Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 493-502
493
A GENERALIZATION OF MULTIVALENT FUNCTIONS
WITH NEGATIVE COEFFICIENTS
M.K. AOUF
Faculty of Science
Department of Mathematics
University of Mansoura
Mansoura, Egypt
(Received January 6, 1987)
T
Let
ABSTRACT.
the
be
P
class of
analytic and p-valent functions which can be
expressed in the form
n=l
Tp
The subclasses
,
Cp(A,B,a)of Tp
(A,B,a) and
have been considered.
Sharp results
concerning coefficient estimates, distortion and covering theorems are obtained.
radius of convexity for the class
,
T (A,B,) and
that the classes
P
C
P
The
It is further proved
T (A,B,) is determined.
P
(A,B,) are closed under arithmetic mean and convex
linear combinations.
KEY WORDS AND PHRASES.
P-valent, Analytic, Radius of Convexity.
30A32, 30A36.
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
INTRODUCTION.
S (p
Let
f(z)
I) denote the class of functions of the form
P
z
p
+
ap+n
n=l
U
there
{z:Iz <
I}.
exists
an
f..
For A,B fixed,
-I
z
p+n
which are analytic and p-valent in the unit disc
< F) if
U, such that
A function f is said to be subordinate to a function F(f
analytic
function
(z) with
l(z)l
f
A
<
B
l, and 0
<
p, we say that
if
zf’(z)
f( <
] +
[pB+(A-B)(p,a.)]z
’+
Bz
z e U,
Izl,
,
f e S
P
z
(A,B,u)
if and only
494
M.K. AOUF
,
or equivalently
S
f
(A,B,)
P
zf’(z)
P
f(z_)
B
Further
zf’(z)
P
f
S
--I
belong
to
U.
K (A,B,s) if
class
the
to
P
T denote the subclass of
P
which can be expressed in the form
f(z)
,
z
.
p
S
P
P
[I]
have
studied
if
Kp(A,B,a)
T
P
[2] and Silverman and Silvia [4,5] have studied
Jain
certain
only
and we set
z
Certain subclasses of univalent functions with negative coefficients.
Sohi
and
consisting of functions analytic and p-valent
P
T and C (A,B,a)
P
P
(A,B,a)
[3], Gupta and
Silverman
S
ap+n
n=l
,
(A,B,a)
P
I, z
* (A,B,).
Let
T
<
[pB+(A-B) (p-e)
f(z)
said
is
if and only if
subclasses
of
mmltivalent
functions
Also Goel and
with
negative
In this paper we obtain coefficient estimates, distortion and covering
theorems for the classes T (A,B,e) and C (A,B,e). We also determine the radius of
P
P
It is further shown that the classes
convexity for the class
T (A,B,e).
P
T (A,B,) and C (A,B,) are closed under arithmetic mean and convex linear
P
P
combinations.
By taking =0, we get results due to Goel and Sohi [I] and by
assigning specific values to A and B and taking p=l, we get results due to Silverman
cbefflcients.
,
,
,
[3] and Gupta and Jain [2].
2.
COEFFICIENT INEQUALITIES.
THEOREM I.
A function
f(z)
z
p
[
ap+n
n-I
n=l
[(l+B)n+(B-A)(p-e)]lap+nl..
z
p+n
is in
T * (A,B,=) if and only if
P
(B-A)(p-e).
(2.1)
The result is sharp.
zl
I, then
[zf’(z)
-JB zf’(z)
p+n
-n ap+nl z
n=l
[pB+(A-B)(p-a)] f(z)
-pf(z)
"
(B-A)(p-a)z p
[nB + (B-A)(p-)]
n=l
g
(l+B)n +
n=l
(B-A)(p-a)]]ap+nl_.
Hence by the principle of maximum modulus
Conversely, suppose that
lap+hi z p+n]
(B-A)(p-a)
,
f(z)
P
g 0.
