I nternat. J. h. & Math. S ci.
Vol. 6 No. 3 (1983) 483-486
483
UNIVALENT FUNCTIONS DEFINED BY
RUSCHEWEYH DERIVATIVES
S.L. SHUKLA and VINOD KUMAR
Department of Mathematics
Janta College, Bakewar
Etawah 206124, India
(Received September 28, 1982 and in revised form August 8, 1983)
ABSTRACT.
We study some radii problems concerning the integral operator
zy
o
uY-I f(u) du
and M (a), of univalent functions defined by Ruscheweyh
for certain classes, namely K
n
derivatives.
z
y+l
F(z)-
n
Infact, we obtain the converse of Ruscheweyh’s result and improve a
result of Goel and Sohi for complex
by a different technique.
The results are sharp.
KEY WORDS AND PHRASES. Hadamard product, starlike, univalent.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 30C45.
I
INTRODUCTION
Let S denote the class of functions of the form f(z)
{z
regular in the unit disc U
-
zl
f(.z,).}
{Dn+l.
n
D
> i
f (z)
and
n
D
+ E a,.
z
k
which are
k=2
< i}.
A function f of S is said to belong to the class K
Re
z
n
U, n e N O
where z
z
f(z)
(l-z)
n+l
if
{0,i,2,...},
* f(z),
and the operation (*) stands for the convolution or Hadamard product of the power series.
Ruscheweyh [1] introduced the classes K
Kn+
K
1
n
univalent.
and showed, via the inclusion relation
n
that the functions in these classes are starlike of order 1/2 and hence are
He also observed that
D
n
f(z)
Following A1-Amiri [2], we also
of f.
z(z
n-I
refer to
f(z))(n)/n.
Dnf
as the nth order Ruscheweyh derivative
484
S.L. SHUKLA and V. KUMAR
A function f of S is said to belong to the class Mn (), 0 <
n+l
{D
< i, if
,
f{z)}
>
N
z
U, n
O
z
Goel and Sohi [3] introduced the classes M (c) and showed, via the inclusion
n
that the functions in these classes are univalent.
relation
Re
Mn+l(a) Mn(a),
Ruscheweyh [1] proved that the function F defined by
F(z)
z
y+l
uY-I f(u)
o
zY
du
if f
K and y is a complex number such that Re(y)>(n-l)/2. Goel and
n
n
Sohi [3] obtained an analogous result for the class M (). Conversely, they [3,
n
M () when F g M () and
Theorem 4] determined the radius of the disc in which f
n
n
y is a real number such that y > -i.
belongs to K
In the present paper we obtain the converse of Ruscheweyh’s [i] result.
obtain the above mentioned result of Goel and Sohi
technique, for complex y
2.
[3, Theorem 4],
We also
by using a different
The results are shown to be sharp.
PRELIMINARY LEMMA.
Let P denote the class of functions of the form P(z)
i
o
regular in U and satisfy Re
{p(z)}
+ l
k=l
> 0 for z & U.
bkzk
which are
We require the following lemma which follows from a result of Yoshlkawa and
Yoshikai
[4, Theorem i]:
LEMMA 2.1.
Let p
If b is a non-negative real number and c is a complex
P0"
number such that c+b # 0, then
+ zp’(z)/(c+bp(z))} >
0
{ici2+2+4b+b2-k}I/2/[e-b[,where
E
Re {p(z)
holds in
izi
< R(c,b)
Re(c2)+4(l+b2)(l+2b).
3.
2(2+4b+b
2) Ici2
+
2b
2
(l+z)/(l-z).
The result is sharp with the extremal function p(z)
MAIN RESULTS.
In the following theorem we study the converse of Ruscheweyh’s [I] result
Let y be a complex number such that Re(y) > -i. If F e K
THEOREM 3.1
n
then the function f defined by
F(z)
satisfies
b
Re {D
n+l
z
y+l
n
f(z)/D f(z)} > 1/2
(n+l)/2, and R(c,b)
u
o
zY
Y-I f(u)
Izl
in
is given by Lemma 2.1.
(3.1)
du
< R(c,b) where c
(y-n)
+ (n+l)/2,
The result is sharp.
For the existence of the integral in (3.1), the power represents principle branch.
We note that the integral operator under consideration can also be written as
i
F(z)
(y+l)
o
t
Y-I f(tz)
which solves the question of principal branch.
dt
UNIVALENT FUNCTIONS DEFINED BY RUSCHEWEYH DERIVATIVES
PROOF.
485
It is easy to verify the identity
z(DnF(z))
(n+l) D
n+l
n
F(z) -nD
F(z).
(3.2)
F(z).
