A NOTE ON CERTAIN INEQUALITIES FOR
p-VALENT FUNCTIONS
H.A. AL-KHARSANI AND S.S. AL-HAJIRY
Department of Mathematics
Faculty of Science, Girls College, Dammam
Saudi Arabia.
EMail: ssmh1@hotmail.com
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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Received:
05 February, 2008
Accepted:
01 July, 2008
Communicated by:
N.K. Govil
2000 AMS Sub. Class.:
30C45.
Key words:
p-valent functions.
Abstract:
We use a parabolic region to prove certain inequalities for uniformly p-valent
functions in the open unit disk D.
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Contents
1
Introduction
3
2
Certain Results for the Multivalent Functions
5
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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1.
Introduction
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)
0
Replacing f in (1.1) by zf (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.
We set
o
n
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 .
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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By the properties of the domain Ωα , we have
(1.3)
<e(q(z)) > <e(Qα (z)) >
where
2(p − α)
Qα (z) = p +
π2
log
p+α
,
2
√ 2
1+ z
√
.
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.
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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2.
Certain Results for the Multivalent Functions
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:


zf ” (z)
1 + f 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.
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
Proof. We define w(z) by
p
zf 0 (z)
− p = w(z),
f (z)
2
(2.2)
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(p ∈ N, z ∈ D).
Then w(z) is analytic in D and w(0) = 0.Furthermore, by logarithmically differentiating (2.2), we find that
1+
zf ” (z)
p
zw0 (z)
−
p
=
w(z)
+
,
f 0 (z)
2
2 + w(z)
(p ∈ N, z ∈ D)
zf ” (z)
−
f 0 (z)
0
zf (z)
−p
f (z)
(2.3)
p
zw0 (z)
,
=1+ p
w(z)(2 + w(z))
2
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(p ∈ N, z ∈ D).
Suppose now that there exists a point z0 ∈ D such that
max |w(z)| : |z| ≤ |z0 | = |w(z0 )| = 1;
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which, in view of (2.1), readily yields
1+
Contents
(w(z0 ) 6= 1);
and, let w(z0 ) = eiθ (θ 6= −π). Then, applying the Lemma 1.1, we have
(2.4)
z0 w0 (z0 ) = cw(z0 ),
c ≥ 1.
From (2.3) – (2.4), we obtain


”
0
1 + z0ff0 (z(z0 )0 ) − 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 ))
1
2c
= 1 + <e
p
(2 + w(z0 ))
1
2c
= 1 + <e
(θ 6= −π)
p
(2 + eiθ )
2
2c
=1+
≥1+
3p
3p
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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Contents
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
which
0 (z)
implies that zff (z)
lie in a circle
zf 0 (z)
means that f (z) ∈ Ω, and so
(2.5)
<e
zf 0 (z)
f (z)
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which is centered at p and whose radius is
0
zf (z)
≥ − p
f (z)
p
2
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
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
∞
X
(2k + p − α) |ak+p | < p − α.
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k=2
It is sufficient to show that
0
zf (z)
p+α
<
−
(p
+
α)
.
f (z)
2
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We find that
(2.6)
(2.7)
0
P
k−1 zf (z)
−α + ∞
(k
−
α)a
z
k+p
k=2
P
f (z) − (p + α) = k−1
1+ ∞
k=2 ak+p z
P∞
α + k=2 (k − α) |ak+p |
P
<
,
1− ∞
k=2 |ak+p |
2α +
∞
X
k=2
(2k + p − α) |ak+p | < p + α.
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This shows that the values of the function
Φ(z) =
(2.8)
zf 0 (z)
f (z)
lie in a circle which is centered at (p + α) and whose radius is
zf 0 (z)
∈ Ωα . Hence f (z) ∈ SPp (α).
f (z)
p+α
,which
2
means that
The following diagram shows Ω 1 when p = 3 and the circle is centered at 72 with
2
radius 74 :
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)
1 + zff 00 (z)
−p
2
(2.9)
<e
<1+
,
zf 00 (z)
3p − 2
1+ 0
−p
f (z)
then f (z) is uniformly p−valent convex in D.
1+
zf 00 (z)
p
−
p
=
w(z),
f 0 (z)
2
vol. 9, iss. 3, art. 90, 2008
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Proof. If we define w(z) by
(2.10)
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
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(p ∈ N, z ∈ D),
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.
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7
2
with radius
7
4
:
y
5
2.5
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
0
2
4
6
8
10
vol. 9, iss. 3, art. 90, 2008
x
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-2.5
-5
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Figure 1.
6
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Theorem 2.4. Let f (z) ∈ A(p). If
(2.11)
∞
X
k=2
then f (z) ∈ U CVp (α).
(k + p)(2k + p − α) |ak+p | < p(p − α),
Close
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 closeto-convex functions.
Theorem 2.5. Let f (z) ∈ A(p). If f (z) satisfies the following inequality
00 zf (z)
2
<p− ,
(2.12)
<e
0
f (z)
3
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
Title Page
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
(2.14)
zf 00 (z)
zw0 (z)
=
(p
−
1)
+
.
f 0 (z)
2 + w(z)
Therefore, by using the conditions of Jack’s Lemma and (2.11), we have
z0 f 00 (z0 )
w(z0 )
<e
= (p − 1) + c<e
f 0 (z0 )
2 + w(z0 )
c
2
=p−1+ >p−
3
3
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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
(p ∈ N, z ∈ D)
z p−1
2
which implies that
f 0 (z)
z p−1
∈ Ω, which means
0 0
f (z)
f (z)
<e
≥
−
p
z p−1
z p−1
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
and, hence f (z) is uniformly p-valent close-to-convex in D.
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Theorem 2.6. Let f (z) ∈ A(p). If
∞
X
p−α
(k + p) |ak+p | <
,
2
k=2
then f (z) ∈ U CCp (α).
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By taking p = 1 in Theorems 2.2 and 2.6 respectively, we have
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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.
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Corollary 2.8. Let f (z) ∈ A(1). If f (z) satisfy the following inequality
00 zf (z)
1
<e
,
<
f 0 (z)
3
then f (z) is uniformly close-to-convex in D.
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
vol. 9, iss. 3, art. 90, 2008
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References
[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].
Inequalities for p-Valent
Functions
H.A. Al-Kharsani and
S.S. Al-Hajiry
[3] A.W. GOODMAN, On the Schwarz-Cristoffel transformation and p-valent functions, Trans. Amer. Math. Soc., 68 (1950), 204–223.
vol. 9, iss. 3, art. 90, 2008
[4] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl., 155
(1991), 364–370.
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[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.
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