Volume 9 (2008), Issue 3, Article 90, 6 pp.
A NOTE ON CERTAIN INEQUALITIES FOR p-VALENT FUNCTIONS
H.A. AL-KHARSANI AND S.S. AL-HAJIRY
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE
G IRLS C OLLEGE , DAMMAM
S AUDI A RABIA .
ssmh1@hotmail.com
Received 05 February, 2008; accepted 01 July, 2008
Communicated by N.K. Govil
A BSTRACT. We use a parabolic region to prove certain inequalities for uniformly p-valent functions in the open unit disk D.
Key words and phrases: p-valent functions.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Let A(p) denote the class of functions f (z) of the form
p
f (z) = z +
∞
X
ak z k ,
(p ∈ N = 1, 2, 3, . . . ),
k=p+1
which are analytic and multivalent in the open unit disk D = {z : z ∈ C; |z| < 1}.
A function f (z) ∈ A(p) is said to be in SPp (α), the class of uniformly p-valent starlike
functions (or, uniformly starlike when p = 1) of order α if it satisfies the condition
0
0
zf (z)
zf (z)
(1.1)
<e
−α ≥
− p .
f (z)
f (z)
Replacing f in (1.1) by zf 0 (z), we obtain the condition
”
zf (z)
zf ” (z)
(1.2)
<e 1 + 0
− α ≥ 0
− (p − 1)
f (z)
f (z)
required for the function f to be in the subclass U CVp of uniformly p-valent convex functions
(or, uniformly convex when p = 1) of order α. Uniformly p-valent starlike and p-valent convex
functions were first introduced [4] when p = 1, α = 0 and [2] when p ≥ 1, p ∈ N and then
studied by various authors.
040-08
2
H.A. A L -K HARSANI AND S.S. A L -H AJIRY
We set
n
o
p
Ωα = u + iv, u − α > (u − p)2 + v 2
0
”
(z)
(z)
with q(z) = zff (z)
or q(z) = 1 + zff 0 (z)
and consider the functions which map D onto the
parabolic domain Ωα such that q(z) ∈ Ωα .
By the properties of the domain Ωα , we have
p+α
(1.3)
<e(q(z)) > <e(Qα (z)) >
,
2
where
√ 2
2(p − α)
1+ z
√
.
Qα (z) = p +
log
π2
1− z
Futhermore, a function f (z) ∈ A(p) is said to be uniformly p-valent close-to-convex (or, uniformly close-to-convex when p = 1) of order α in D if it also satisfies the inequality
0
0
zf (z)
zf (z)
<e
−α ≥
− p
g(z)
g(z)
for some g(z) ∈ SPp (α).
We note that a function h(z) is p-valent convex in D if and only if zh0 (z) is p-valent starlike
in D (see, for details, [1], [3], and [6]).
In order to obtain our main results, we need the following lemma:
Lemma 1.1 (Jack’s Lemma [5]). Let the function w(z) be (non-constant) analytic in D with
w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 , then
z0 w0 (z0 ) = cw(z0 ),
c is real and c ≥ 1.
2. C ERTAIN R ESULTS FOR THE M ULTIVALENT F UNCTIONS
Making use of Lemma 1.1, we first give the following theorem:
Theorem 2.1. Let f (z) ∈ A(p). If f (z) satisfies the following inequality:


” (z)
1 + zff 0 (z)
−p
<1+ 2 ,
(2.1)
<e  zf 0 (z)
3p
−p
f (z)
then f (z) is uniformly p-valent starlike in D.
Proof. We define w(z) by
zf 0 (z)
p
− p = w(z),
f (z)
2
(2.2)
(p ∈ N, z ∈ D).
Then w(z) is analytic in D and w(0) = 0.Furthermore, by logarithmically differentiating (2.2),
we find that
zf ” (z)
p
zw0 (z)
1+ 0
− p = w(z) +
,
(p ∈ N, z ∈ D)
f (z)
2
2 + w(z)
which, in view of (2.1), readily yields
zf ” (z)
−
f 0 (z)
0
zf (z)
−p
f (z)
1+
(2.3)
p
zw0 (z)
=1+ p
,
w(z)(2 + w(z))
2
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 90, 6 pp.
