I ntern. J. Math. Mh. Si.
Vol. 5 No. 2 (1982) 289-299
289
GENERALIZATIONS OF p-VALENT FUNCTIONS
VIA THE HADAMARD PRODUCT
ANIL K. $ONI
Department of Mathematics and Statistics
Bowling Green State University
Bowling Green, Ohio 43403, U.S.A.
(Received July I, 1981)
ABSTRACT.
The classes of univalent prestarlike functions
[i] and a certain generalization of
Rwere
R,
>-i, of Ruscheweyh
studied recently by Ai-Amlrl
[2].
The
+ p
i)
author studies, aong other things, the classes of p-valent functions R(
where p is a positive integer and
+ p)
in particular that R(
for the class R(
is any integer with
R( + p
+ p > 0.
The author shows
i) and also obtains the radius of R( + p)
+ p- i).
KEY WORDS AND PHRASES. p-valent stake functns, p-valent ose-t-convex
functions, Hadamd product.
AMS (MOS) SUBJECT CLASSIFICATION (1980) CODES.
i.
Primary 30C45.
INTRODUCT ION.
The classes of univalent prestarlike functions R
various authors [1,2].
The author extends these classes to the classes of p-valent
+
starlike functions R(
greater that -p.
> -i, were studied by
i), where p is a positive integer and
p
is any integer
The present studies give, along with othe results, a method to
+ p)
determine the radius of R(
for the class R(
+ p
i).
{z:
Let A denote the class of regular functions in the unit disc D
P
having the power series
f (z)
z
p +
n=p+l
anzn,
p
a positive integer,
z
D.
Izl
< i}
(i.i)
A.K. SONI
290
We denote by S*(), the subclass of A
I
whose members are starlike of order
O_<S <i.
Ruscheweyh [I] introduced the following classes
’K
of univalent prestarlike
functions
{f(z)If(z)
K
e e N
{0,1,2
O
};
e
A
where F
K
observed by Ruscheweyh, f e
z
f(z)*
(l-z)
Here
+i
’*’
Ruscheweyh proved that
S*(1/2).
subclass of
if and only if Re
n
a z
D+If (z)
Df (z)
K+1 K
n0 bnzn
and g(z)
n
D},
z e
1
>_
2
z
D where
{f(z)If(z)
R is
e
As
Def(z)
S*(1/2).
and K
0
n=
then f(z)*g(z)
Hence for each
N 0,
K
a b z
Ka+1
A
I
s
K, a O
classes R, where
to the
Re
and
a subclass of
D
+I f
N
in
(z) >
+
Df(z)
S*(0).
1
n
n n
is a
Recently, Ai-Amiri [2] studied a certain generalization of
Singh [3] extended the classes K
They observed that
1
2
denotes the n-tH derivative of the function F.
in particular he obtained the radius of
R
+
>
(z-if(z)) ()
denotes the Hadamard product of two regular functions,
n0
that is to say if f(z)
(n)
(zf(z))(+l)
Re
and
1
K,
Further Singh and
z e
D}
e N
O
In this note, we extend their ideas
to the class of p-valent functions.
We call a function f(z)
Re
zf’ (z)
f(z)
>
0, z
D.
A
P
to be p-valent starlike if it satisfies
Further, we say that a function f(z) e A
is p-valent
P
close-to-convex if there exists a p-valent starlike function g(z) for which
Re zf’ (z)
0, z e D.
g
Let R( + p- i) denote the class of functions f(z) e A satisfying
P
(a+p)
(zaf(z))
>
Re
+ p I, Z e D,
I
where
(Z-if(z))(+p-l)
is any integer greater that -p.
R( + p)
In Section 2 we shall show that
R(e + p
1).
Since R(0) is the class of functions which satisfy
Re
f(z)
(1.2)
> p
i
0
(1.3)
291
GENERALIZATIONS OF p-VALENT FUNCTIONS
[4] that such functions are p-valent
it follows by our definition taken from
like.
Hence (1.3) implies that R( + p
We denote by H(
condition
+ p
I) contains p-valent starlike functions.
i), the classes of functions f(z) e A that satisfy the
P
F (zaf(z)) (z-igz))a(za-lf(z)
(a+P)
Re
(a+p-l)
(+p-l)
R(a + p
for some g(z)
star-
]
> a
]
+ p
+
1
p
D
z
(1.4)
i), a integer greater that -p.
