Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 321-330
321
ON CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER
H.S. AL-AMIRI and THOTAGE S. FERNANDO
Department of Mathematics and Statistics
Bowling Green State University
Bowling Green, OH 43403, USA
(Received November 15, 1988)
ABSTRACT.
its generalization,
K(b)
of starlike functions of complex order b was introduced and
Aouf and M.A. Nasr.
The authors using the Ruscheweyh derivatives
0 and
the class K(b) of functions close-to-convex of complex order b, b
introduce
c
S*(b)
The class
studied by M.K.
Ko(b).
the classes
Kn(b)
where n is a nonnegatlve integer.
Sharp coefficient bounds are determined for
sufficient conditions for functions to belong to
distortion and covering theorems
for
Kn(b)
n
KEY WORDS AND PHRASES.
The authors also obtain some
All results are sharp.
n
Starlike
Kn(b)
S*(b)
and determine the radius of the largest
K (b) belongs to K (I).
disk in which every f
Kn(b).
Here
as well as several
close-to-convex functions
functions,
of
complex
order, Ruscheweyh derivatives, Hadamard product.
1980 AMS SUBJECT CLASSIFICATION CODE. Primary 30C45.
I.
INTRODUCTION.
E
Let A denote the class of functions f(z) analytic in the unit disk
{z:
< I} having the power series
zl
f(z)
z
+
[. amZ
(1.1)
z e E.
m=2
Aouf and Nasr [1] introduced the class
S*(b)
of starlike functions of order b, where b
is a nonzero complex number, as follows:
S (b)
+
f: f e A and Re
f(z)
>
0, z e E}.
We define the class K(b) of close-to-convex functions of complex order b as
K(b) if and only if f e A and
follows: f
Re {I
+ zf’g;(z)
for some starlike function g.
I)}>
0, z
E,
(1.2)
H.S. AL-AMIRI AND T.S. FERNANDO
322
classes
The
Rn,
N
n
O
introduced by Singh and Singh
and where N O is the set of nonnegatlve
[2], f Rn if and only if f e A and
integers,
were
z(Dnf(z))
Re
>
Dnf(z)
O, z e E,
(1.3)
where
Dnf(z)
z
f(z) *
(1.4)
z) n+l’
(1
. .
and (*) stands for the Hadamard product of power series, i.e., if
f(z)
0
n
a z
n
g(z)
0
b z
n
n
then f(z)* g(z)
0
a b z
nn
n
The operator Dn is referred to in Ai-Amiri
[3] as the Ruscheweyh derivative of
Note that R0 is the familiar class of starlike functions, S*. More, it is
order n.
known [2] that
Rn+ 1CRn,
NO,
n
and consequently R consists of functions starlike in
n
E.
Let
f
g
Kn(b)
n
E
NO,
b is a nonzero complex number, denote the class of functions
A satisfying
for some g
Here
R
n
K0(b)
K(b).
Many authors have studied various classes of univalent and multivalent functions
using the Ruscheweyh derivatives D
n,
n
N
O
In particular one can look at the work
of Ruscheweyh [4].
In
Kn(b, n e N O
for Kn(b) and several
Section 2 determines coefficient estimates of functions in
3, we obtain some distortion and covering theorems
section
sufficient conditions for functions to be in
Kn(b).
The radius of close-to-convexlty
for the class of close-to-convex of complex order b is also determined in section 3.
2.
COEFFICIENT ESTIMATES.
In this section, sharp estimates for the coefficients of functions in
determined in Theorem 2.1. First, we need the following lemmas.
LEMMA 2.1. For n
NO, let
+ (2b- 1)z
(Dnf(z)),
(1
Then f
E
z)
3
K (b).
PROOF.
n
Let g
A be defined so that
z
Dng(z)
(1
z) 2
Kn(b)
(2.1)
are
CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER
A brief computation gives
The definition of Rn implies g e R
n
[
+
n
The
2. I.
function
.
representation in E
f(z) --z +
mffi
2
.
m--2
c
Then
4
m
z e E.
as
f
defined
c z
m
m
E
R
n
(2.
in
1)b + l]z
where n
has
the
m.
power
series
(2.3)
NO
n! m!
(n+mA brief computation gives
PROOF.
D n g(z)
(n + m
n: (m
z +
mffi 2
Since g E
+ z
,
z
n! <M- I)!
1)! [(m
(n + m
z +
Let g(z)
LEMMA 2.2.