GENERALIZATION OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
zf" (z)
P
f(z)
f’(z)
B
[pB + (A-B)(p-a)]
f(z)
z
n--I
n=l
Re
<
(B-A) (p-=) z p-
nB+(B-A)(P-)]
n=l
nlp+nl
i
n--
U.
ap+n
z
(2.2)
I.
p+n
zf’(z)
Upon clearing the
real.
f(z) is
through real values, we obtain
z on the real axis so that
denominator in (2.2) and letting
E
[nB+(B-A)(P-e)]Ia p+nl zp+n
(B-A)(p-a)z p
Choose values of
495
z
(B-A)(p-=)-
[nB + (B-A)(p-=)]
n--
lp+nl)
which implies that
lap+n[
[(l+B)n + (B-A)Cp-a)]
.
The function
z p-
f(z)
(B-A)Cp-a).
(B-A)(p-a)
(l+B)n + (B-A)(p-a)
n=
z
p+n
is an extremal function.
COROLLARY I.
f e T
If
P
f(z)
equality only for functions of the form
A function
COROLLARY 2.
(B-A) (p-a)
n, -(l+B)n+(BA)(p_a),
,,lap+
(A,B,=) then
f(z)
z p-
n=l
if
z
p
with
(B-A)(p-a)
z
(l+B)n + (B-A)(p-)
lap+nlZ p+n is
in
p+n
C (A,B,a) if and only
P
n-1
PROOF
Since
It is well known that
zf’(z)
P
we may replace
z
p
[
n=l
ap+nl
f e C (A,B,) if and only if
P
ap+n
lp+nl
z
in Theorem I.
zf’(z)
P
E
T
P
(A,B ,a).
M.K. AOUF
96
3.
REPRESENTATION FORMULA.
lan+plZ n+p
zp
f(z)
A function
THEOREM 2.
n=l
f(z)
where
z
p
exp {(B-A)(p-a)
z
(t)
-B t@(t)
P
(A,B,a) if and only If
(3.1)
I, z e U.
*(A,B,a),
PROOF.
zf’(z)
B
P
f(z)-zf(z)
[pB+(A-B)(p-a)]’[
f(.)
<
I, z
Since the absolute value vanishes for z
zf’(z)
if(z)
zf’(z)
B.
where h(z)
o, we have
(3.2)
h(z)
pB+( A-B (p-a)
analytic
is
U.
P
f (z)
h(z)
T
dt}
l#(z)l <
#(z) is analytic in U and satisfies
Let f(z) e T
then
P
is in
in U
and
lh(z) <
for
z e U.
Integrating
(3.2) with
z(z) we find that
f(z)
z
p.
exp {(B-A)(p-a)
’:
(t)
dt}
1-Bt#(t)
The converse is obtained by differentiating (3.1).
4.
DISTORTION AND COVERING THEOREMS FOR
,
THEOREM 3.
If
P
rp
(B-A) (p-a)
I+B+(-A) (p’a)
rP
(B-A) (p-a)
+ I+B+(B-A)(p--a)
with equality for
PROOF.
f(z) e T
f(z)
z
,
T
P
(A,B,a) and C
P
(A,B,a).
(A,B,a), then
rp+
f(z)l
rp+ (Izl
r),
(B-A)(p-a)
p
I+B+(B-A) (p-a)
(4.1)
zp+l(z.+r)
From Theorem I, we have
n--I
n-i
This implies that
Z ap+n
n=l
(B-&) (p-a)
I+B+(B-A) (p-a)
Thu s
If(z)l
Izl p + n=l>.’ lap+nl Izl p+n
n=l
(4.2)
GENERALIZATION OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
r
p
(B-A) (p-a)
+
’I-+B+(B-A) (-a)
r
497
p+l
Similarly,
rp
)
COROLLARY 3.
f(z)
If
rp
rP
(B-A)
(p.) p+l
I+B+(B-A)(p-a) r
z
p
Z Jan+p[
rp+
(B-A)(p-a)(p+l)
p
+
z
I+B+(B-A)(p-*)]
(B-A)(p-)(p+1)
n+p
p
(A,B,)
then
If(z)
rP+l ([z[--
pII+B+(B-A)(p-(x)]
e C
r),
with equality for
z p-
f(z)
THEOREM 4.
wl
disc
<
The disc
(B-A)(p-a)(p+1)
z
p[I+B+(B-A)(p-a)]
[z[ <
p+l
(z
+/-r).
is mapped onto a domain that contains the disc
,
I+B
I+B+(B-A)(p-a)
by any
[w[ < p+[pB+(A-B)(p-a)]
p. [I+B+(B-A) (p-a)
f e T (A,B,), and onto a domain that contains the
P
by any fe
Cp(A B,a)
The theorem is sharp, with extremal functions
zp
zp
PROOF.
