Internat. J. Math. & Math. Sci.
Vol. I0 No. 4 (1987) 733-744
733
ON A CLASS OF p-VALENT STARLIKE FUNCTIONS
OF ORDER c
M.K. AOUF
Dept. of Maths., Faculty of Science
University of Mansoura
Mansoura, Egypt
(Received October 9, 1986)
ABSTRACT.
Let
in the unit disc U
B
<
s
1 and o
<
b
<
denote the class of functions w(z), w(o)
o, lw(z) <
analytic
<
1}. For arbitrary fixed numbers A, B, -1 < A < 1, -1
{z" zl
< p, denote by P(A, B, p, ) the class of functions P(z)
p +
n
n z analytic in U such that P(z)
n=1
p + [p B +
(A-B) (p-)] w(z)
1 + Bw(z)
w
.
n=p+l
condition that
U. Moreover, let S(A, B, p, ) denote
z
zp +
the class of functions f(z)
P(A, B, p, ) if and only if P(z)
a n z n analytic in U and satisfying the
S(A B p ) if and only if zf’(z)
f(z)
f(z)
P(z) for some P(z)
P(A, B, p, ) and all z in U.
In this paper we determine the bounds for
f(z)
and
arg
f-zz
in S(A, B,
p, ), we investigate the coefficient estimates for functions of the class S(A, B,
p, ) and we study some properties of the class P(A, B, p, ).
KEY WORDS AND PHRASES.
p- Valent, analytic, bounds, starlike functions of order
1980 AMS SUBJECT CLASSIFICATION CODES.
30A32, 30A36.
INTRODUCTION.
Let A (p a fixed integer greater than zero) denote the class of functions
P
p
+
which are analytic in U {z
f(z) z
Izl <1}. We use to denote
k=p+l ak
the class of bounded analytic functions w(z) in U satisfies the conditions w(o)
o
ana w(z)
U.
z
for z
Let P(A, B)(-1 B < A 1) denote the class of functions having the form
1.
zk
T.
bn zn
n=l
which are analytic in U and such that
Pl(Z)
P1 (z)
1 +
+ Awz
Bw(z)
+
w
,
z
(1.1)
Pl(Z)
P(A, B) if and only if
U.
The class P(A, B) was introduced by Janowski [I].
(1.2)
734
M.K. AOUF
For -I
B
<
A
and o
<
< a
<
p, denote by P(A, B, p, ) the class of
functions
p + z c k z k which are analytic in U and which satisfy that P(z)
k=l
p, ) if and only if
P(z)
P(z)
(p -)
Pl(Z)
+
Pl(Z)
P(A, B).
Using (1.2) in (1.3), one can show that P(z)
p +
P(z)
+
(A-
1+
It was shown in [1] that
P(A, B,
(1.3)
P(A, B, P, ) if and onIv if
)IzP
w
Bw.
(1.4)
P(A, B) if and only if
Pl(Z)
(1.5)
-B
p(z +
I +
P (the class of functions of form (1.1) which are
P(I, -I)
for some p(z)
analytic in U and have a positive real part in U). Thus, from (1.3) and (1.5), we
have
P(z)
P(A, B, p, ) if and only if
[p+pB+(A-B)(p-)]p(z)+[p-pB-(A-B)(p-)]
P(z)
(l+B)p(z) + I-B
p(z)
e
(1.6)
P.
Moreover, let S(A, B, p, ) denote the class of functions f(z)
Ap
which
satisfy
zf’(z)
(1.7)
P(z)
f(z)
for some P(z) in P(A, B, p, ) and all z in U.
We note that S(A, B, 1, o)
S*(A, B), is the class of functions
which satisfy
zf’l(Z)
PI (z)’ PI
fl(z)
P(A, B).
f1(z)
AI
(1.8)
The class S*(A, B) was introduced by Janowski [I]. Also, S(1, -1, p, )
S(p), is
os
<p, investigated by
the class of p-valent starlike functions of order
Goluzine [2].
From (1.3), (1.7) and (1.8), it is easy to show that f S(A, B, p, ) if and
only if for z U
,
f(z)
2.
zp [
fl(z) ]
’z
(P
)
fl
THE ESTIMATION OF f(z)l AND arg
LEMMA 1.