I]
z(Dnf(z))’
Dng(z)
K (b).
This proves that f
REMARK
323
Rn,
D
n
g(z)
S
E
Thus,
1)!
m
1)’ cz
m
the well known coefficient estimates for
using
starlike functions one gets
(n+m-1)!l cm
m, m > 2,
n! (m- 1)!
and the proof is complete.
Imam
z +
Let f(z)
LEMMA 2.3.
Cm 12
4
[
1)!
(m
1)!
(n + m
kffi 2
PROOF.
z +
Let f(z)
mffi2
+
for some g e R
n
a z
[
z(D
n
If f
K (b), n
E
NO,
then
12
(k
a z
m
f(z))’
D g(z)
n
m
I)
12[
be in
Kn(b).
1
l-
Then (1.5) implies
+ w(z)
w(z)
and where w e A such that w(0)
(2.5)
z e E,
0, w(z)
and
Iw(z)l
for
324
z
H.S. AL-AMIRI AND T.S. FERNANDO
E.
Let g(z)
z +
w(z)
. .
Then (2.5) and the Definition 1.4 imply
c z
m
m= 2
(n + k
I)!
(k
I)!
k=2
{n!
2bz +
[kak
l)Ck]Z k}
+ (2b
(n + k -I)..!
(k- I)’.
k=2
Ck)zk
(kak
(2.6)
Using Clunle’s method, that is to examine the bracketed quantity of the left-hand
side in (2.6) and keep only those terms that involve z k for k
m-
for some fixed
m, moving the other terms to the right side, one obtains
m-1
l[n!2bz
+
Lr
kffi2
(n + k- I)’.
(k
l)!
.
m
[’kak
+ (2b
(n + k
I)’
(k-l)’
k=2
I.c kz kl,
(kak- Ck)Z
k
+
kffim+l
Akzk"
Let
m-1
l[n!2bz
Lr
+
k=2
(n + k
I)!
1)!
(k
m
(n + k
I)’
I)
(k
k=2
t8
Let z
0 < r
re
in (2.7) and using
.
<
1.
k=2
n!2
1-and
mam
In particular, when m
(z)
z
k
+
2
AkZ
k=m+l
-l)Ck]Z k}
k
(2.7)
Ck 12 r2k
[kak
ml
41b12r2 + kffi2
-
+ (2b
(z) dz for both expressions of
0
we get
(n + k- I)!
(k-l)
<
Upon letting r
f
Computing
Iw(z)l <
(kak Ck
[kak
[
(n + k(k- 1)!
1)’12
Ika k+
(2b
1)ckl 2 r2k.
after some easy computations we obtain
Cm
2
2 we have
k=2
(k- 1)!
325
CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER
The proof of the lemma is complete.
[
z +
Let f(z)
THEOREM 2.1.
m= 2
a z
m
m
If f e K (b) where n e
n
NO
then
(m- I)’
< (nn! / m- 1)! [(m
1),b, + 1].
This result is sharp. An etremal function is given by (2.3).
lam
[
z +
Let f(z)
PROOF.
mffi
z +
g(z)
I cmzm"
a z
m
2
m
be in K n (b)
We claim that for m
)
Let the associate function of f,
2 and n e
m=2
2lb[
(n + m- I)
m
(n + k- I)’
n! (k-l)!
[.
+
NO,
k= 2
(2.9)
-Ck
We use the second principle of finite induction on m to prove (2.9).
For m
12a 2 -c21
2,
(
<n
n!
+ I),
assume (2.9) is true for all m (p.
I(P
+
l)ap+
Cp+
=2
n! p!
(n + p)
n! p! |
(n+p)!
+ 2
--2
P
k-2
-
!
I)!
(n +
n!(-1)’
2
+
Taking m
in
(2.4), we get
p, the above yields
Cp+ll 2 , 4
4
p +
I12
Now using (2.9) since k
l(p+l)ap+
2(b)
is true as shown in (2.8)
(n + I)
21bl
J
Ibl 2
(n +k
i)!
"! (k- )!
(n + k
1)
n’ <k- ),
P
1512
(n + k- 1)
n! (k- I)
.C.
(n+k- I)!
kffi2
2
,ck
iCkl2
kffi2
I+2
[ I%1
(n + k- I)!
k=2
k-I
I
(" +
-
]
Now
H.S. AL-AMIRI AND T.S. FERNANDO
326
Applying the principle of mathematical induction on p, it is easily seen that the sum
of the last two terms appearing in the bracketed expression in the right hand slde
of the above is equal to
!2
n!(n (k+
I)!