In Section 4 we shall show that
H( + p)
H(a + p
Again since H(O) is the class of functions
is starlike,
(1.5) implies that H(a + p
(1.5)
i).
zf’(z) > 0 where
f that satisfy Re
g
g(z)
i) contains p-valent close-to-convex
functions.
For f
6
A
P
define
D+P-lf(z)
zp
f(z)*
(i
is any integer greater than -p.
where
ma+P-lf(z)
z
(1.6)
z)a+P
Then
p
(za-I f(z)) (a+p- i)
(a + p
(1.7)
1)!
It can be shown that (1.6) yields the following identity
z(Da+P-lf(z))’
(a +
a(Da+P-lf(z)).
p)D+Pf(z)
(l.8)
From (1.2) and (1.7) it follows that a function f in A belongs to R(a + p- I) if
P
and only if
Re
Note that for p
and Singh [3].
D+P f(z)
Da+P-lf(z)
i, the classes R(
+ p
> a
+ p
+
1
(1.9)
p
i) reduce to the classes R
Hence our results are generalizations of Singh and Singh.
From (1.4) and (1.7), it follows that a function f in A
P
H(a + p
belongs to
i) if and only if
Re
for some g
of Singh
R(e + p
i).
-|
z(D+p-lf(z))’]
D+P-I g(z)
]
>
+ P 1
+ P
(i.io)
A.K. SONI
292
In Sections 3 and 4 we shall describe some special elements of R( + p
and H(
+ p
i), respectively.
mined.
+ p)
+ p
in R(
R1/2(
( + p
+ p
i) via the Hadamard product.
P( +
+ p)
R1/2(
i) and of
In Section 6, the classes
classes
in
R1/2( +
i,) which are
p
extensions of the
Many authors have considered a variation of these
i), are given.
note basically uses the techniques given by AI-Amlri
THE CLASSES R( + p
Also
i) are deter-
p
classes, notably Ruscheweyh [i], Suffridge [5], Goel and Sohi [6].
2.
In
These elements have integral representations.
Section 5, we introduce the classes
the radii of R(
i)
However, this
[2].
i).
We shall prove the following:
THEOREM i.
PROOF.
R(e + p)
R( + p
c
i).
R( + p). Define w(z) by
Let f
D+Pf(z)
+ p
+
D+P-lf(z)
i
+
p
f
E
R( + p
lw(z)
< i, z
i
w(z)
p 1
+ w(z)
(2.1)
0, w(z) # -i for z
Here w(z) is a regular function in D with w(0)
suffices to show that
1
+
D.
It
D, since then (2.1) would imply that
i).
Taking logarithmic derivative of both sides of (2.1) and using the identity
(1.8) the following is obtained.
D+P+if(z)
1
D+pf(z)
( + p + i)
I
1
+ (e+p)"i ++ (+p-2)w(z)
w(z)
2zw’ (z)
(l+w(z)) (+p+(+p-2)w (z))
The above equation must yield
lw(z)
lemma of Jack [7] one can obtain z
and K > i.
0
< 1 for all z
e D such that
]
(2.2)
J
D, for otherwise by using a
z0w’ (z 0) Kw(z0) lw(z0)
Consequently (2.2) would yield
Da+P+I f (z0)
D+p f (z0)
1
(+p+l)
-
(a+P) + (a+P-2)w (z0)
(+p+l) (l+w (z 0)
0)
lo+p+(c+p-2)W(Zo)
(+p+ (+p-2)w (z
2
2Kw
(z0)
(+p+l) (l+w (z 0)
i
293
GENERALIZATIONS OF p-VALENT FUNCTIONS
Since
i
- 1
i
Re
+ w(z 0)
the above equation implies
D
Re
a+p+l
w(z0)
Re
f (z
0)
<
DCrl.pf(zo
1
+ w(z 0)
1
a + p
c
+
+
p
1
R(a + p).
This is a contradiction to the assumption that f
R(e + p
Hence f
This completes the proof of Theorem i.
3.
SPECIAL ELEMENTS OF R(a
+ p
i).
In this section we form special elements of the classes R(a + p
+ p
Hadamard product of elements of R(a
Y + p zj
h (z)
Y
THEOREM 2.
Let f
Re
hT(z),
i) and
j=p
i) via the
where
Re 7 >-p.