Let P(z)
(1.9)
S*(A, B).
’z-Ll
FOR THE CLASS S(A, B, p, ).
P(A, B, p, ). Then, for
Izl <
r, we have
CLASS OF p-VALENT STARLIKE FUNCTIONS OF ORDER e
P-[PB+(A-B)(P..-=)]Br2
2
P(z)-
PROOF.
(A-B) (p.-=) r
B2r 2
It is easy to see that the transformation
1+Aw(z)
i+Bw(z) maps lw(z)l
P1 (z)
_<
l_B2r
?35
<
r onto the circle
2
1-ABr
Pl(Z) 1-B2r
2
(2.1)
Then the result follows from (1.3) and (2.1).
THEOREM i.
If f(z)
S(A, B, p, ), then for
Izl
r, o
<
r
<
I,
C(r; -A,-B, p, )< f(z)I -< c(r; A, B, p, ),
where
e
(A-B)(p
rP(l
B
+ Br)
(2.2)
for Bo,
C(r; A, B, p, )
rP.eA(P
)r
for B=o.
re i, o <
These bounds are sharp, being attained at the point z
f,(z)
zp
fo
f*(z)
zp
fo(Z;
2, by
(z; -A, -B, p, )
(2.3)
A, B, p, ),
(2.4)
and
respectively, where
(I +
f(z;
Be-iQz)
B
forB# o
A, B, p,)
)e -iCz
eA(P
PROOF.
f(z)
From (1.9), we have f(z)
zp [
fl(z)
’z
]
(P
fl
P1
o
S(A, B, p, ) if and only if
=)
It was shown by Janowski [I] that for
z
() -1
fl(z) z exp j"
d
o
forB
S*(A, B)
fl(z)
P (z)
(2.5)
S*(A, B)
P(A B)
(2.6)
M.K. AOUF
3
Thus from (2.5) and (2.6), we have for f(z)
z
I
z p exp
f (z)
p
() -P
S(A, B, p, )
d ; ), P
(z)
P(A, B, p, ).
o
Therefore
Izl p
If(z)I
z
I
exp (Re
o
zt, we obtain
substituting
f()l
zl p
exp (Re I
o
zlP
exp
P(zt) -p dr).
t
Hence
f(z)l
<
(Re P(zt)t
I max
o
-P)dt)
From Lemma I, it follows that
P(zt) -p
max
ztl
rt
(A-B)(p
t
e)r
l+Brt
hen, after integration, we obtain the upper bounds in (2.2). Similarly, we obtain
the bounds on the left-hand side of (2.2) which ends the proof.
REMARKS ON THEOREM I.
1.
Choosing A
1 and B
-1 in Theorem 1, we get the result due to Goluzina [2].
2.
P
and
Choosing
o in Theorem 1, we get the result due to Janowski [I].
3.
Choosing p
1, A 1 and B =-1 in Theorem 1, we get the result due to
Robertson [3].
4.
Choosing p
1, A 1 and B
o in Theorem 1, we get the result due to
=
Singh
[4].
THEOREM 2.
If f(z)
arg
(fz_p)I
arg
(fz-p)I
<_
S(A, B, p, ), then for
(A-B)(p-B )
sin -1
A(p-)r, B
o.
Izl
fo(Z)
I
(A
(S
z)
<
I
(Br), Bfo,
(2.7)
(2.8)
These bounds are sharp, being attained by the function
[ zP(1 + B
r
B)(p
B
fo(Z)
defined by
)
B f o,
(2.9)
z p exp(A(p
)
z)
B
o,
II
1.
CLASS OF p-VALENT STARLIKE FUNCTIONS OF ORDER
It was shown by Goel and Mehrok [5] that for
PROOF.
fl(z)
arg
fl(z)
arg
(A
<
B
<
z
B)sin-1
B
Ar
(Br), B
fl
=
737
S*(A, B)
(2.10)
o,
(2.11)
o.
Therefore, the proof of Theorem 2 is an immediate consequence of (1.9), (2.10) and
(2.11).
REMARK ON THEOREM 2.
1, A 1 and B =-1 in Theorem 2, we get the result due to
Choosing p
Pinchuk [6].
