Journal of Inequalities in Pure and
Applied Mathematics
SUBORDINATION RESULTS FOR THE FAMILY OF UNIFORMLY
CONVEX p−VALENT FUNCTIONS
volume 7, issue 1, article 20,
2006.
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 16 May, 2005;
accepted 02 October, 2005.
Communicated by: G. Kohr
Abstract
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2000
Victoria University
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Abstract
The object of the present paper is to introduce a class of p−valent uniformly
functions UCVp . We deduce a criteria for functions to lie in the class UCVp and
derive several interesting properties such as distortion inequalities and coefficients estimates. We confirm our results using the Mathematica program by
drawing diagrams of extremal functions of this class.
2000 Mathematics Subject Classification: 30C45.
Key words: p−valent, Uniformly convex functions, Subordination.
We express our thanks to Dr.M.K.Aouf for his helpful comments.
Contents
1
2
3
4
5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Class P ARp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Characterization of U CVp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Subordination Theorem and Consequences . . . . . . . . . . . . . . . 9
General Properties of Functions in U CVp . . . . . . . . . . . . . . . . 15
5.1
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
H.A. Al-Kharsani and S.S.
Al-Hajiry
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1.
Introduction
Denote by A(p, n) the class of normalized functions
(1.1)
p
f (z) = z +
∞
X
ak+p−1 z k+p−1
k=2
regular in the unit disk D = {z : |z| < 1} and p ∈ N, consider also its
subclasses C(p), S ∗ (p) consisting of p−valent convex and starlike functions
respectively, where C(1) ≡ C, S ∗ (1) ≡ S ∗ , the classes of univalent convex and
starlike functions .
It is well known that for any f ∈ C, not only f (D) but the images of all
circles centered at 0 and lying in D are convex arcs. B. Pinchuk posed a question
whether this property is still valid for circles centered at other points of D. A.W.
Goodman [1] gave a negative answer to this question and introduced the class
U CV of univalent uniformly convex functions, f ∈ C such that any circular
arc γ lying in D, having the center ζ ∈ D is carried by f into a convex arc.
A.W.Goodman [1] stated the criterion
f 00 (z)
(1.2)
Re 1 + (z − ζ) 0
> 0, ∀z, ζ ∈ D ⇐⇒ f ∈ U CV.
f (z)
Later F. Ronning (and independently W. Ma and D. Minda) [7] obtained a
more suitable form of the criterion , namely
00 zf (z) zf 00 (z)
, ∀z ∈ D ⇐⇒ f ∈ U CV.
> 0
(1.3)
Re 1 + 0
f (z)
f (z) This criterion was used to find some sharp coefficients estimates and distortion theorems for functions in the class U CV .
Subordination Results for the
Family of Uniformly Convex
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H.A. Al-Kharsani and S.S.
Al-Hajiry
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4
1
2
0
0
-1
2
3
-2
4
5
6
2.
The Class P ARp
We now introduce a subfamily P ARp of P . Let
υ2
(2.1)
Ω = w = µ + iυ :
< 2µ − p
p
(2.2)
= {w : Re w > |w − p|}.
Note that Ω is the interior of a parabola in the right half-plane which is
symmetric about the real axis and has vertex at (p/2, 0). The following diagram
shows Ω when p = 3:
Let
(2.3)
Subordination Results for the
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P ARp = {h ∈ p : h(D) ⊆ Ω}.
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√ Example 2.1. It is known that z = − tan2 2√π2p w maps
ν2
w = µ + iν :
< p − 2µ
p
√
conformally onto D. Hence, z = − tan2 2√π2p p − w maps Ω conformally
onto D. Let w = Q(z) be the inverse function. Then Q(z) is a Riemann mapping function from D to Ω which satisfies Q(0) = p; more explicitly,
√ 2 X
∞
1+ z
2p
√
(2.4)
=
Bn z n
Q(z) = p + 2 log
π
1− z
n=0
(2.5)
8p
16p
184p 3
= p + 2 z + 2 z2 +
z + ···
π
3π
45π 2
Obviously, Q(z) belongs to the class P ARp . Geometrically, P ARp consists
of those holomorphic functions h(z) (h(0) = p) defined on D which are subordinate to Q(z), written h(z) ≺ Q(z).
The analytic characterization of the class P ARp is shown in the following
relation:
(2.6)
h(z) ∈ P ARp ⇔ Re{h(z)} ≥ |h(z) − p|
such that h(z) is a p−valent analytic function on D.
Now, we can derive the following definition.
Definition 2.1. Let f (z) ∈ A(p, n). Then f (z) ∈ U CVp if f (z) ∈ C(p) and
00 (z)
1 + z ff 0 (z)
∈ P ARp .
