Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 435-438
435
ON ALPHA-CLOSE-TO-CONVEX FUNCTIONS OF
ORDER BETA
M.A. NASR
Faculty of Science
University of Mansoura
Mansoura, Egypt
(Received January 9, 1985)
MB()
ABSTRACT Let
[
0 and B -> O]
>
f(z)
denote the class of all functions
z +
z an z
n--2
U
analytic in the unit disc
the condition
for all
m when
e2
o
s I.
>
< (
.
) zf’
f(
(1
Re
f’(z)f(z)/z
with
0
KEY WORDS AND PHRASES.
Close-to-staa,
-
z=re
zf"(z))
f’(z)
do
fMB()
is close-to-star of order
+ (I +
In this note we show that each
and which satisfy for
>
functions, close-to-convex functions,
a-convex
functions.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
30 C45
NTRODUCTI ON.
A shall denote the class of all functions
f(z) =z + z
n=2ar, z
n
analytic in the unit disc U
{z-lzl<1}
tions in A which are univalent in U.
Let a_>o, _>o, and let f A with
(I- ) zf’(z)
f(z)
J (, f)
If for z re
whenever
order
i(
U"
le2 {
0-< 01
or f
<
Re d(a,
e2
MB().
2,
f)l
de
then f
The class
>
-
and
S
shall denote the subclass of func-
f’(z)f(z)/z
+
(i +
0
in
zf"(z)).
f"(z)
b,
and let
(I I)
(1.2)
-close-to-convex function of
is said to be an
was introduced by Nasr[l].
MB()
M.A. NASR
436
It was shown [1] that MB()CS if and only if 0-< B < o.
KB, is the class of close-to-convex functions of order B
Note that M B (1)
and by Pommerenke [4]
introduced by Reade [2] and studied by Goodman [3] for B ->
R is the class of close-to-star functions of order
for 0 _< B < 1, and MB(O)
B
M(o) is the class of s-convex funcintroduced by Goodman [3]. Moreover Mo(o)
tions introduced by Mocanu [5] and My/o
) By(o), o > O, is the class of
Bazilevi functions of order y introduced by Nasr [6].
In this note we continue the investigation of o-close-to-convex functions of
order B studied in [I].
2.
RESULTS
In this section we show that each
MB(o)
is close-to-star of order B when
f
() is close-to-convex of order
when
we show each f
<
<
0
_<
when
f
-<
then
<
B y
o and if f
r
We assume, unless otherwise stated, that o is a real number, that 0
iO
< o
Also that 0
and that z=re
We shall need the following result.
0
0
For
< o.
M
>_
LEMMA I"
If
given by
h
(2.1)
(The powers taken are the principal values).
RB
PROOF"
then the function
(zf’(z)/f(z))
f(z)
h(z)
belongs to
MB(y),
f
.
M(y)
M(),
Let
If we choose the branch of (zf’(z)/f(z))
a simple calculation shows that the function h
MB().
f
when z
equal o
(2.1) belongs to
0
which is
defined by
RB
THEOREM iPROOF"
Since
foe
Re d--
MB(o)
f
If
MB(o)
f
}
then
R
f
B
it follows from Lenma
d0
fz Re I
zh ,( z)
h(z)
In (2.2) we choose the branches for f(z) 1/a and
Of102
for
Re
w
pe
{h(z)/f(z) 1/}dO
to obtain
(2.2)
8
h(z) I/a for which
(2.3)
If we use (2.1), (2.2) and (2.3) it is easy to prove that
z=O.
In fact, since
do
(h(z)/f(z)) I/
h(z)I/o/f(z)I/
with value
}
that
f
(2.4)
TI.
is univalent, we can let
w
f(z),
z
z(w)
f- (w)
and
ALPHA-CLOSE-TO-CONVEX FUNCTIONS OF ORDER BETA
h(z). 1/a
1/6
w
I
437
dw
f(z)/
/P
I/
d_ [ h{z(pei@))]
d (pe i@ -/
o
II
I/
i/a
I/
TT-- [P
o
P
dp
1/a
[ h(z)
[f(z)] II
d
d
p
l/a
p
dp.
Hece
o
h(z) I/
{fi)i/a I
Re
dO
{ d[h(z)]l/4
Re
dp.
The result now follows frGm (2.1) and (2.4).
COROLLARY 1PROOF"
O
If
Let f
S2
Re
MI(),
f
M 6(),
-> 1,
-> 1,
K.
f
then
zf"(z)
{(I+ f,--z)
} de
Now from THEOREM I, we have
then
e- 1)
42
zf’(z_)}
f(z)
Re
Re zf’(z)/f(z)
do
dO
76.
-I,
and therefore
(1+
Re
el
de
and the proof of Corollary
COROLLARY 2"
f
If
f
Re
<
r, 13
M6(),
0
sO2
6 -< y < &,
-6
thep
Suppose there exists a
y
M
f
0
<
6
y -< e,
such that
U for which
+ 1-
01
6
is complete.
PROOFBy THEOREN 1, f R
6.
N (y). Then there is
6
01
(-1)13
>
f’(z)
{ [
Re
f
f’()
f-TT)---
]I
de
dO
(z.5)
However, for f
2
Re
+
f()
(2.6)
438
M.A. NASR
Substituting (2.5) into (2.6), we obtain
0
But (1
(I- T) [B +
<
e
)<0
tion that
f
REMARKFor B
implies
R B.
2
B1B2 {
Thus
f e
f’()
Re
Re
[
Tf’(T)
f()
dO]
]}de
<
-B, which contradicts the assump-
MB(y).
0 we obtain results due to Miller, Mocanu and Reade [7].
ACKNOWLEDGMENT
The author would like to thank Professor M. O. Reade for reading the manuscript
and giving some helpful suggestions.
REFERENCES
i.
NASR, M. A., Alpha-Close-to-Convex Functions of Order Beta, Mathematica 24
79-83.
2.
READE, M. 0., The Coefficients of Close-to-Convex Functions, Duke Math. Journal
23 (1950), 459-462.
3.
GOODMAN, A. W., On Close-to-Convex Functions of Higher Order, Ann. Univ. Sci.
Budapest Eotvos sect. Math. 2__5 (1972), 17-30.
4.
POMMERENKE, C., On the Coefficients of Close-to-Convex Functions, Michigan Math
Journal, 9 (1962), 259-269.
5.
MOCANU, P. T., Une Properiete de Convexite Generalisee dans la Theorie de la
Rpresentation Conform, Mathematica Ii (1969) 127-133.
6.
NASR, M. A., Bazilevivc Functions of Higher Order, Rev. Roum. Math. Pures er
A_p_p_.
7.
28
(1983) 709-714.
MILLER, S. S., MOCANU, P. T., AND READE, M. A., All s-Convex Functions are Univalent and Starlike, Proc. Amer. Math Sco. 37 (1973) 552-554.