[(i
I"t
l+r
%b
and
r
ze
(I
ze
)U b
-it (1
0 _<
<_
0 < b <-
where
2
+ (l-b)r
r
(3.14)
H(r)
For equality in (2.2)
(ii)
(1-z)2bkl //
f(z)
2b (l+k
(l+z)
(3.15)
For equality in (2.3)
(iii)
’(z)
f3
(1-ze
-it)
2bk
(l-z)
(l+bk)H (r)+r (b-l)-b (l+kl)
(l+z)
(l+bkl) H (r)+(b- l)+b (l+kl)
(3.16)
where
l+r
(iv)
2
(l-r2) 2
-2r cos t
[l+r2-2rH(r)],
/
(3.17)
For equality in (2.4)
2bk
f4(z)
(l+z)
2b
/ (l-z)
Gl(Z)/Z
(3.18)
Taking into consideration (1.3), (1.5), (1.6), (3.3) and using
PROOF OF THEOREM 2.
the notation
(kl+2)
u
G2(z)/z
and
v
we find that for
u
v lie in the
and
interval
[I/(l-r)
2b,
I/(l+r)
2b],
(3.19)
we need to obtain the minimum of
k
F l(u,v)
l+k
[(l+kl)A(r,u)-klB(r,v)]/ru
v
(3.20)
and the maximum of
l+k
k
H l(u,v)
V
(l+k 1)B(r,u)-k
A(r,v)]/ru
(3.21)
It is readily verified in the case of (3.20) that the equations:
3F
F
3u
3v
do not give the minimum and that minimum is attained for
v
I/(l+r)
2b
and this value is given by (2.5).
of equality for the function
In order to maximize
3H
3H
3u
3v
f(z)
Hl(U,V)
f4(z)
given by
u
I/(l-r)
Simple calculation
2b
and
confirms the case
given by (3.18).
(3.21),
it is found that the equations:
M.A. NASR
70
give
[2r/(l-r2) 2
v
log
(l+r)/(1-r)]b
(3.22)
and
l-r
loF
2b
u
l+bk
l+bkl
l+k
l+k
The value of
H(r)
v
+
2br
-I---
(l-r2)
2r
log
(l+r
---1
(3.23)
given by (3.22) lies in the interval (3.19) because,
0
u given by (3.23) lies in the interval
but the value of
This proves (2.6).
(3.19) if
H(r)+[b(k-2)(1-r)/(b(k+2)-4)]
r-[b(k-2)(l+r)/(b(k+2)-4)]
(3.25)
The case of equality can be directly confirmed by the function f(z)
given by
H(r)b(l+k )+bk +r(1-b)
(l+z)
’(z)
f5
H (r) b’-l+k )-bk
(l-z)
l+r (l-b)
(1-ze
-it
-2b(l+kl)
(3.26)
where
i+r2_2 r
cos t
2
(1-r)
2/ [1+r2-2rH(r.]]
when (3.15) does not hold, the maximum value of
u
1/(I+r)
by (2.7).
2b
v
1/(1-r)
2b
(3.27)
H1(u,v)
is attained for
and the corresponding value of sup K
Vk(l-b)
r
is given
Simple calculations confirm the case of equality for the functions given by
(3.15).
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I.
2.
3.
4.
5.
6.
7.
8.
9.
order(to appea.
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of the
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GOLUZIN, G. M. Geometric theory of functions of a complex variable, Amer.
Math.
Soc. Transl. Math. Mono, Providence, R. J., 1969.
ZEDERKIEWICEZ, J. Sur la courbure des lignes de niveau dans la classe des functions
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EENIGENBLIRG, P. On the radius of curvature of convex analytic functions, Can. J.
Math. 23(1970), 486-491.
ZMOROVIC, V. A. On certain variation problems of the theory of Schlict functions.
Ukranian Math. J., 4(1952), 76-298.
KEOGH, F. R. Some inequalities for convex and starshaped domains, J.
London Math.
Soc., 29(1954), 121-123.
NOONAN, J. W. Curvature and radius of curvature for functions with obunded boundary rotation. Can. J. Math. 25(1973), 1015-1023.
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