Subordination Results for the
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H.A. Al-Kharsani and S.S.
Al-Hajiry
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3.
Characterization of U CVp
We present the nessesary and sufficient condition to belong to the class U CVp
in the following theorem:
Theorem 3.1. Let f (z) ∈ A(p, n). Then
00 00
f (z)
f (z)
(3.1) f (z) ∈ U CVp ⇔ 1 + Re z 0
≥ z 0
− (p − 1) ,
f (z)
f (z)
z ∈ D.
00
(z)
Proof. Let f (z) ∈ U CVp and h(z) = 1 + z ff 0 (z)
. Then h(z) ∈ P ARp , that is,
Re{h(z)} ≥ |h(z) − p|. Then
00
f (z)
f 00 (z)
Re 1 + z 0
≥ z 0
− (p − 1) .
f (z)
f (z)
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
H.A. Al-Kharsani and S.S.
Al-Hajiry
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Example 3.1. We now specify a holomorphic function K(z) in D by
(3.2)
K 00 (z)
= Q(z),
1+z 0
K (z)
where Q(z) is the conformal mapping onto Ω given in Example 2.1. Then it is
clear from Theorem 3.1 that K(z) is in U CVp .
Let
(3.3)
p
K(z) = z +
∞
X
k=2
Ak z k+p−1 .
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From the relationship between the functions Q(z) and K(z), we obtain
(3.4)
(p + n − 1)(n − 1)An =
n−1
X
(k + p − 1)Ak Bn−k .
k=1
Since all the coefficients Bn are positive, it follows that all of the coefficients An
are also positive. In particular,
A2 =
(3.5)
8p2
,
π 2 (p + 1)
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
and
(3.6)
H.A. Al-Kharsani and S.S.
Al-Hajiry
A3 =
p2
2(p + 2)
16
64p
+ 4
2
3π
π
.
Note that
(3.7)
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log
k 0 (z)
=
z p−1
Z
0
z
Q(ς) − p
dς.
ς
By computing some coefficients of K(z) when p = 3, we can obtain the
following diagram
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1
0.5
0
-0.5
-1
-2
2
0
-2
-1
Subordination Results for the
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H.A. Al-Kharsani and S.S.
Al-Hajiry
0
1
2
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4.
Subordination Theorem and Consequences
In this section, we first derive some subordination results from Theorem 4.1; as
corollaries we obtain sharp distortion, growth, covering and rotation theorems
from the family U CVp .
00
00
(z)
(z)
Theorem 4.1. Assume that f (z) ∈ U CVp . Then 1 + z ff 0 (z)
≺ 1 + zK
and
K 0 (z)
f 0 (z)
z p−1
≺
K 0 (z)
.
z p−1
00
00
(z)
(z)
Proof. Let f (z) ∈ U CVp . Then h(z) = 1 + z ff 0 (z)
≺ 1 + zK
is the same
K 0 (z)
as h(z) ≺ Q(z). Note that Q(z) − p is a convex univalent function in D. By
using a result of Goluzin, we may conclude that
Z z
Z z
f 0 (z)
h(ς) − 1
Q(ς) − p
K 0 (z)
(4.1)
log p−1 =
dς ≺
dς = log p−1 .
z
ς
ς
z
0
0
Equivalently,
f 0 (z)
z p−1
≺
K 0 (z)
z p−1
.
Corollary 4.2 (Distortion Theorem). Assume f (z) ∈ U CVp and |z| = r < 1.
Then K 0 (−r) ≤ |f 0 (z)| ≤ K 0 (r).
Subordination Results for the
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Equality holds for some z 6= 0 if and only if f (z) is a rotation of K(z).
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Proof. Since Q(z) − p is convex univalent in D, it follows that log K 0 (z) is
also convex univalent in D. In fact, the power series for log K 0 (z) has positive
coefficients, so the image of D under this convex function is symmetric about
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the real axis. As log
f 0 (z)
z p−1
0
(z)
≺ log Kzp−1
, the subordination principle shows that
(4.2)
0
K 0 (−r) = e{ log K (−r)} = e
min Re{log K 0 (z)}
|z|=r
{Re log K 0 (z)}
≤e
= |f 0 (z)| ≤ e
max Re{log K 0 (z)}
|z|=r
0
= e{log K (r)} = K 0 (r).
Note that for |z0 | = r, either
Re{log f 0 (z0 )} = min Re{log K 0 (z)}
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
|z|=r
H.A. Al-Kharsani and S.S.