3.
COEFFICIENT ESTIMATES FOR THE CLASS S(A, B, P, ).
< p and
LEMMA 2. If integers p and m are greater than zero; o
1. Then
-1 < B < A
m-1
n I(B
j=o
m-1
[k2(B 2
z
-
2
)2+ B
(j + 1)
A)(p
-I)+ (B- A) 2 (p- ) 2 + 2kB(B
k=l
k-1
A) 2 (p
(B
](B
A)(p
(B
A) (p -)
)2
m
A)(p
)] x
12
) + B.
(3.)
+
(j
j:o
PROOF. We prove the lemma by induction on m. For m 1, the lemma is obvious.
Next, suppose that the result is true for m q-l. We have
2
I) +
+
R
q2
I)2
A)2(P
(B
(B
)2
(B
A)2(P )2
+ [(q
q-2
j=o
1) 2 (B 2
I(B-
k-1
+ 2kB(B -A)(p -=)] x n
A)2(p
q
,Lk2B
k:l
(B
j=o
)2
q-2
+ I:
k =1
[k2(B 2
A)(p
)] x
2
1) + (B- A) (P
+ B
A){p-=I
(j + I)
BI
1) +
k-1
+ 2kB(B
A)(p -) +
(j + 1) 2
I(B- A)(p-) +
(j +
j=o
-=)2
/
2(q
I)2
B
12
1)B(B -A)(p- )] x
738
M.K. AOUF
q-2
I( B
II
j=o
A)(p- )
(j +
q-1
I(B
+
)
+
I)2
THEOREM 3
.
k=p+1
n-(p+l)
<
for n
n.
q.
This proves the lemma.
ak zk
S(A, B, p, ), then
A)(p -) + Bk
(B
11
+
12
B
zp +
If f(z)
-)2
q2
(j +
Showing that the result is valid for m
j=o
+ (B- A) 2 (p
1)B(B A)(p -o)
2(q
x
1)2
A)(p
1)2B2
(q
2
Bj
(3.2)
k+l
p + 1, and these bounds are sharp for all admissible A, B and
>
PROOF. As f
zf’(z)
S(A, B, p, ), from (1.4) and (1.7), we have
p + [pB +
(A- B)(p -)]w(z)
w
A)(p -)]f(z)
w(z)
+ Bw z)
f(z)
This may be written as
Bzf’(z) + [-pB + (B
pf(z)
zf’(z).
Hence
pz p + I: (p +
k=l
[B
Z
k=l
ap+ k z p+k}
pz p +
]w(z)
z (p +
k=l
or
k)ap+ k z p+k}
[pB + [-pB + (B
+ [-pB + (B
z p + z (p +
k=l
p
A)(p -)]
k)ap+ k z p+k}
k)ap+ k zP+k
A)(p -)] +
I:
(p + k)B +
k=l
[-pB + (B
z
A)(p -)]
p- (p + k)
k=l
which may be written as
ap+ k z k]
ap+ k
z
w(z)
(P
p) +
zp +
and for each
CLASS OF p-VALENT STARLIKE FUNCTIONS OF ORDER
[{(p + k) B + [-pB + (B
z
ap
w(z)
and
739
w(z)
(3.3)
a p+ k z
-k
k=o
where
ap+ k z k]
A)(p -)]
k:o
=
z
k=o
z k+1
bk+l
Equating coefficients of z m on both sides of (3.3), we obtain
m-1
z
(p + k) B + [-pB + (B
k=o
A)(p -)]
ap+ k bm_ k
-m} a
p+m
ap+m on
which shows that
ap, ap+
right-hand side depends only on
ap+(m_1
of left-hand side.
