403
Internat. J. Math. & Math. Sci.
VOL. 19 NO. 2 (1996) 403-406
A REMARK ON CERTAIN p-VALENT FUNCTIONS
M. K. AOUF and H. E. DARWlSH
Department of Mathematics, Faculty of Science
University of Mansoura, Mansoura Egypt
(Received May 3, 1994)
ABSTILACT. The object of the present paper is to prove an interesting result for
certain analytic and p-valent functions in the umt disc U { z
z-I
KEY WORDS AND PHRASES. Anal.tic, p-valent, Ruscheweyh derivative.
1991 AMS SUBJECT CLASSIFICATION CODES. 30C45.
1. INTRODUCTION.
Let A(p) denote the class of functions of the form
f(z)-z p+
o
E
akz
k
(1.1)
(pN-{1,2,...})
k=p+l
which are analytic and p-valent in the unit disc
f(z) (j: ,2) defined by
U--{zlz l< 1}.
Fo, fimctions
fj(z)-z p + Z ak.jZ k,
(1.2)
k=p+l
we define the convolution product f, *g(z) of functions f (z) and g(z) by,
fl*f2( z)- zp + Z ak,lak,2 zk.
(1.3)
k=p+l
Using the above convolution product, we define
,(1- z) +p
*f(z) (f(z) eA(p))
(1.4)
where n is any integer greater than p. We note that
z P (zn-lf(z))(n+p-1)
(1.5)
(n + p- 1)!
The symbol D n+p-1 when p
was introduced by Rtmcheweyh [8], and the
n+p-1
was introduced by God and Sohi [5]. This symbol was named the
syrnbol D
Ruscheweyh derivative of (n + p -1) th order by Chert and Owa [4].
It follows from (1.5) that
Dn+p-lf(z)
z(Dn+p-lf(z))
(n+ p)Dn+Pf(z)
nDn+p-lf(z) (cf.[4]and [5]).
(1.6)
Recently, Chen and Lan ([ 1], [2]), Chen, Lee and Owa [3], Chert and Owa [4]
and Snvastava, Owa and Pashkouleva [9] have been proved some interesting results
of analytic functions revolving Ruscheweyh derivatives. In the present paper, we
prove an interesting result for functions f(z) E A(p) satisfying
M. K. AOUF AND H. E. DARWISH
404
Re
Dn+p+lf(z) ,..:,or.
0ot-:
and
n6N0-N,.[0 }.
2. MAIN RESULT.
In order to prove our main restdt, we recall here the following lemma:
LEMMA (Miller 161; Miller and Mocanu 171).
Let qffu, v) be a complex- valued ftmctlon,
C, D C C
C is the complex plane),
q): D
and let u u + iu z v--v,, + iv:.. Suppose that the function 9(u. v)satisfies the
following conditiots:
(i) (u. v) is continuous in D:
>- 0;
(ii) 0) D and
-
(iii)Re {(iuz, v) } 0 for all(iuz,v)D and such that v -::2
+ ql z + q2z 2 +-.- be regular m the unit disc U such that (q(z), zq’(z))
_D forallz U. If
Re { qiq(z), zq’(z)) } > 0 (z _U),
Let q(z)
then
Re(q(z)) > 0
(z U).
Applying the .,above Lemma, we derive the following:
THEOREM. Let the function f(z) be in the class A(p) satisfy
Re
for 0 _< ct <
Dn+p+lf(z)
}
>"
Zp
(
and n e N o. Then
(z U)
(2.1)
(z u),
(2.2)
where
+ x/1 + 4t(n + p + 1)(n + p + 2)
2(n+ p+ 2)
PROOF, For f(z) m A), we define the function q(z) by
’"f(z)
13+ (1-13)q(z),
(2.3)
(2.4)
is given by (2.3). Then q(z) is regular in U and q(z)- + qz + q2z 2 +
Taking the derivatives of both sides m (2.4), we have
where
t3
z(Dn+Pf(z))’- p(Dn+Pf(z)) 2(1-13)[13+ (1- 13)q(z)]zq’(z).
p
z
(2.5)
CERTAIN p-VALENT FUNCTIONS
Smce the dentty
405
5) tmphes
z(Dn+pfI’z))’--(n p+l)Dn’p*lt’z)
In+ l)Dn*Pftz),
(2 5) becomes
Dn+p+lf(z)
zp
-[[3+ (1- [3)q(z)] 2
*
2(1-13)[13+ (1- l)q(z)]zq’(,’.)
(n p+ 1)
(2.7)
or
Taking q(z)- u- u + m z and zq’(z)-v- v + v,." z, we define the function qKu, v)
by
13)[13 +
Du]v -or
v) ---[13 +
13)ul *
(2.9)
(n+ p+ 1)
Then it follows from (2.9) that
(i) q(u, v) is continuous in D C x C,
(ii) (1, O) D and Re(q(l,0)}- 1- .,0,
(iii) fbr all (iu 2,vl) D such that Vl .213(1- 13’Iv
+
Re{q)(lu2,vl)}- ]3 2 (1
(n + p 1)
32
u
<0
for 0 : a < 1. n N0
n>-p and 13 is given by (2.3). Therefore, the function (u,v)
satisfies the conditions m the lernma. Thus we have Re{q(z)} > 0 (z U) that is,
+ q + 4et(n + p+ l)(n+ p + 2)
2(n+ p+ 2)
which complaes th proof of the Theorem.
Re
Letting
1
(2.10)
0, the theorem gives:
ot
COROLLARY 1. Let the function f(z) be in the class A(p) satis
Re! Dn+P+lf(z) !
zp
>0
(z
(2 11)
for n e N o and n .>-p. Then
i
Re, ,i:Dn+Pf(z)
f ">
p
(z U).
(n+p+2)
z
1- p in the above theorem, we have
’,
Taking n
(2.12)
M. K. AOUF AND H. E. DARWISH
406
C()ROLI,ARY 2.
Let the function ffz) be tn the class A(p! satis’
Re{D2f(z) f
Z
for 0 --_-ot <
p
oad p -N. Then
Re
-P(z)+ (- p)r(z)
t.,
REMARK 1. Putting p
(7hen, Lee and Owa [3].
z’
.>
,/ + 24c
(2.14)
6
in the above results, we gel the results obtained by
REMARK 2. Using the same technique as m the theorem (or putting zf’(z) instead
P
of f(z) in the theorem), we have the following result
COROLLARY 3. Let the function f(z) be m the class A(p) satisf3’
pz p-I
tbrOct(1 and
nNo.
(z u)
Then
[(Dn+pf(z))
where
fl
ven by (2.3)
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functions defined on the trait disc, Bull. Austral. M.ath. Sot:. 35 (1987), 387 396.
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type, Intemat. J. Math. Math. Sci. 12 (1989), 107- 112.
3. CHEN, M.-P., LEE, S.-K. and O WA, S. Are.mark on certain regular functions,
Simon Stevin 65..__(199 I), no. 1-2, 23- 30.
4. CHEN, M.-P. and OWA, S. A property of certmn analytic functions involving
Ruscheweyh derivatives, Proc. Japan Acad. 65, Ser.___.A (1989), no.10,333335.
5. GOEL, R. M. and SOHI, N. S. A new criterion for p- valent functions,
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115.
(1975)
Soc.: 49
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