Al-Hajiry
or
Re{log f 0 (z0 )} = max Re{log K 0 (z)}
|z|=r
for some z0 6= 0 if and only if log f 0 (z) = log K 0 (eiθ z) for some θ ∈ R.
Theorem 4.3. Let f (z) ∈ U CVp . Then
14p
(4.3)
|f 0 (z)| ≤ z p−1 e π2 ς(3) = z p−1 Lp
for |z| < 1. (L ≈ 5.502, ς(t) is the Riemann Zeta function.)
Proof. Let φ(z)
means that φ(z)
0 (z)
= zgg(z)
, where g(z) =
√ z
2p
√
≺ p + π2 log 1+
.
1− z
g(z)
log p =
z
Z
0
z
0
zf (z). Then φ(z) ≺ Q(z) which
Moreover ,
φ(s) − p
s
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and therefore, if z = reiθ and |z| = 1,
√ Z r
Z r
g(z) 1
+
2p
1
t
dt
iθ
√ dt
log p =
log
<e(φ(te ) − p) ≤ 2
z
t
π 0 t
1− t
0
√ Z 1
2p
1
1+ t
2p
√ dt = 2 (7ς(3)),
≤ 2
log
π 0 t
π
1− t
where
Z
0
1
1
log
t
Then we find that
√ 1+ t
√ dt = 7ς(3)
1− t
0 zf (z) 2p
2 (7ς(3))
.
zp ≤ e π
[8].
Subordination Results for the
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H.A. Al-Kharsani and S.S.
Al-Hajiry
The following diagram shows the boundary of K(z)’s dervative when p = 2
in a circle has the radius (5.5)2 :
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0
-20
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Corollary 4.4 (Growth Theorem). Let f (z) ∈ U CVp and |z| = r < 1. Then
−K(−r) ≤ |f (z)| ≤ K(r).
Equality holds for some z 6= 0 if and only if f (z) is a rotation of K(z).
Corollary 4.5 (Covering Theorem). Suppose f (z) ∈ U CVp . Then either f (z)
is a rotation of K(z) or {w : |w| ≤ −K(−1)} ⊆ f (D).
Corollary 4.6 (Rotation Theorem). Let f (z) ∈ U CVp and |z0 | = r < 1.
Then
|Arg{f 0 (z0 )}| ≤ max Arg{K 0 (z).
(4.4)
|z|=r
Equality holds for some z 6= 0 if and only if f (z) is a rotation of K(z).
P
k+p−1
Theorem 4.7. Let f (z) = z p + ∞
and f (z) ∈ U CVp , and let
k=2 ak+p−1 z
An+p−1 = max |an+p−1 |. Then
f (z)∈U CVp
(4.5)
Ap+1 =
8p2
.
π 2 (p + 1)
The result is sharp. Further, we get
n Y
8p2
8p
(4.6)
An+p−1 ≤
1+
.
(n + p − 1)(n − 1)π 2 k=3
(k − 2)π 2
P
k+p−1
Proof. Let f (z) = z p + ∞
and f (z) ∈ U CVp , and define
k=2 ak+p−1 z
∞
X
zf 00 (z)
=p+
ck z k+p−1 .
φ(z) = 1 + 0
f (z)
k=2
Subordination Results for the
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Then φ(z) ≺ Q(z). Q(z) is univalent in D and Q(D) is a convex region, so
Rogosinski’s theorem applies.
8p
16p
184p 3
z + ··· ,
Q(z) = p + 2 z + 2 z 2 +
π
3π
45π 2
so we have |cn | ≤ |B1 | = π8p2 := B. Now, from the relationship between functions f (z) and Q(z), we obtain
(n + p − 1)(n − 1)an+p−1 =
n−1
X
(k + p − 1)ak+p−1 cn−k .
k=1
2
pB
= π2 8p
. If we choose f (z) to be that function
(p+1)
(p+1)
2
zf 00 (z)
, then f (z) ∈ U CVp with ap+1 = π2 8p
, which
f 0 (z)
(p+1)
From this we get |ap+1 | =
for which Q(z) = 1 +
shows that this result is sharp. Now, when we put |c1 | = B, then
pap c2 + (p + 1)ap+1 c1
2(p + 2)
pB(1 + Bp)
.
|ap+2 | ≤
2(p + 2)
ap+2 =
When n = 3
pap c3 + (p + 1)ap+1 c2 + (p + 2)ap+2 c1
3(p + 3)
1 pB(1 + Bp)(2 + Bp)
|ap+3 | ≤
2
3(p + 3)
1
Bp
=
pB(1 + Bp) 1 +
.