Hence we can write
m-1
z
[
(p + k) B + [-pB + (B
k=o
.
m
ap+ k
-k
k=o
for m
1, 2, 3,
mr1
ap+ k
Akzk
z
k=m+l
(P + k) B + [ -pB + (B
A)(p -)]
k=o
121
ap+k 12
m-1
{(p + k) B + [-pB + (B
k=o
A)(p -)]}
{(p + k) B + [-pB + (B
k=o
m
z {-k}
k=o
.
m
k=o
m
2
z
k:o
l-kl
2
ap+ k rkeik +
2
ap+kl
k2 ap+k
2
r 2k
r 2k +
z
k=m+1
A) (p -)]}
s
k=m+1
Akl
Ak
rkeik
r 2k
ap+ k rk eie k
2
e io k
2
m-1
z
z k ] w(z)
and a proper choise of A k(k_> o).
o < r < 1, o_< o_< 211, then
rei
Let z
zk +
A)(p -)]
ap+ k
ie)l 2de
2
de
2 2k
r
(3.4)
M.K. AOUF
7zO
Setting
as
1 in (3.4), the inequality (3.4) may be written
r/
m-1
z
2
k2}lapk
)[(B
A)(p
A)(p -)]
(p+k) B + [-pB + (B
k=o
lap+m 12
m2
>
Simplification of (3.5) leads to
m-I
ap+m
k2 (B 2-
z
<-
A)(p
1)+ (B
) +
2kB]}lap+k
12
(3.6)
k=o
-
Replacing p + m by n in (3.6), we are led to
2
I
lanl
n-(p+l)
(n_p)
{k2(B 2
k=o
2
(3.7)
ap+kl
A)(p -) + 2kB]
[B
I) + (B- A)(p- )x
where n > p + 1.
For n p + 1, (3.7) reduces to
ap+ll
2
<
)2
2
(B- A)(p-
or
lap+t1
(A
B)(p
e)
which is equivalent to (3.2).
To establish (3.2) for n
(3.8)
>
p + 1, we will apply induction argument.
Fix n, n > p + 1, and suppose
(3.2) holds for k
I, 2,
n-(p+1). Then
2
anl
(B
(n_p)2
A)(p -e)[(B
(B
A)2(p )2
n-(p+l)
k2 (B 2
:
+
A)(p -e) + 2kB]
k-1 B-A)(p-) +
]I
(J +
j:o
1)2
Thus, from (3.7), (3.9) and Lenma 2 with m
2
I) +
k=1
n-(p+l)
j=o
I(B
A)(p
e)
(j + 1) 2
+ Bj
n
12
B
12
p, we obtain
(3.)
CLASS OF p-VALENT STARLIKE FUNCTIONS OF ORDER
This completes the proof of
Clunie [7].
,
741
(3.2). This proof is based on a technique found in
For sharpness of (3.2) consider
zp
(B-A)(p)
B
z)
f(z)
(1
B
,I
1, B # O.
REMARK ON THEOREM 3.
1, A
1, and B =-1 in Theorem 3, we get the result due to
Choosing p
Robertson [3] and Schild [8].
DISTORTION AND COEFFICIENT BOUNDS FOR FUNCTIONS IN P(A, B, p, c:).
4.
THEOREM 4. If P(z) P(A, B, p, ), then
larg p(z)
sin-1
<
The bound is sharp.
The proof follows from Lemma 1.
PROOF.
P(z)
p
-([B + (A-B)(1
r
i=
/1
-- -
+ [B + (A-B)(1
1 + 51 Bz
Putting
(4.1), we have
arg P(z)
sin -I
P
p
To see that the result is sharp, let
I11
)] +B)r
e)]Br2
B)(p -)]Br 2
An immediate consequence of Lemma 1 is
COROLLARY 1. If P(z) is in P(A, B, p, ), then for
p
(A-B)(p
)r-[pB + (A-B)(p
1
p + (A-B)(p
)]Br 2
B2r2
)r-[pB + (A-B)(p
1
(4.1)
1.
]Br2
B)(p
[pB + (A
(A
r.
)]2r /1
I + [B + (A-B)(I
in
z
()](1
P
I + [B + (A-B)(1
[B + (A-B)(1
Izl
p-[pB + (A-B)(p-)]Br 2’
B2r
_<
Izl
<
r
I
<
P(z)l
)]Br 2
2
and
p
(A-B)(p
)r-[pB + (A-B)(p
B2r
1
p + (A-B)(p
)r-[pB + (A-B)(p
1
)]Br 2
2
B2r
2
)]Br 2
<
Re {P(z)}
<
M.K. AOUF
742
REMARK ON COROLLARY 1.
