Intn. J. ath. & Math. Sci.
Vol. 6 No. 3 (1983) 449-458
449
SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS
E.M. $1LVIA
Department of Mathematics
University of California, Davis
Davis, California 95616
(Received lanuary 24,
Let [C,D], -1 (_ D
ABSTRACT.
$(z), S(0)
I
S’(0)
0,
(
C
i,
1983)
denote the class of functions
analytic in the unit disk U
such that I + (zg’’(z)/g’(z))
is subordinate to
(z:
Jz
<
1]
(l+Cz)/(1+Dz), z s U.
We investigate the subclasses of close-to-convex functions f(z),
f(0)
I
f’(0)
for which there exists S s }([C,D]
0,
f’/g’ is subordinate to (l+Az)l(l+Bz), -I (_ B
(
A
I.
rotation theorems and coefficient bounds are obtained.
shown that these classes are preserved under certain
such that
Distortion and
It is also
intesral
operators.
WORDS AND PHRASES,
Univalent, convex,
convolution.
starlike,
AMS (MOS) SUBJECT CLASSIFICATION (1980) CODES:
subordination,
30C45, 30C55.
INTRODUCTION,
1
Let
S
z +
denote the class of functions f(z)
n--.
tic and univalent in the unit disk U
and
G
analytic in
< G, if there
with w(0)
is univalent
we say that
exists a Schwarz
]w(z)
0 and
in
U
U
(
then S
I in
(z:
S
is
]z
analy-
For functions
subordinate to
function w(z),
U,
I).
a z
n
G,
denoted
w(z) analytic in
such that g(z)
G if and only if (0)
G(w(z)).
G(0) and
U
If
G
s(U) c
450
E. M. SILVIA
A
For
G(U).
B,
and
-1
<_ B
A _< 1,
function
I is in the class [A,B] if p(z)
with p(O)
class was introduced by Janowski [4].
p
analytic in
(l+Az)/(l+Bz).
C
Given
D,
and
-I i D
C
I, ][C,D] and J [C,D] denote the classes of functions
in
U
with f(0)
f’(0) -I
and zf’(z)/f(z)
s
troduced by Janowsci [4]
and [3]).
For C
-I, }[i,-I]
well-known subclass of convex (starlike)
z +
the class C[A,B;C,D], -I
g s ){[C,D]
a z
(
were in-
e .[A,B].
U
in
C
is
,
if and only if
z
0
the
said to be
in
I, if there exists
!
The well-known (Kaplan [5])
class of close-to-convex functions is C[I,-I;I,-I]
and [A,B] c [i,-I]
,/ ),
functions.
analytic
n--. n
(,/ [I,-I]
<_ B < A <_ I, -I <_ D
such that f’/g’
g s ,/ [C,D]
anlyric
s [C,D]
JS[C,D]
The classes
<
and studied further by Goel and Mehrok ([1]
i and D
A function f(z)
f
0 such that I + zf’(z)/f’(z)
[C,D], respectively.
This
shows C[A,B;C,D]
g() -1 d
C
C while
C c S.
Since
e [C,D], we also note that
C[I,-I;C,D] was studied by Goel and Mehrok ([2] and [3]).
In Section 2 Of this paper we obtain distortion and rotation
theorems for f’(z) wheever f s C[A,B;C,D] and a subordination result
relating C[A,B;C,D]
class C[A,B;C,D]
and [A,B].
In Section 3, it is shown that the
We
is preserved under certain integral operators.
conclude with coefficient inequalities.
2.
DISTORTION
_
AND ROTATION THEOREMS.
Unless otherwise mentioned in the sequel,
on the real constanta A, B,
-1 .i B
<
A .i 1.
THEOREM 1.
(1-At) (1-Dr)
C and D are that -i
f s C[A,B;C,D],
(C-D)/D
1-Br
<
the only restrictions
if (z)l <
Izl
i
r
(l+Ar)(l+Dr)
I+Br
<
D
<
C
I and
1,
(C-D)/D
D
0
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS
451
and
(-A) x,-Cr)
l-Br
< If’(z)l <
1+Ar)ex{Cr)
D
O.
Th___e bounds are sharp.
For f
PROOF.
p e [A,B]
8
C[A,B;C,D], there exists a g
such that
Since g e [C,D]
(1-Dr)
exp(-Cr) <_
For p z [A,B],
l-At
I-Br
_
if and only if zg’
(C-D)/D
Iz]
! Ig’(z)l
Ig’(z)l _<
r,
< Ip(’-)l
(2.1)
g’(z)p(z).
f’(z)
e
" [C,D],
(l+Dr)
for
(C-D)/D
z
D
!
