Inequalities forp-Valent Functions H.A. Al-Kharsani and
S.S. Al-Hajiry vol. 9, iss. 3, art. 90, 2008
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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
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 uniformlyp-valent functions in the open unit diskD.
Inequalities forp-Valent Functions H.A. Al-Kharsani and
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Contents
1 Introduction 3
2 Certain Results for the Multivalent Functions 5
Inequalities forp-Valent Functions H.A. Al-Kharsani and
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1. Introduction
LetA(p)denote the class of functionsf(z)of the form f(z) =zp+
∞
X
k=p+1
akzk, (p∈N= 1,2,3, . . .),
which are analytic and multivalent in the open unit diskD={z :z ∈C; |z|<1}.
A functionf(z) ∈A(p)is said to be in SPp(α), the class of uniformlyp-valent starlike functions (or, uniformly starlike when p = 1) of orderα if it satisfies the condition
(1.1) <e
zf0(z) f(z) −α
≥
zf0(z) f(z) −p
.
Replacingf in (1.1) byzf0(z), we obtain the condition
(1.2) <e
1 + zf”(z) f0(z) −α
≥
zf”(z)
f0(z) −(p−1)
required for the functionf to be in the subclassU CVpof uniformlyp-valent convex functions (or, uniformly convex whenp= 1) of orderα. Uniformlyp-valent starlike andp-valent convex functions were first introduced [4] whenp = 1, α = 0and [2]
when p≥1, p∈Nand then studied by various authors.
We set
Ωα =n
u+iv, u−α >p
(u−p)2+v2o
with q(z) = zff(z)0(z) or q(z) = 1 + zff0”(z)(z) and consider the functions which map D onto the parabolic domainΩαsuch thatq(z)∈Ωα.
Inequalities forp-Valent Functions H.A. Al-Kharsani and
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By the properties of the domainΩα, we have
(1.3) <e(q(z))><e(Qα(z))> p+α 2 , where
Qα(z) =p+2(p−α) π2
log
1 +√ z 1−√
z 2
.
Futhermore, a functionf(z)∈A(p)is said to be uniformlyp-valent close-to-convex (or, uniformly close-to-convex whenp = 1) of order α inD if it also satisfies the inequality
<e
zf0(z) g(z) −α
≥
zf0(z) g(z) −p
for someg(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 inD(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 functionw(z)be (non-constant) analytic inDwithw(0) = 0. If |w(z)|attains its maximum value on the circle|z| =r < 1 at a pointz0, then
z0w0(z0) = cw(z0), cis real andc≥1.
Inequalities forp-Valent Functions H.A. Al-Kharsani and
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2. Certain Results for the Multivalent Functions
Making use of Lemma1.1, we first give the following theorem:
Theorem 2.1. Letf(z)∈A(p).Iff(z)satisfies the following inequality:
(2.1) <e
1 + zff0”(z)(z) −p
zf0(z) f(z) −p
<1 + 2 3p, thenf(z)is uniformlyp-valent starlike inD.
Proof. We definew(z)by
(2.2) zf0(z)
f(z) −p= p
2w(z), (p∈N, z ∈D).
Thenw(z)is analytic inDandw(0) = 0.Furthermore, by logarithmically differen- tiating (2.2), we find that
1 + zf”(z)
f0(z) −p= p
2w(z) + zw0(z)
2 +w(z), (p∈N, z ∈D) which, in view of (2.1), readily yields
(2.3) 1 + zff0”(z)(z) −p
zf0(z)
f(z) −p = 1 + zw0(z)
p
2w(z)(2 +w(z)), (p∈N, z ∈D).
Suppose now that there exists a pointz0 ∈Dsuch that
max|w(z)|:|z| ≤ |z0|=|w(z0)|= 1; (w(z0)6= 1);
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and, letw(z0) = eiθ(θ 6=−π). Then, applying the Lemma1.1, we have (2.4) z0w0(z0) = cw(z0), c≥1.
From (2.3) – (2.4), we obtain
<e
1 + z0ff0(z”(z0)0)−p
z0f0(z0) f(z0) −p
=<e
1 + z0w0(z0)
p
2w(z0)(2 +w(z0))
=<e
1 + 2c p
1 (2 +w(z0))
= 1 +2c p<e
1 (2 +w(z0))
= 1 +2c p<e
1 (2 +eiθ)
(θ 6=−π)
= 1 + 2c
3p ≥1 + 2 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
zf0(z) f(z) −p
< p
2, (p∈N, z ∈D)
which implies that zff(z)0(z) lie in a circle which is centered atpand whose radius is p2 which means that zff(z)0(z) ∈Ω, and so
(2.5) <e
zf0(z) f(z)
≥
zf0(z) f(z) −p
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i.e.f(z)is uniformlyp-valent starlike inD.
Using (2.5), we introduce a sufficient coefficient bound for uniformly p-valent starlike functions in the following theorem:
Theorem 2.2. Letf(z)∈A(p).If
∞
X
k=2
(2k+p−α)|ak+p|< p−α.
thenf(z)∈SPp(α).
