MATEMATIQKI VESNIK
UDC 517.546
research paper
46 (1994), 71{75
originalni nauqni rad
ON A SUBFAMILY OF p-VALENT FUNCTIONS
WITH NEGATIVE COEFFICIENTS
S. R. Kulkarni, M. K. Aouf, S. B. Joshi
Abstract. Sharp results concerning coecients, distortion theorem and the radius of convexity for the class Pp (; ; ) are determined. Furthermore it is shown that the class Pp(; ; )
is closed under convex linear combinations. The extreme points of Pp(; ; ) are also determined
1. Introduction
Let Sp (p a xed
integer greater than 0) denote the class of functions of the
P
p+n that are holomorphic and p-valent in the unit
form f (z ) = z p + 1
n=1 ap+n z
disc jz j < 1. Also let Tp denote the subclass
of Sp consisting of functions that can
P
p+n . A function f 2 T is in
be expressed in the form f (z ) = z p ; 1
p
n=1 jap+n jz
Pp (; ; ) if and only if
f 0 (z )z 1;p ; p
0
1
;
p
0
1
;
p
2 (f (z )z ; ) ; (f (z )z ; p) < ;
jz j < 1, for 0 6 < p=2 , 0 < 6 1, 1=2 < 6 1.
Such type of study was carried out by Aouf [1] for Pp (; ). We note that
P1 () = P1 (0; ; 1) is precisely the class of functions in E studied by Caplinger
[2]. The class P1 (; 1; ) = P1 (; ) is the class of holomorphic functions discussed
by Juneja-Mogra [4]. Gupta-Jain [3] studied
the family of holomorphic univalent
P
n
functions that have the form f (z ) = z ; 1
n=2 jan jz and satisfy the condition
f 0 (z ) ; 1 < ; 0 6 < 1; 0 < 6 1:
f 0 (z ) + (1 ; 2) Kulkarni [5] has studied above mentioned P
properties for the functions having
n
Taylor series expansion of the type f (z ) = z + 1
n=2 an z
In this paper sharp results concerning coecients, distortion theorem and the
radius of convexity for the class Pp (; ; ) are determined. Finally we prove that
AMS Subject Classication (1980): Primary 30C45
71
72
S.R.Kulkarni, M.K.Aouf, S.B.Joshi
the class Pp (; ; ) is closed under the arithmetic mean and convex linear combinations.
2. Coecient Theorem
P
p+n is in P (; ; ) if and
Theorem 1. A function f (z ) = z p ; 1
p
n=1 jap+n jz
only if
1
X
(p + n)[1 + (2 ; 1)]jap+n j 6 2 (p ; ):
n=1
The result is sharp, the extremal function being
2 (p ; )
p+n
f (z ) = z p ;
(p + n)(2 ; 1) + 1 z :
(1)
Proof. Let jz j = 1. Then
jf 0 (z )z 1;p ; pj ; j2 (f 0 (z )z 1;p ; ) ; (f 0 (z )z 1;p ; p)j
= ;
1
X
n=1
(p + n)jap+n jz n ; 2 (p ; ) ; (2 ; 1)
6
1
X
n=1
1
X
(p + n)jap+n jz n
n=1
(p + n)[1 + (2 ; 1) ]jap+n j ; 2 (p ; ) 6 0
by hypothesis. Hence, by the maximum modulus theorem f 2 Pp (; ; ).
For the converse we assume that
f 0 (z )z 1;p ; p
2 (f 0 (z )z 1;p ; ) ; (f 0 (z )z 1;p ; p) n
; PP1
n=1 (p + n)jap+n jz
< :
= 1
n
2 (p ; ) ; n=1 (p + n)jap+n j(2 ; 1)z Since j<(z )j 6 jz j for all z we have
P1
(p +P
n)jap+n jz n
n
=1
< 2 (p ; ) ; (2 ; 1) 1 (p + n)ja jz n < :
p+n
n=1
We select the values of z on the real axis so that f 0 (z )z 1;p is real. Simplifying
the denominator in the above expression and letting z ! 1 through real values, we
obtain
1
1
X
X
(p + n)jap+n j 6 2 (p ; ) ; (2 ; 1) (p + n)jap+n j;
n=1
and it results in the required condition.
The result is sharp for the function (1).
n=1
On a subfamily of p-valent functions with negative coecients
73
3. Distortion Theorem
Theorem 2. If f 2 Pp (; ), then for jz j = 1,
2 (p ; )
2 (p ; )
p+1
p
p+1
rp ;
(p + 1)[1 + (2 ; 1)] r 6 jf (z )j 6 r + (p + 1)[1 + (2 ; 1)] r ; (2)
and
prp;1 ;
2 (p ; ) p
2 (p ; ) p
0
p;1
1 + (2 ; 1) r 6 jf (z )j 6 pr + 1 + (2 ; 1) r ;
(3)
Proof. In view of theorem 1 we have
1
2 (p ; )
jap+n j 6 (p + 1)[1
+ (2 ; 1)] :
X
n=1
Hence
jf (z )j 6 rp +
and
jf (z )j > rp ;
1
2 (p ; )
p+1
jap+n jrp+n 6 rp + (p + 1)[1
+ (2 ; 1)] r
X
n=1
1
X
n=1
2 (p ; )
p+1
jap+n jrp+n > rp ; (p + 1)[1
+ (2 ; 1)] r :
In the same way we have
jf 0 (z )j 6 prp;1 +
and
1
p ; ) p
r
(p + n)jap+n jrp+n;1 6 prp;1 + 12+((2
; 1)
X
n=1
1
X
0
p;1
jf (z )j > pr ; (p + n)jap+n jrp+n;1 > prp;1 ; 12+((2p ;;)1) rp :
n=1
This completes the proof of the theorem.