Y+j
A satisfy the condition
P
D+Pf(z)
> 2(y+p-l)(c+p-l)
2 (+p) (y+p-l)
n+P-lf,z,
1
z
E
any integer greater than -p and y > -p
p a positive integer,
Then
(3.1)
D,
+ 2.
z
F(z)
f(z)*hy
I tY-if’(t)dt
Y + p
(z)
zY
(3.2)
0
belongs to R( + p- I).
PROOF.
A satisfy the condition (3.!).
P
Let f
z(DC+PF(z))’
+
y(DY+PF(z))
From (3.2) we obtain
(p+y)De+Pf(z),
(3.3)
and
z(D+P-IF(z))’
+
(D+P-IF(z))
(p+)Da+P-lf(z).
(3.4)
Define w(z) by
Da+P F z
D+P-IF(z)
c + p
+
1
p
+
Here w(z) is a regular function in D with w(0)
suffices to show that
lw(z)
< i, z
a
1
+
1
1
w(z)
(3.5)
+ w(z)
0, w(z) # -i for z
D.
It
D.
Taking the logarithmic derivative of (3.5) and using (1.8) for F(z) one can
get
!).
A.K. SONI
294
I(+p)
D+PF(z)
z(D+PF(z))’
D(+PF (z)
2zw’ (z)
(l+w (z)) (+p+(+p-2)w (z))
Dcp-IF(z)
(3.6)
Now (3.3) and (3.6) yield
_y_
D+PF(z)
(p+,,)D+pf(z)
(+p)+(+p-2)w(z)l
+ w(z)
+
2zw’ (z)
(l+w (z)) (C+p+(+p-2)w (z))
(3 7)
Use (3.4) and (1.8) to eliminate t.he derivative and then apply (3.5) to get
(p+y)
D+P-I f (z) D+P-iF(z)
+ (+p) i ++ (+p-2)w(z)|-]
w(z)
F|y
(3 8)
I
Therefore (3.7), (3.8) and (3.5) give
D(+Pf(z)
DC+p-lf(z)
+ p
e + p
c
i
w(z)
(l+w(z))
1
i
+
+ p
(Y+D) + (y+p-2)w(z),
z
2zw’ (z)
((+p) (l+w(z))
lw(z)
Equation (3.9) should yield
there exists z
to
D with
0
< i for all z
z0w’(z0)
D, for otherwise by Jack’s lemma
lW(Zo)
K >- I, and
Kw(z0)
(3.9)
I+p+(y+p-2)w(z)
i.
Applying this
(3.9) it follows that
Re
D+pf(zO)
D+P-lf(zo
I
+ P
+
y+p
2
1
1
(e+p) 4 (y+p-l)
p
2(y+p-l) (c+p-l)
1
2 ((+p) (y+p-l)
This contradicts the assumption on f given by
Hence F
(3.1).
R( + p
i).
This
completes the proof of Theorem 2.
REMARK I.
For y
i and p
i, Theorem 2 reduces to a result given in [3].
The following special cases of Theorem 2 represent some improvement on theorems
due to Libera [8] in the sense that much weaker assumptions produce the same results.
By taking
O, p
COROLLARY i.
Let f e A
is starlike in D, where
1 in Theorem 2 we get
I
be such that Re
zf’ (z)
f(z)
>
-i
Y
i, z
D.
Then F
295
GENERALIZATIONS OF p-VALENT FUNCTIONS
z
I tY-If(t)dt"
+ 1
F(z)
z
For
(3.10)
0
i, Theorem 2 reduces to
i, p
Let f e A be such that Re[l + zf" (z)
COROLLARY 2.
(z)
1
i
>
2y
D.
y>l, z
Then F(z) as given in (3.10) above is convex in D.
Using the technique employed in the proof of Theorem i and Corollary 2 we
obtain the following result.
COROLLARY 3.
zg"(z)
Re[l + g’
(z)
be such that Re f’(z) > 0, z e D and g be such that
g (z)
Let f e A
I
1
2y
>
> i, z
F’ (z) > 0,
convex, i.e., Re
G’(z)
D.
Then F(z) as given by (3.10), is close-to-
D and where G(z) is the convex function given by
z
g
G(z)
y + 1
tY-ig(t)dt.
zY
0
We state without proof the following theorem since its method of proof is
similar to that of Theorem 2.
THEOREM 3.
Let p be a positive integer and
and let Re y >-p
+
Then F(z)
i.