Proof. Let
∞
X
k=2
(2k+p−α)|ak+p|< p−α.
It is sufficient to show that
zf0(z)
f(z) −(p+α)
< p+α 2 . We find that
zf0(z)
f(z) −(p+α)
=
−α+P∞
k=2(k−α)ak+pzk−1 1 +P∞
k=2ak+pzk−1 (2.6)
< α+P∞
k=2(k−α)|ak+p| 1−P∞
k=2|ak+p| ,
(2.7) 2α+
∞
X
k=2
(2k+p−α)|ak+p|< p+α.
Inequalities forp-Valent Functions H.A. Al-Kharsani and
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This shows that the values of the function
(2.8) Φ(z) = zf0(z)
f(z)
lie in a circle which is centered at(p+α)and whose radius is p+α2 ,which means that
zf0(z)
f(z) ∈Ωα.Hencef(z)∈SPp(α).
The following diagram shows Ω1
2 whenp= 3and the circle is centered at 72 with radius 74 :
Next, we determine the sufficient coefficient bound for uniformlyp-valent convex functions.
Theorem 2.3. Letf(z)∈A(p).Iff(z)satisfies the following inequality (2.9) <e 1 + zff00000(z)(z) −p
1 + zff000(z)(z)−p
!
<1 + 2 3p−2, thenf(z)is uniformlyp−valent convex inD.
Proof. If we definew(z)by (2.10) 1 + zf00(z)
f0(z) −p= p
2w(z), (p∈N, z∈D),
thenw(z)satisfies the conditions of Jack’s Lemma. Making use of the same tech- nique as in the proof of Theorem2.2, we can easily get the desired proof of Theorem 2.4.
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We …nd that
zf0(z)f(z) (p+ ) = +P1
k=2(k )ak+pzk 1 1 +P1
k=2ak+pzk 1 < +P1
k=2(k )jak+pj 1 P1
k=2jak+pj
(1)
2 + X1
k=2
(2k+p )jak+pj < p+ :
(2)
This shows that the values of the function
(z) = zf0(z)
f(z)
(3)
lie in a circle which is centered at
(p+ )and whose radius is
p+2 ;whichmeans that
zff(z)0(z) 2 :Hence
f(z) 2SPp( ):The following diagram shows
12
when
p= 3and the circle centered at
7
2
with radius
74 :10 8
6 4
2 5
2.5
0
-2.5
-5
x y
x y
6
Figure 1.
Theorem 2.4. Letf(z)∈A(p).If (2.11)
∞
X
k=2
(k+p)(2k+p−α)|ak+p|< p(p−α), thenf(z)∈U CVp(α).
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Proof. It is sufficient to show that
1 + zf00(z)
f0(z) −(p+α)
< p+α 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 uniformlyp-valent close- to-convex functions.
Theorem 2.5. Letf(z)∈A(p).If f(z)satisfies the following inequality
(2.12) <e
zf00(z) f0(z)
< p− 2 3, thenf(z)is uniformly p-valent close-to-convex inD.
Proof. If we definew(z)by
(2.13) f0(z)
zp−1 −p= p
2w(z), (p∈N, z ∈D),
then clearly, w(z)is analytic inD andw(0) = 0.Furthermore, by logarithmically differentiating (2.10), we find that
(2.14) zf00(z)
f0(z) = (p−1) + zw0(z) 2 +w(z).
Therefore, by using the conditions of Jack’s Lemma and (2.11), we have
<e
z0f00(z0) f0(z0)
= (p−1) +c<e
w(z0) 2 +w(z0)
=p−1 + c
3 > p− 2 3
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which contradicts the hypotheses (2.9). Thus, we conclude that |w(z)| < 1for all z ∈D; and equation (2.10) yields the inequality
f0(z) zp−1 −p
< p
2, (p∈N, z ∈D) which implies that fzp−10(z) ∈Ω, which means
<e
f0(z) zp−1
≥
f0(z) zp−1 −p
and, hencef(z)is uniformlyp-valent close-to-convex inD.
Theorem 2.6. Letf(z)∈A(p).If
∞
X
k=2
(k+p)|ak+p|< p−α 2 , thenf(z)∈U CCp(α).
By takingp= 1in Theorems2.2and2.6respectively, we have
Corollary 2.7. Letf(z)∈A(1).Iff(z)satisfies the following inequality:
<e
zf”(z) f0(z) zf0(z)
f(z) −1
< 5 3, thenf(z)is uniformly starlike inD.
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Corollary 2.8. Letf(z)∈A(1).If f(z)satisfy the following inequality
<e
zf00(z) f0(z)
< 1 3, thenf(z)is uniformly close-to-convex inD.
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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-KHARSANIANDS.S. AL-HAJIRY, Subordination results for the fam- ily 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 andp-valent func- tions, 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.