(3.3)
Also, from the definition of F it can be verified that
z(D
Since F
g
K
n
F(z))’
(y+l) D
there exists a function p in P
n
D
n+l
D
n
I
F(z)
n
f(z)
yD
n
such that
o
(I + p(z)).
(3.4)
F(z)
Using (3.2), (3.3), and (3.4), we get
(y+l)
Dn+l
f(z)
n+l
F(z))’
F(z) + z (D
n+l
- -
yD
DnF(z) + i zp’(z) DnF(z)
(l+p(z)) z(nnF(z))’
+
(l+p(z))nnF(z) + 1 zp’(z)DnF(z)
2 1(l+p(z))
21
+
Thus,
(y+l)Dn+if(z)
i
(l+p(z)){ (n+l)
Dn+iF(z)-nDnF(z) }.
n+l.
2
DnF(z).
(3.5)
F(z)+(n+l)Dn+iF(z)-nDnF(z)
(y-n) + 1/2(n+l) (l+p(z)) DnF(z).
(3.6)
[(y-n)(l+p(z))+zp’(z)+(--)(l+p(z))
Also,
(y+l)
Dnf(z)
yD
yD
n
F(z) +
z(DnF(z))
n
From (3.5) and (3.6), we obtain
[D
where c
(y-n) + (n+l)/2 and b
n+l
I
f(z)
Dnf(z)
zp’(z)
p(z) + c+bp(z)
]/(1/2)
(n+l)/2.
The required result now follows by using Lemma 2.1.
To establish sharpness, we take F(z)
z/(l-z).
Then,
Dn+iF(z)
n+2
z/(l_z)
z/(1-z) n+l
1
l+z)
(3.7)
(i +
Iz
F(z)
From (3.4) and (3.7), we get p(z)
(l+z)/(l-z); hence, the sharpness of the result
D
n
follows from that of Lemma 2.1.
In the following theorem, we obtain the converse of the result of Goel and Sohi
[3, Theorem 2]
for complex y.
THEOREM 3.2.
Let F e M (s) and y be a complex number such that Re(y)> -i.
n
486
S.L. SHUKLA and V. KUMAR
If f is defined by (3 i)
{Dn+if (z)}
then Re
>
Izl
in
(IYi12+I)
< R*
i
The result is sharp.
PROOF.
Since F e M
n
(), there exists a function p in P o such that
n+l
F(z)
D
z
+ (l-)zp(z).
(3.8)
Differentiating (3.8) and using (3.3), we get
D
Using Lemma 2.1 for c
n+l
f (z)/z-e
l-e
0,
y+l and b
p(z) +
w6 find
.zp’ (z)
that the real part of right hand side
2+1)
of (3.9) is greater than zero in
=
in
Izl
(3 9)
y+l
i
Hence, Re{
y+i|
Dn+lzf(Z)}
>
< R*.
The sharpness of the result follows easily by taking the function F defined by
D
n+l
F(z)
+ (l-a)
ez
z
(l+z.
i_--).
M (e), then the function F
proved that, if f
Goel and Sohi [3, Theorem
defined by (3.1) also belongs to
Mn(e)
n
provided that Re(y)
> -i.
In this direction,
the following theorem provides a better result for suitable choices of y
M (s) and y is a real number such that -i < y < n+l, then the
n
function F defined by (3.1) belongs to
THEOREM 3.3.
If f
Mn+l(e).
PROOF.
Since
n+l
F(z))’
and, by the definition of F,
z (D
z
(n+2) D
(Dn+IF (z))
n+2
(y+l)
F(z)
Dn+if (z)
we have
Re{(n+2)
Since F
Dn+2F (z)
Z
(n+l-y)
(n+l) D
Dn+IF (z)}
Z
-
n+l
F(z)
Dn+IF (z)
(y+l)Re
{Dn+if (z)}
>
Z
M (e), the above inequality leads us to
n
Dn+2F(z)
(n+2) Re{}
Z
>
>
Hence, F e
ACKNOWLEDGEMENT.
(n+l-y) Re{
Dn+IF (z)} +
Z
(n+l-y) + (y+l)e
(y+l)
(n+2).
Mn+l(a).
The authors are thankful to the referee for his valuable suggestions
about the earlier version of tM[s paper.
REFERENCES
.
I.
3.
4.
RUSCHEWEYH, S. New Criteria for Univalent Functions, Proc. Amer. Math. Soc. 49
(1975), 109-115.
AL-AMIRI, H.S. On Ruscheweyh Derivatives, Ann. Polon. Math. 38 (1980), 87-94.
GOEL, R.M. and SOHI, N.S. Subclasses of Univalent Functions, Tamkang .J..Math. ii
(1980), 77-81.
YOSHIKAWA, H. and YOSHIKAI, T. Some Notes on Bazilevlc Functions, J. London Math.
Soc. (2_) 2__0 (1979), 79-85.