R((z + p- i) for all f
f(z)*hy(z),
be an integer greater than -p
,
THEOREM 4.
Let p be a positive integer, and
Theorem 3 can be improved as follows:
R( + p
F(z)
i),
f(z)*h(z)
Let f(z) e R(
PROOF.
the operators
to
R(( + p- i).
In case y
Then for f(z)
(3.2), belongs
as given by
+
D+P D+P-I we
( + p)
p
P
z
i).
+ I
be any integer greater than -p.
z
t-if(t)dt
R( + p).
(3.11)
0
Differentiating (3.11) and then applying
get, respectively, by using (1.8)
D+Pf(z)
( + p +
I)D+P+IF(z)
D+PF(z)
and
( +
Therefore
Re
FL
.
+ P + 1
+ p
p)D+P-lf(z)
D+P+IF(z)
D+PF(z)
( +
1
+
]
p]
p)D+PF(z).
Re
D+Pf(z)
D+P-lf(z)
>
+ p
+ p
l
A.K. SONI
296
This implies that
Re
+P
+ p +
>
D+PF(z)
z e D.
i
R( + p), and this completes the proof of Theorem 4.
Hence F(z)
REMARK 2.
4.
D+P+IF (z)
For p
i, Theorem 4 reduces to a result of Singh and Singh [3].
THE CLASSES H( + p
i).
We state without proof Theorems 5 and 6 since their proofs use the same techSee Section 1 for the definition of the classes
nique employed in Theorem i.
H(e + p
i).
H( + p)
THEOREM 6.
If p is any positive integer, a is any integer greater than -p,
and Re 7 > -p
+
z
f(z)*hy(z)
H(e + p
whenever f(z)
I).
i, then
F(z)
5.
H(e + p
THEOREM 5.
f
p + Y
t-if(t)dt
H( + p
i)
0
i).
RADII OF THE CLASSES R(a + p) AND
P(a + p).
Because discussing the problem concerning the radii of the classes R(a + p)
and
R1/2( +
f(z) e A
p) we define the classes
R1/2(
+ p
+ p
R1/2(
i).
i) conta, ins functions
that satisfy the condition
P
Re
(zf(z)) (+P)
(z-if (z)) (+p-l)
where e is any integer greater than -p.
1
>
+ p
2
z
D,
(5.1)
These classes have been studied by Goel
and Sohi [6].
P(
From (1.7) and (5.1), it follows that a function f in A belongs to
P
+ p i) if and only if
Re
DOc+P f (z)
D(x+P-lf (z)
> 1
2
(5.2)
Our main interest is to determine the radius of the largest disc D(r)
{z:
Izl
8 > a, z
<
r},
0 < r
D(r).
1 so that if f
R(e + p
i) then Re
D+Pfz_
D+P-lf(z)
>
+ p
+
i.
p
A partial answer to this problem can be deduced by a simple appli-
297
GENERALIZATIONS OF p-VALENT FUNCTIONS
[9]:
cation of a lemma due to (Ruscheweyh and Singh)
If p(z) is an analytic function in the unit disc D with p(0)
LEM}A i.
I,
Re p(z) > 0 and also
2(S + l)
A
(5.3)
I2 112)1/2] 1/2
2
[A + (A
2
II 2
+
i.
Then we have
(z)+ p(z)ZP’(Z)0"+
]
Re
>
S
The bound given by (5.3) is best possible.
THEOREM 7.
Let p be any positive integer,
any integer greater than -p.
If
f(z) e R( + p- i) then
Re
Da+P+I f (z)
D(+pf(z)
o + p
+ p +
>
for
1
Izl
< r
,p
where
r
a+p
(’P
(5.4)
2 + / 3 + ((+p-l)
This result is sharp.
PROOF.
+ p
Let f(z) e R(a
De+Pf (z)
Da+P-lf(z)
Therefore q(0)
i).
regular
We define the
i
(q(z) + a + p
(a + p)
i),
function q(z) on D by
z
(5.5)
D
i and Re q(z) > 0 in D.
Taking logarithmic derivative of (5.5) and using (1.8) we get
D+P+if(z)
e + p
p + i
Using Lemma (i) with S
1
+
D+pf(z)
+ p
l,
Izl
e+p f (z)
I
>
e + P
e+p+l
+ P
<
[A + (A
zq’(z)
+ p
q(z) +
2
((+p-l)2
(( + p)
]
(5.6)
2
for
i)2)1/2]1/2
where
A
1
I, (5.6) and (5.3) show that
Da+p+I f (z)
D
(z) +
(+p+l)
2( + p) + 8.