I, A
and B =-I in the above corollary we get the following
Choosing p
distortion bounds studied by Libera and Livingston
[9] stated in the following
corollary.
<
If P(z) is in P(1, -1, a)= P(a), o<
(the class of
functions P(z) with positive real part of order
o < < 1), then for zl _< r < I
COROLLARY 2.
1
2(1
,
)r+(l
I -r"2
2
)r+(2
-r
2
a)r 2
1 +
P(z)l-<
-<
2C! )r+(1
2
)r 2
I -r 2
and
2(1.-
)r 2 <Re {P(z)}
+
<
2(1
)r+(1.-
2
I -r 2
a)r 2
The coefficient bounds which follow are derived by using the method of Clunie
[7].
THEOREM 5.
If P(z)
p +
.
k=l
Ibnl
_<
(A
B)(p -), n
bkzk
is in
P(A, B, p, a), then
(4.2)
1, 2
these bounds are sharp.
PROOF. The representation P(z) in (1.4) is equivalent to
[B + (A
B)(p -) -B P(z)]w(z)
[B + (A
B)(p -a) -B
P(z) -p, w
(4.3)
R
or
.
k=o
bkzk]w(z)
z
k=l
bk zk
bo
.
q
p
(4.4)
This can be written as
n-I
[(A- B)(p -) -B :
k=l
bkzk]w(z)
n
k=l
bk zk +
k=n+l
kzk
(4.5)
the last tern also being absolutely and unifomly convergent in compacta on U.
re ie, perfoming the indicated integration and making use of the bound
Writing z
Iw(z)l
zl < 1 for z in u gives
CLASS OF p-VALENT STARLIKE FUNCTIONS OF ORDER
B)2(p =)2
(A
B2
+
nl
bkl2
k=l
;2111
lo, 2111
1
2 o
>
n-1
(A- B)(p -) + B z b k
k=1
n-I
B)(p -) + B
{(A
k=l
o
n
:=
k 1
n
bk rk eike
2
k=1
k=n+II:
qkl
z
k=n+l
eikol2
rk
qk
2
r2k +
bkl
y.
+
743
r 2k
dB
b kr k e iOk} w(re
y.
,
rkei ko
io)
2
2
r 2k
The last term is non-negative and r < 1, therefore
(A-
)2
B)2(p
+ B2
2
n
z
nl
Ib
ib
kl
k12
k=l
k=l
(4.6)
>_
or
Ibn 12
Since
-1
Ibnl
<
<
B
<
=) 2 + (B 2- 1)
B) 2 (p
(A
1, we have B 2 -I
-< (A
B)(p
<
p + (A
:
k=l
ib k 12
(4.7)
Hence
(4.8)
=)
and this is equivalent to (4.2).
P(z)
o.
n-1
B)(p
If w(z)
z
n,
then
=)z n +
which makes (4.2) sharp.
REMARK ON THEOREM 5.
1, and B -I in Theorem 5, we get the result due to Libera
1, A
Choosing p
[10] stated in the following corollary.
COROLLARY 3.
If P(z)
1 +
z
k=l
then
Ibnl
<2(1
=), n
these bounds are sharp.
1, 2, 3,
b
k
zk
E
P(1, -1, 1, =)
P(=), o
<
<
I,
M.K. AOUF
744
REFERENCES
I.
2.
JANOWSKI, W. Some external problems for certain families of analytic functions, Ann. Polon. Math., 28 (1973), 297-326.
GOLUZINA, E. G. On the coefficients of a class of functions, regular in a disk
and having an integral representation in it, J. of Soviet Math. (21 6_ (1974),
606-617.
3.
4.
5.
6.
7.
8.
9.
10.
ROBERTSON, M. S.
On the theory of univalent functions, Ann. of Math., 37
169-185.
(1936),
SINGH, R. On a class of starlike functions, J. Indian Math Soc., 32 (1968),
208-213.
GOEL, R. M. and MEHROK, B. S. On a class of clQse-to-convex functions, Indian
J. Pure Appl. Math., I_2(5) (1981), 648-658.
PINCHUK, B. On starlike and convex functions of order a, Duke Math. J., 35
(1968), 721-734.
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