(B-A) (p-a)
I+B+(B-A)(p-a)
(B-A)(p-a) (p+l)
p[ I+B+(B-A) (p-a)
zP+
Tp
zp+1
THEOREM 5.
If
f
C (A,B,a).
P
The results follow upon letting
g
(A,B,e) and
r
T (A,B,a), then
P
(P+I)(B-A),(P-a) r
pr p-I- I+B+(B-A)
(p-e)
p
[f’(z)[
in Theorem 3 and Corollary 3.
M.K. AOUF
498
pr
p-I
(p+l) (B-A) (p-a) r p
+ --I+B+(B-A)(p-a)
(Iz]
r).
Equality holds for
f(z)
PROOF.
z
(B-A)(p-)
p
zp+l
(z
I,p+nl
r
I+B+(B-A)(p-a)
+/-r).
We have
If’(z)
pr
p-1 +
Z
(p+n)
n--1
P rp-I +
rP-I
rP Z
n;1
I
[p+r
(p+n)
p+n-1
ap+nl
lap+n
(p+n)
n--
(4.3)
In view of Theorem I,
Z
](I+B-)+-(A-B)=(P-a)]
la n+p
I+B
(I+B) in+p-
n;1
(B-A)(p-(x)
or
Z
(l+B)(n+p)
lan+pl
p(I+B)+(A-B) (p-)
.
n--1
(B-A)(p-=) +
Jan+p[
(4.4)
(4.4) with the help of (4.2) implies that
Z
(n+P)
an+pl
!p+l) (B-A) (p-a)
I+B+(B-A)(p=F
A substitution of (4.5) into (4.3) yields the rlght-hand inequality.
On the other-hand
f’(z)
)
PrP-I
This completes the proof.
rP-I[P
-rn=ZI
(p+n)
(p+l) (B-A) (p-a)
I+B+(B-A) (p-a)
rp
lap+nl
(4.5)
GENERALIZATION OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
COROLLARY 4.
f(z)
If
z
499
p
then
prP-I
(B-A)(p_-S_)(_+I)
p-I +
5.
f(z) e
If
Izl
<
R
If’(z)
(B-A)(p-a)(p+l)2 rP (I
zl
r).
p+1 (z
(B-A)(p-.)(p_+l)
p[l+B+(B-A)(p-a)] z
p
RADIUS OF CONVEXITY FOR THE CLASS
THEOREM 6.
p
p I+B+(B-A) (p-a)
f(z)=z
Equality holds for
r
+B+(B-A) (p-e)
p
pr
2
Tp
inf
n
p
,
+/-r).
,
(A,B,a).
(A,B,a), then f(z) is p-valently convex
T
P
[(l+B)n+(B-A)(p-a)
(B-A) (p-a)
(n_p)2
in the disc
n
(n=l’2’ "’’)"
The result is sharp, with the extremal function
f(z)
z
(B-A) (P-a)
(I+B) n + (B-A)(p-a)
p
zp+n
We have
-n
z f" (z)
n(n+p)
an+ p zn
p-n_[l (n+p)lan+plZl n
n
Izl n
P-nl (n+p)In+pllzl n
n(n+p)fan+
Thus
(I +
zf’(z)
"- Pl
p if
nl (n+p)21 an+ p Izln p
or
nl(n-p )2 lan+pl Izl n
According to Theorem 1,
n.l
(5.2)
I"
+B )n+(B-A) (p,a)
(B-A)(p-a)
an+
pl
M.K. AOUF
500
Hence (5.2) will be true if
(B-A) (p-a)
or if
n+-p)2]n
[(l+B)n+(B-A)(p-a)
(B-A) (--)
Iz
(5.3)
(n=l’2’’’’)"
The theorem follows easily from (5.3).