(5.7)
A.K. SONI
298
Minor computations yield the following:
A + (A
2
((+p-l)
Thus (5.7) yields the radius r
2
1)2) 1/2
(2 +
as given by
,P
3
)2.
+ (+p-l) 2
(5.8)
(5.4).
The method of AI-Amiri [2] is used to determine the extremal functions.
The
extremal functions thus obtained for this theorem are rotations of f(z) where f(z)
is given by
D+Pf(z)
D+P-lf(z)
i
( + p)
rl + zz +
i-
i
+ P
z e D.
This completes the proof of Theorem 7.
REMARK 3.
For
0, p
i, Theorem 7 gives the well-known radius of convexity
for the class of starlike functions:
r0,1 --2-/f.
Now an easy modification of the method used by Ai-Amiri [2, Theorem 4] gives
the following result.
THEOREM 8.
F(
f(z)
Izl
< r
Let p be any positive integer,
+ p
any integer greater than -p.
i), then f(z) satisfies (5.2) with
+
replaced by
If
i for
where
r
’P
1/2
i) + 2(e + p + 2)
(a + p
1/2
( + p + 3) + 2( + p + 2)
1/2
This result is sharp.
REMARK 4.
Ai-Amiri
6.
[2, Theorem 4].
P(
THE CLASSES
By R(e
+
p
Re
for some
i, Theorem 8 becomes a special case of a result due to
For p
+ p
I,),
i,).
we denote the classes of functions f(z)
(i- )
D+Pf(z)
D+P_if(z)
+
-> 0, p any positive integer and
D
+p+l f
(z)
D+pf(z)
I
> 1
A that satisfy
P
z e D,
any integer greater than -p.
(6.1)
Again
using the technique employed in [2], the following theorem is obtained.
THEOREM 9.
f(z)
+
p
R2(
Let p be any positive integer,
any integer greater than -p.
i), then f(z) satisfies (6.1) for
]z]
<
r,p,
where
If
299
GENERALIZATIONS OF p-VALENT FUNCTIONS
I(
+ p + 1
B))1/2) 1/21 /2
2B) + 2(B(a + p + 1 +
(a + p + 1 + 2) + 2((a + p + 1 +
This result is sharp.
REMARK 5.
For
I, Theorem 9 reduces
to Theorem 8.
Also for p
i, Theorem
9 represents a special case of a theorem due to Al-Amiri [2, Theorem 8].
ACKNOWLEDGEMENTS.
This paper forms a part of the
author’s doctoral thesis written
at Bowling Green State University of Ohio at Bowling Green.
The author would llke
to thank Professor Hassoon S. AI-Amiri for his guidance and direction.
REFERENCES
1.
RbSCHEWEYH, S. New Criteria for Univalent Functions, Proc. Amer. Math. Soc.
49 (1975), 109-115.
2.
AL-AMIRI, H.S.
Certain Generalizations of Prestarlike Functions, J. Aust. Math.
Soc. (Series A) 28 (1979), 325-334.
and SINGH, S.
Integrals of Certain Univalent Functions, Proc.
Amer. Math. Soc. 77 (19 79), 336-340.
3.
SINGH, R.
4.
UMEZAWA, T. Multivalently Close-to-Convex Functions, Proc. Amer. Math.. Soc. 8
(1957), 869-874.
5.
SUFFRIDGE, T.J.
6.
GOEL, R.M. nd SOHI, N.S. A New Criterion for p-Valent Functions, Proc. Amer.
Math. Soc. 78 (1980), 353-357.
7.
JACK, I.S. Functions Starlike and Convex of Order
3 (1971), 469-474.
8.
LIBERA, R.J.
9.
On Certain Extremal Problems for Functions with
RUSCHEWEYH, S. and SINGH, V.
Positive Real Part, Proc. Amer. Math. Soc. 61 (1976), 329-334.
Starlike Functions as Limits of Polynomials, Advances in Complex Function Theory, (Lecture Notes in Mathematics 505, Springer-Verlag,
Berlin) (1976), 164-202.
, J.
London Math. Soc. (2)
Some Classes of Regular Univalent Functions, Proc. Amer. Math.
Soc. 16 (1965), 755-758.