The above bounds are sharp. Equalities are attended for the following function
2 (p ; )
p+1
(4)
f (z ) = z p ;
(p + 1)(2 ; 1) + 1 z ; z = r:
Theorem 3. Let f
2 P ; p (; ; ). Then the disc jz j < 1 is mapped onto a
domain that contains the disc
[(2 ; 1) ; p + 2]
jwj < (p + (1)p ++ 1)[1
:
+ (2 ; 1)]
The result is sharp with extremal function (4).
Proof. The result follows upon letting r ! 1 in (2).
74
S.R.Kulkarni, M.K.Aouf, S.B.Joshi
Theorem 4. f
r(p; ; ; ), where
2 Pp (; ; ), then f is convex in the disc jz j < r =
p2 [1 + (2 ; 1)] 1=n
r(p; ; ; ) = inf
:
n2N (p + n)2 (p ; )
The result is sharp, the extremal function being of the form (1).
Proof. It is enough to show that
zf 00 (z )
6 p for jz j < 1.
1+
;
p
f 0 (z )
First
we note that
P1
n
zf 00 (z ) + (1 ; p)f 0 (z ) zf 00 (z )
nP
=11n(p + n)jap+n jjz j :
1+
;
p
=
6
f 0 (z )
f 0 (z )
p ; n=1 (p + n)jap+n jjz jn
Thus, the result follows if
1
X
n=1
n(p + n)jap+n jjz jn 6 p p ;
1 ( p+n )2 ja jjz jn 6 1.
p+n
n=1 p
P
or, equivalently,
But, in view of Theorem 1, we have
1
X
1
X
n=1
(p + n)jap+n jjz jn ;
(p + n)[1 + (2 ; 1)]jap+n j 6 2 (p ; ):
n=1
Thus f is convex if
p + n 2 n (p + n)[1 + (2 ; 1)]
jz j 6
; n = 1; 2; 3; . . . ;
p
2 (p ; )
i.e.
2
(2 ; 1)] 1=n ; n = 1; 2; 3; . . . ;
jz j 6 (pp +[1 n+)2
(p ; )
which completes the proof.
4. Closure Theorem
Next two results respectively show that the family Pp (; ; ) is closed under
taking "arithmetic mean" and "convex linear combinations".
P
p+n
Theorem 5. If f (z ) = z p ; 1
jap+n jz p+n and g(z ) = z p ;P1
n=1
n=1 jbp+n jz
P1
1
p
p
+
n
are in Pp (; ; ), then h(z ) = z ; 2 n=1 jap+n + bp+n jz is also in Pp (; ; ).
Proof. f and g both being members of Pp (; ; ), we have in accordance with
Theorem 1
1
X
(p + n)[1 + (2 ; 1)]jap+n j 6 2 (p ; )
n=1
and
1
X
(p + n)[1 + (2 ; 1)]jbp+n j 6 2 (p ; ):
n=1
(5)
(6)
On a subfamily of p-valent functions with negative coecients
75
To show that h is a member of Pp (; ; ) it is enough to show that
1
1X
2 n=1(p + n)[1 + (2 ; 1)]jap+n + bp+n j 6 2 (p ; ):
This is exactly an immediate consequence of (5) and (6).
Theorem 6. Let fp (z ) = z p and
2 (p ; )
p+n
fp+n (z ) = z p ;
(p + n)(2 ; 1) + 1 z ; n = 1; 2; 3; . . . :
Then f 2 Pp (; ; ) if and only if itPcan be expressed in the form f (z ) =
P1
1 = 1.
n=0 p+n fp+n (z ), where p+n > 0 and
n=0 p+n
Proof. The proof of this theorem follows along the same lines as the proof of
Theorem 3.3 in Kulkarni [5]. The details are omiited.
Corollary. The extreme points of Pp (; ; ) are the functions fp (z ) = z p
2 (p ; )
p+n
and fp+n (z ) = z p ;
(p + n)(2 ; 1) + 1 z , n = 1, 2, . . . .
REFERENCES
[1] M. K. Aouf, Certain classes of p-valent functions with negative coecients II, Indian J. Pure
appl. Math. 19(8), 1988, 761{767
[2] T. R. Caplinger, On certain classes of analytic functions, Ph.D. dissertation, University of
Mississipi, 1972
[3] V. P. Gupta, P. K. Jain, Certain classes of univalent functions with negative coecients, Bull.
Austr. Math. Soc 15, 1976, 467{473
[4] O. P. Juneja, M. L. Mogra, Radii of convexity for certain classes of univalent analytic functions, Pacic Jour. Math. 78, 1978, 359{368
[5] S. R. Kulkarni, Some problems connected with univalent functions, Ph.D. Thesis, Shivaji
University, Kolhapur 1981, unpublished
(received 02.06.1993.)
Dept. of Mathematics, Willingdon College, Sangli { 416 415, India
Dept. of Mathematics, Faculty of Science, University of Mansoura, Mansoura, Egypt
Dept. of Mathematics, Walchand College of Eng., Sangli { 416 415, India