6.
CLOSURE THEOREMS.
In this section we shall prove that the classes
T
P
(A,B,a) and
C
P
(A,B,a) are
closed under convex linear combinations.
THEOREM 7.
f(z)
If
z
p
n--I
T
P
(A,B,=)
h(z)
then
z
p
.
n=l
a
n--I
PROOF.
Since
n+ p
f(z) and g(z) are in
T
P
+ b
n+ p
(A,B,a),
n_l [(l+B)n+(B-A)(p-a)]lan+pl
n+p
z
is also in
T
p
(A,B,a).
we have
(B-A)Cp-a)
(6.1)
(B-A)Cp-a).
(6.2)
and
n_l
[(l+B)n+(B-A)Cp-a)
]Ibn+pl
From (6.1) and (6.2) we get
I/2n_
[(l+B)n+(B-A)(p-a)]
h(z)
which implies that
T
lan+p+bn+pl
6
(B-A)(p-)
(A,B,a).
P
The following theorem can be proven similarly.
n+p and g(z)fz p
Cp(A,B,a),
THEOREM 9.
Then f
Let
T (A,B,a)
P
f(z)
PROOF.
zP-I/2nl lan+p+bn+p]
h(z)
then
l_
f (z)
p
z
p
f
n+p (z)
z
n+p
is also in
Cp(A.B.a).
n+p (n=1,2,3,...).
(B-A)(p-a)
(l+B)n+(B-A)(p-a) z
zp
if and only if it can be expressed in the form
nL kn+p fn+p (z)
Suppose f(z)
n=o
Xn+p
f
where
n+p (z)
Xn+ p
z
p
.
o and
n=o
Jo Xn+p=I"
(B-A) (p-a)
(l+B)n+(B-A)(p-a)
An+p
z
n+p
GENERALIZATION OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS
501
The n
Y.
n
So by Theorem I,
n+p
kn+ p
I.
p
(A,B,a).
P
f(z) e T
Conversely suppose
Setting
(I->,
k
f(z) e T
lan+p
(B-A) (p-a)
(I+B)n+(B-A) (p-)
IX
P
(A,B,).
Then
(B-A) (p-a)
(I+B)n+(B-A)(p-)
<
(l+B)n+(B-A)(p-a)
(BgA)(p’a)
[an+p
(n=l 2 ...)
and
P
n=l
n+p’
we have
f(z)
Z kn+p fn+p (z)"
n--o
This completes the proof of the theorem.
(I) Putting a=o in the above theorems we get the results obtained by
R.M. Goel and N.S. Sohi [I].
< I, in the above theorems we
(2) Putting p=l and taking A= -B, B=, where o <
get the results obtained by Gupta and Jain [2].
(3) Putting p=l and taking A= -I, B=I in the above theorems we get the results
obtained by Silverman [3].
REMARKS.
In conclusion, I would like to thank Professor Dr. H. Silverman
for reading the manuscript and for helpful suggestions.
ACKNOWLEDGMENT.
REFERENCES
I.
GOEL,
R.M.
and
SOHI,
N.S.
Multivalent
Indian J. Pure appl. Math., 12(7)
2.
GUPTA, V.P. and JAIN,
P.K.
negative coefficients
Certain
functions
with negative
coefficlents,
(1981), 844-853.
classes
of
univalent
functions
Bull. Austr. Math. Soc., 14 (1976), 409-416.
with
502
3.
M.K. AOUF
SILVERMAN, H.
Univalent functions with negative coefficients, Proc. Amer. Math.
Soc. 51 (1975), I09-I16.
4.
5.
SILVERMAN, H. and SILVIA, E.M. Convex families of starlike functions,
Houston J. Math., 4, (1978), 263-268.
SILVERMAN, H. and SILVIA, E.M. Pre-Starlike functions with negative
coefficients Int. J. Math. Scl., 2 (1979), 427-439.