0,
r
i
[4]
and
(2.2)
exp(Cr)
(I+Az)/(I+B) gives
the univalence of
+A
(2.)
I+Br
The result follows immediately upon applying
Equality is obtained for f e C[A,B;C,D]
_l+Az) (1+Dz) (C-D)/D
I + Bz
D
(2.3) and (2.2) to (2.1).
satisfying
0
f’(z)
(2.4)
I+Az) exp(Cz)
I + Bz
and z
[C,D] and
8
D
0
+ r.
REMARK.
For A
Gee1 and Mehrok [2].
1 and B
-1, Theorem 1 agrees with Theorem 3 of
452
E. M. SILVIA
THEOREM 2.
,
For f
arcsin(Dr)
Jars
Izl <--
CtA,B;C,D],
r
<
I,
+ arcsin (A-B);
I_ABr2
D
0
D
0
f’ (z)
+ arcsin (A-B)r
arcsin(Cr)
PROOF.
Since zg
I-ABr 2
From (2.1) we have
[C,D], we know [2] that for
e
arcsin(Dr)
]zl
_
(2.5)
r
<
1
D 4 0
(2.6)
For p s [A,B],
I" --I-AB;21
<
r)
is
contained in the disk
(A-B)r
frm which it f11’’
that2
2
1-B r
1-Br
larg
Substituting
P(IZl <
P(z)
!
arcsin
(2.6) and (2.7)
REMARKS 1.
For A
1, B
(A-B)r
2"
I-ABr
into
(2.7)
(2.5) gives the result.
-1, Theorem 2 agrees with Theorem 4 of
Goel and Mehrok [2].
2.
For A
result of Krzyz [7]
C
1, B
D
-1, Theorem 2 reduces to the
that
]arg f’(z)] !
2(arcsin r + arctan r),
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS
453
The convolution or Hadamard product of two power series
f(z)
and g(z)
a z
n
n=O
n=O
(f’g) (z)
a b
z
n
b z
n
is
defined as
the power series
In order to obtain a subordination result
n=O
linking C[A,B;C,D]
and [A,B]
we need the
following
.
LEMMA A (Ruscheweyh and Shell-Small, [11]).
and
U
b_e convex i__n
THEOREM 3.
that for all
and
Then
C[A,B;C,D] then there exists p e [A,B] such
I___f f
s
and suppose f <
t
with
I+Dsz!C-D)/D
()
D
0
f’ (sz)p(tz)_
f’(tz)p(sz)
(2.8)
exp[C(s-t)z]
D
0.
PROOF. We will use an approach due to Ruscheweyh [10]. From (2.1)
zg’’(z)
we have zf’’(z
+ zv’ (z) for g e [C,D] and p s [A,B].
f’(z)
g’(z)
p(z)
Therefore,
zf’’ z.L_ z’(z)
f’(z)
p(z)
For
s
and
t
such that
_
(1 + zx"(z)
Isl
1 <
g’(z)
I,
tl
!
1 + Dz
(2 9)
I, the function
Z
h(z)
0
with this
(l-su
h,
------)du is convex in
1-tu
U.
Applying Lemma A to (2.9)
we have
(zf’’(z)
f’ (z)
Given any function
zp’(z))
p(z)
.
h(z)
with
(0)
* h(z) < 1(C-D)z
+ Dz
(z) analytic in
U
(2.10)
O, we have
E. M. SILVIA
454
SZ
(’h)(z)
(u) du
z
so that
U,
8
(2.10) reduces to
tz
SZ
’(sz)v(tz))-)
lg(p(sz)f’
(tz
< (C-D)
tz
du
(2.11)
1 + Du
and exponentiating both sides
Integrating the righthand side of (2.11)
leads to (2.8).
COROLLARY 1.
If f
8
C[A,B;C,D] then there exists a_ p e f[A,B]
and a Schwarz unction w(z)
p(z)
such that
(1 + Dw(z))
(C-D)/v
D
0
f’(z)
p(z)exp(Cw(z)
PROOF.
The result follows directly upon substituting
1 and
0 into Theorem 3.
t
COROLLARY 2.
z +
If f(z)
n=2
1"21
!
e C[A,B;C,D]
a z
n
then
(C-D) + (A-B)
2
PROOF.
we
s
Ig’(O)
If g < F then
[8]
From Corollary I,
f’(z)/p(z) and
take g(z)
l+Dz)
(c-D)/D
D
0
D
0
F(z)
xp(Cz)
Then g’(0)
Therefore
2a
2
c
I
21a21- l1{
as claimed.
for p(z)
1 +
c z
n=l
<- {C-D{
n
(C-D)
and
la2l
<_
and F’(0)
+I Cll
2
<-
C
D.