(p ∈ N, z ∈ D).
http://jipam.vu.edu.au/
I NEQUALITIES FOR p-VALENT F UNCTIONS
3
Suppose now that there exists a point z0 ∈ D such that
max |w(z)| : |z| ≤ |z0 | = |w(z0 )| = 1;
(w(z0 ) 6= 1);
and, let w(z0 ) = eiθ (θ 6= −π). Then, applying the Lemma 1.1, we have
z0 w0 (z0 ) = cw(z0 ),
(2.4)
c ≥ 1.
From (2.3) – (2.4), we obtain


z0 f ” (z0 )
0
1 + f 0 (z0 ) − p
z
w
(z
)
0
0
 = <e 1 + p
<e  z0 f 0 (z0 )
w(z0 )(2 + w(z0 ))
−
p
2
f (z0 )
2c
1
= <e 1 +
p (2 + w(z0 ))
2c
1
= 1 + <e
p
(2 + w(z0 ))
2c
1
(θ 6= −π)
= 1 + <e
p
(2 + eiθ )
2
2c
≥1+
=1+
3p
3p
which contradicts the hypothesis (2.1). Thus, we conclude that |w(z)| < 1 for all z ∈ D; and
equation (2.2) yields the inequality
0
zf (z)
p
< ,
−
p
(p ∈ N, z ∈ D)
f (z)
2
which implies that
that
zf 0 (z)
f (z)
zf 0 (z)
f (z)
lie in a circle which is centered at p and whose radius is p2 which means
∈ Ω, and so
(2.5)
<e
zf 0 (z)
f (z)
0
zf (z)
≥
− p
f (z)
i.e. f (z) is uniformly p-valent starlike in D.
Using (2.5), we introduce a sufficient coefficient bound for uniformly p-valent starlike functions in the following theorem:
Theorem 2.2. Let f (z) ∈ A(p). If
∞
X
(2k + p − α) |ak+p | < p − α.
k=2
then f (z) ∈ SPp (α).
Proof. Let
∞
X
(2k + p − α) |ak+p | < p − α.
k=2
It is sufficient to show that
0
zf (z)
p+α
<
−
(p
+
α)
.
f (z)
2
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 90, 6 pp.
http://jipam.vu.edu.au/
f (z)
1
k=2
1+
ak+p z k
1
k=2
1
1
(1)
4
2 +
1
X
jak+p j
H.A. A L -K HARSANI AND S.S. A L -H AJIRY
) jak+p j < p + :
(2k + p
k=2
(2)
We find that
of the
P function
P
This shows
that the values
−α + ∞
(k − α) |ak+p |
(k − α)ak+p z k−1 α + ∞
zf 0 (z)
k=2
k=2
P∞
P∞
(2.6) − (p + α) = <
,
f (z)
1 + k=2 ak+p z k−1
1 − k=2 |ak+p |
zf 0(z)
∞
(3)
(z) =
X
f (z)
(2.7)
2α +
(2k + p − α)
|ak+p | < p + α.
k=2
lieshows
in a that
circle
which ofisthe
centered
This
the values
functionat (p + ) and whose radius is
zf 0 (z)
zf 0(z)
(2.8)
Φ(z)
=
means that f (z) 2 : Hence f (z) 2 SP
( ):
f p(z)
p+
2
;which
0
(z)
lie in a circle which is centered at (p+α) and whose radius is p+α
,which means that zff (z)
∈ Ωα .
2
1 when p = 3 and the circle centered at following
Hence fThe
(z) ∈
SPp (α). diagram shows
2
The
shows Ω 1 when p = 3 and the circle is centered at
7
7 following diagram
2
with radius
4
:
y
2
7
2
with radius
7
4
:
5
2.5
0
2
4
6
8
10
x
-2.5
-5
Figure 1.
6
Next, we determine the sufficient coefficient bound for uniformly p-valent convex functions.
Theorem 2.3. Let f (z) ∈ A(p). If f (z) satisfies the following inequality
!
000 (z)
−p
1 + zff 00 (z)
2
<
1
+
,
(2.9)
<e
00
3p − 2
1 + zf0 (z) − p
f (z)
then f (z) is uniformly p−valent convex in D.