I)!
k
Consequently
it follows that
This shows
that
p + I.
(2.9) is valid for m
I.m -%.
< (n +
I)!
m
Hence, by the second principle of
From Lemma 2.2 and 2.9 it follows that
finite induction, the claim is correct
II
(2.,0)
.>2.
Finally from Lemma 2.2 and 2.10 we deduce that
Hence the proof of the Theorem 2.1 is complete
0 in Theorem 2.1 we have the following corollary.
Putting n
COROLLARY 2
z +
If f(z)
a
mffi 2
I%1 For <-
m
is a close-to-convex function of complex
)lbl
+ I.
This result Is sharp.
1, Corollary 2.1 is reduced to the well known coefficient
bounds for the close-to-convex functions due to Reade [5].
order b, then
REMARK 2.2.
b
Next we have two theorems that provide sufficient conditions for a function to be
in
Kn(b).
THEOREM
satisfied in
Let f e A and n
E, then f Kn (b).
2.2.
Re
Re
Re
(iv)
Re
E
N
O
If
any
of
n
+ [(D f(z))’ I]} > O,
{I + [(I -z)(D f(z))’ I]} > 0,
{I + [(l
I]} > 0,
-z)
f(z))’
{I +
1]} >
{1
n
2
z )(D
[(I
2
n
(D n
f(z))’
0.
the
following
conditions
is
327
CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER
PROOF.
The proofs follow by choosing g as below:
g(z)
(i)
z,
n! (m
(n + m
1)! m
1) z
n! (2m
(n + 2m
2)!
2m-1
2)! z
z +
g(z)
(ii)
m2
[
z +
g(z)
(iii)
m=2
n! m!
1)
(n + m
+
g()
(iv)
m2
HgORgN
+
Let f()
2.3.
a
m-2
m=2
[
(ii)
m=2
1)!
1)! m
(n + m
n! (m
I)!
1)’
2(n +
(iii)
am+1,l
(n / m
n! (m
lbl,
(iv)
N0’
each of the following
Kn(b).
[am[ [b[.
(n + m)(m + 1) a
Ima m
m
+m-2[ n!(n +(mm_-2)
m+l
[(m- 1)am_l-
Ib[
(n +
m)(nm(m+ m_ -l)l)(m + 1)
I,
where a
21(n + 1)a2 al
(n + m)(n + m
PROOF.
1)la2[
respectively.
For n e
m
conditions is sufficient for f to be in
(i)
m
and
1)(m + 1)
I)
+
(n + m- 2)!
2)!
n! (m
[
m=2
,Ibl’
am/l!,
[(m 1)am_l
where a
2(n + m
m-
l)
a
m
1.
We prove the sufficiency of part (1) since the proofs of the remaining
parts are slmilar to the proof of (1).
From (1) of Theorem 2.2, f c
Re
{1
Kn(b)
+ [(Dnf(z))
if f satisfies the condition
1]} >
0, z c E.
(2.11)
Condition (2.11) would be satisfied if
[(Dnf(z))’
is true.
<
2, z c Z
However upon substituting
(Dnf(z))
in
I]
(2.12) one needs only show
+
[
m(n + m
n’ (m
I)
m-1
a z
I)’ m
(2.12)
H.S. AL-AMIRI AND T.S. FERNANDO
328
II
-
m(n + m
n. (m
[m =^Z
I)!
,,,(.+m- ):
mn, (m
I)! am z
m= 2
-*I
m(n + m
)’
n! (m- l)t.
m= 2
Ib
(2.13)
2, z eE.
.
Assuming (1) of this theorem we have
l
<
a z
m
)!
,am’
<
+
2.
Thus (2.13) is established and the proof of the sufficiency of part (i) is complete.
REMARK 2.3.
I, Theorems 2.2 and 2.3 are reduced
0 and b
For n
to theorems of
Ozakl [6].
3.
DISTORTION THEOREMS.
The objective of this section is to obtain some distortion theorems for the class
{z/Is <
The radius of the largest disk E(r)
Kn(b).
r},
0
<
r 4
such that if
K (I) can be determined as a consequence of one of those results.
f e K (b) then f
n
n
Let f e K (b)
THEOREM 3.1.
n e N
n
1-i2b- II=
3
(1 + r)
Izl r
12b- 3!It
Then for
O
.,.