(C-D) + (A-B)
2
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS
455
INVARIANCE PROPERTIES.
3.
We will need the followins lemmas.
and
starlike
g
*Fg(u)
,g
i__n
U.
the
in
contained
is
LEMMA C (Silverman and
s
,
convex
Silvia,
hull
U
in
with
l__f cp
1.
[12]).
e
/[C,D]
then for
K, and f e C[A,B;C,D] then 0*f s C[A,B;C,D].
F s [A,B] such that zf’(z)
/ [C,D]
g(z)F(z).
and
(l+Az)/(l+Bz)
Since
is convex
by Lemma B,
0*Fg .< I+Az
z(0sf)
o,g
.
o,g
0 v
0*f
e C[A,B;C,D].
REMARK.
For A
C
i,
B
(3.1)
(z e U)
l+Bz
From Lemma C, 0,g e
for
*[C,D]
D
so that
(3.1)
is equivalent
-1, Theorem 4 was proved by
Ruscheweyh and Sheil-Small [11].
COROLLARY.
(i)
F(0)
F(U).
of
For f s C[A,B;C,D] there exists g
PROOF.
U,
analytic
b_e convex
./ [C,D].
*g
THEOREM 4.
in
F
Then for
q
Le_._t
LEMMA B (Ruscheweyh and Shell-Small, [ii]).
V l(z)
I__.f f e C[A,B;C,D] then so are
z7
rjZ tT-If(t)dt
Re 7
>
x
1
0
0
and
z
(ii) F
2(z)
o
f(_)-f(x.) d,
-x
1.
to
E. M. SILVIA
456
PROOF.
Observe that F. (z)
J
(h.*f) (z), j
1,2, where
n
n i
1
log[
l-x
l-z
n+
l-xz],
>
Re
z
0,
(z)
and
n--1
Ix
by Ruscheweyh [9] and h
I.
x
i,
is
2
Since h
I
clearly convex,
(l-x)n
was shown to be convex,
the result follows
immediately from Theorem 4.
REMARK.
Goel and ]/ehrok [3]
Fl(Z)
preserved under
showed that C[1,-1;C,D] was
1,2,3
when 7
and under
F2(z)
when x =-1
by a different method.
4f,.
COEFFICIENT INE(UALITIES
_-
We begin with coefficient inequalities for ][C,D].
z +
For g(z)
LEblt/A.
n;2
C-D
b21
PROOF.
For g(z)
s E[C,D]
b z
n
z +
n--2
Schwarz function w(z)
n--1
n
7n
and
complex
Ix
and
bnZ
n
e-’-’tU,VJ, there
exists
a
such that 1 + (zg’’(z)/g’(z))
(l+Cw(z))/(l+Dw(z)) or zg’’(z)/g’(z)
(C-D)w(z)/(l+Dw(z)).
Substitution of the series expansions and comparison of the
coefficients leads to
b2Therefore,
C-D
2
Ib 2]
b3
-
and
71
!
Ixb2
C-D
2
b3-
C-D
6
(72
+
(C-2D)71
2
}"
and
C-D
6
{2
+
(C-2D)
-",
Ix(C-D)
]71
).
(4.1)
SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS
We know [6] that for
457
complex
s
2
(4.2)
Combining (4.2) and (4.1) yields the result.
REMARK.
[8]
If we apply the inequality
to
(4.1), the same proof shows that
Ib3-11b 2 2[
2
+
THEOREM 5.
For f(z)
I(C-2D)-
z +
(C-D)+(A-B)
2
i}
n
PROOF.
There exists a g(z)
+
(A-B) (C-D+1)
z +
n=2
function w(z)
n=l
z e U.
b z
n
The bound for
0)
a2
b3
n
+ 2
(A_B)b271
+
(A-B)
3
follows from the Lemma.
to (4.3), we have
a^
C-2D[
I
C-2D
1
R[C,D] and a Schwarz
such that f’(z)Ig’(z)
z
Comparing series expansions, we see
a3
Lemma (p
n
2
and
+ (A-B)(C-D+1)
(C-D) (C-2D)
Ib2l
s C[A,B; C,D]
a z
n=2
,la21
,(C-D)
(l+Aw(z))l(l+Bw(z))
b
2
A-B
+--:--
( 2 -By 1 2 )"
Applying
(4.2)
i
and
(4.3)
and the
458
E.
.
SILVIA
C-D
(A-B)
max(l,
3
IB[
C-D
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2
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3
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Awl. Math.
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4
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