Proof. If we define w(z) by
zf 00 (z)
p
(2.10)
1+ 0
− p = w(z),
(p ∈ N, z ∈ D),
f (z)
2
then w(z) satisfies the conditions of Jack’s Lemma. Making use of the same technique as in the
proof of Theorem 2.2, we can easily get the desired proof of Theorem 2.4.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 90, 6 pp.
http://jipam.vu.edu.au/
I NEQUALITIES FOR p-VALENT F UNCTIONS
5
Theorem 2.4. Let f (z) ∈ A(p). If
∞
X
(2.11)
(k + p)(2k + p − α) |ak+p | < p(p − α),
k=2
then f (z) ∈ U CVp (α).
Proof. It is sufficient to show that
00
p+α
zf
(z)
<
1 +
−
(p
+
α)
.
f 0 (z)
2
Making use of the same technique as in the proof of Theorem 2.3, we can prove inequality
(2.8).
The following theorems give the sufficient conditions for uniformly p-valent close-to-convex
functions.
Theorem 2.5. Let f (z) ∈ A(p). If f (z) satisfies the following inequality
00 zf (z)
2
(2.12)
<e
<p− ,
0
f (z)
3
then f (z) is uniformly p-valent close-to-convex in D.
Proof. If we define w(z) by
f 0 (z)
p
(2.13)
− p = w(z),
(p ∈ N, z ∈ D),
p−1
z
2
then clearly, w(z) is analytic in D and w(0) = 0. Furthermore, by logarithmically differentiating
(2.10), we find that
zf 00 (z)
zw0 (z)
(2.14)
=
(p
−
1)
+
.
f 0 (z)
2 + w(z)
Therefore, by using the conditions of Jack’s Lemma and (2.11), we have
w(z0 )
z0 f 00 (z0 )
= (p − 1) + c<e
<e
f 0 (z0 )
2 + w(z0 )
c
2
=p−1+ >p−
3
3
which contradicts the hypotheses (2.9). Thus, we conclude that |w(z)| < 1 for all z ∈ D; and
equation (2.10) yields the inequality
0
f (z)
p
(p ∈ N, z ∈ D)
z p−1 − p < 2 ,
which implies that
f 0 (z)
z p−1
∈ Ω, which means
0 0
f (z)
f (z)
<e
≥ p−1 − p
p−1
z
z
and, hence f (z) is uniformly p-valent close-to-convex in D.
Theorem 2.6. Let f (z) ∈ A(p). If
∞
X
(k + p) |ak+p | <
k=2
p−α
,
2
then f (z) ∈ U CCp (α).
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 90, 6 pp.
http://jipam.vu.edu.au/
6
H.A. A L -K HARSANI AND S.S. A L -H AJIRY
By taking p = 1 in Theorems 2.2 and 2.6 respectively, we have
Corollary 2.7. Let f (z) ∈ A(1). If f (z) satisfies the following inequality:


zf ” (z)
0
f (z)
 < 5,
<e  zf 0 (z)
3
−1
f (z)
then f (z) is uniformly starlike in D.
Corollary 2.8. Let f (z) ∈ A(1). If f (z) satisfy the following inequality
00 1
zf (z)
<e
< ,
0
f (z)
3
then f (z) is uniformly close-to-convex in D.
R EFERENCES
[1] J.W. ALEXANDER, Functions which map the interior of the unit circle upon simple regions, Ann.
of Math., 17 (1915–1916), 12–22.
[2] H.A. AL-KHARSANI AND S.S. AL-HAJIRY, Subordination results for the family of uniformly
convex p-valent functions, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art. 20. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=649].
[3] A.W. GOODMAN, On the Schwarz-Cristoffel transformation and p-valent functions, Trans. Amer.
Math. Soc., 68 (1950), 204–223.
[4] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl., 155 (1991), 364–370.
[5] I.S. JACK, Functions starlike and convex of order α, J. London Math. Soc., 3 (1971), 469–474.
[6] W. KAPLAN, Close-to-convex schlich functions, Michigan Math. J., 1 (1952), 169–185.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 90, 6 pp.
http://jipam.vu.edu.au/