,, <-1+
i,Dn f.z..’
<
and
,
12b
(3. I)
r)
(1
An extremal function f is given by (2.1).
K (b). Then (I.5) implies for some g e R
n
This result is sharp.
PROOF.
Let f
n
z(D
n
f(z))’
I) w(z)
w(z)
+ (2b
Dng(z)
where w e A and
, Izl
lw(z)
-[2bl[.r
+
r
in E.
z E E,
Izl
This gives for
, iz(Dn f(z))’ ,
vng(z)
+
-r
<
j2b-
(3.2)
r
The definition of Rn implies Dn g(z) is a starlike function. Hence by the well known
bounds on functions which are starlike in E, we get for
r
zl
r
(I +
r) 2
IDn g(z)l
<
r
<
(I
(3.3)
r) 2
Using (3.2) together with (3.3) one can get (3.1) and the proof of the Theorem 3.1 is
complete.
Taking (1) n
0, and (ll) n
0, b
in Theorem
3.1, one can immediately
obtain the followlng corollarles, respectlvely.
COROLLARY 3. I.
If f is a close-to-convex function of complex order b where
COROLLARY 3.2.
If f is a close-to-convex function then for
zl
r
<
I,
329
CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER
1 -r
(1 + r)
l+r
<
3
f’(z)
3
"(1
r)
For the proof of Theorem 3.2, we need the following well known result [7; p. 84]
concerning the class P of functions p(z) which are regular in E such that
Re p(z)
>
p(0)
and
0, z e E.
Let p e P.
LEMMA 3.1.
Then for
+r
,p(z’
-r
2
Izl
r
<
I,
2
2
-r
(3 4)
2
This result is sharp.
Let f e K (b),
n
THEOREM 3.2.
n
D
n
g(z)
Then for some g e R
n
r
2[b[r
2
r
and for
Izl
<
r
I,
(3.5)
2"
K (b) implies that for some g e R
f
n
n
[z(D
+
where p e P.
O
An extremal function is given in (2.1).
This result is sharp.
PROOF.
N
+ (2b- 1)r 2
f(z))’
[z(D
n
n
D
f(z))’
n
p(z),
z e E,
g(z)
Hence (3.5) can be obtained by substituting p(z) in (3.4).
It is interesting to note that the result in Theorem 3.2 does not depend on the
value of n.
in
Also, it can be used to solve the problem concerning the radii of
Kn(b)
Kn(1).
THEOREM 3.3.
Let n e N
O
If f e K (b), then f e K (I) for
n
n
Izl
<
r’ where
r
This result is also sharp.
Let f
PROOF.
K (b).
n
An extremal function
z(D n f(z))’
Dn g(z)
at
+ (2b
-r
l)r
2
and radius
-r
right half plane if r
REMARK 3.1.
<
r’.
Taking n
is given in
(2.1).
Then according to Theorem 3.2 there is some g e R
n
such
lles in the closed disk with center
2
It can be shown that this disk lles in the
This completes the proof of Theorem 3.3.
0 in Theorem 3.3, one can see that, r’ is the sharp radius
of close-to-convexlty for close-to-convex functions of complex order b.
H.S. AL-AMIRI AND T.S. FERNANDO
330
REFERENCES
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AOUF, M.K. and NASR, M.A., Starlike Functions of Complex Order b, J. Natural
Sci. Math.
25(1), (1985), 1-12.
2.
SINGH, S. and SINGH, S., Integrals of Certain Univalent Functions, Proc. Amer.
3.
Math. Soc. 77(3), (1979), 336-340.
AL-AMIRI, R.S., On Ruscheweyh Derivatives, Annales Polanlc Math., 38(I), (1980),
4.
RUSCHEWEYH,
88-94.
S.,
Convolutions
in
Geometric
Function
Theory,
Les
Presses
De
l’Unlversite De Montreal (1982)
5.
6.
7.
8.
READE, M.O., On Close-to-Convex Univalent Functions, Mlchlan Mah. J., 3,
(1955), 59-62.
OZAKI, S., On The Theory of Multlvalent Functions, cl. Pep. Tokyo Burnlka Paig.
A2 (1935), 167-188.
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CLUNIE, J., On Meromorphic Schllcht Functions, J. London Math. Soc. 34 (1952),
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