3(p + 3)
2
ap+3 =
Subordination Results for the
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We now proceed by induction. Assume we have
1
Bp
Bp
|ap+n−1 | ≤
pB(1 + Bp) 1 +
··· 1 +
(n − 1)(p + n − 1)
2
n−2
n
Y
pB
Bp
=
1+
.
(n − 1)(p + n − 1) k=3
k−2
Corollary 4.8. Let
f (z) = z p +
|ap+n−1 | = O n12 .
P∞
k=2
ak+p−1 z k+p−1 and f (z) ∈ U CVp . Then
Subordination Results for the
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General Properties of Functions in U CVp
5.
P
k+p−1
Theorem 5.1. Let f (z) = z p + ∞
and f (z) ∈ U CVp . Then
k=2 ak+p−1 z
f (z) is a p−valently convex function of order β in |z| < r1 = r1 (p, β), where
r1 (p, β) is the largest value of r for which
(5.1)
rk−1 ≤
(p − β)(k − 1)
Qk (k + p − β − 1)B j=3 1 +
pB
j−2
,
(k ∈ N − {1}, 0 ≤ β < p).
Proof. It is sufficient to show that for f (z) ∈ U CVp ,
00
zf
(z)
1 +
≤ p − β, |z| < r1 (p, β),
−
p
f 0 (z)
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
0 ≤ β < p,
where r1 (p, β) is the largest value of r for which the inequality (5.1) holds true.
Observe that
P∞
00
k=2 (k + p − 1)(k − 1)ak+p−1 z k−1 zf
(z)
1 +
.
P
− p = k−1 f 0 (z)
p+ ∞
k=2 (k + p − 1)ak+p−1 z
00 (z)
Then we have 1 + zff 0 (z)
− p ≤ p − β if and only if
P∞
(k + p − 1)(k − 1) |ak+p−1 | rk−1
k=2P
k−1
p− ∞
k=2 (k + p − 1) |ak+p−1 | r
∞
X
≤p−β
(k + p − 1)(k + p − 1 − β) |ak+p−1 | rk−1 ≤ p2 − pβ.
⇒
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Al-Hajiry
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Then from Theorem 4.7 since f (z) ∈ U CVp , we have
k Y
pβ
Bp
|ak+p−1 | ≤
1+
(k + p − 1)(k − 1) j=3
j−2
and we may set
k Y
pβ
Bp
1+
ck+p−1 , ck+p−1 ≥ 0,
|ak+p−1 | =
(k + p − 1)(k − 1) j=3
j−2
(
)
∞
X
ck+p−1 ≤ 1 .
k ∈ N − {1},
Subordination Results for the
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H.A. Al-Kharsani and S.S.
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k=1
Now, for each fixed r, we choose a positive integer k0 = k0 (r) for which
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(k + p − 1 − β) k−1
r
(k − 1)
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is maximal. Then
∞
X
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(k + p − 1)(k + p − β − 1) |ak+p−1 | rk−1
Close
k=2
≤
k Y
Bp
(k0 + p − β − 1) k0 −1
r
1+
(k0 − 1)
j−2
j=3
.
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Consequently, the function f (z) is a p−valently convex function of order β in
|z| < r1 = r1 (p, β) provided that
k (k0 + p − β − 1) k0 −1 Y
Bp
r
1+
≤ p(p − β).
(k0 − 1)
j
−
2
j=3
We find the value r0 = r0 (p, β) and the corresponding integer k0 (r0 ) so that
k (k0 + p − β − 1) k0 −1 Y
Bp
r
1+
= p(p − β),
(k0 − 1)
j
−
2
j=3
(0 ≤ β < p).
Then this value r0 is the radius of p−valent convexity of order β for functions
f (z) ∈ U CVp .
Theorem 5.2. h(z) = z p + bn+p−1 z n+p−1 is in U CVp if and only if
p2
r≤
,
(p + n − 1)(p + 2n − 2)
where |bn+p−1 | = r and bn+p−1 z n−1 = reiθ .
00
(z)
Proof. Let w(z) = 1 + zhh0 (z)
. Then h(z) ∈ U CVp if and only if w(z) ∈ P ARp
which means that Re{w(z)} ≥ |w(z) − p| . On the other side we have
00
zh00 (z)
zh
(z)
≥ 1 + 0
− p ,
Re 1 + 0
h (z)
h (z)
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then
zh00 (z)
p + (n + p − 1)nreiθ
Re 1 + 0
= Re (p − 1) +
h (z)
p + (n + p − 1)reiθ
p3 + p(n + p − 1)(n + 2p − 1)r cos θ + (n + p − 1)3 r2
.
=
|p + (n + p − 1)reiθ |2
The right-hand side is seen to have a minimum for θ = π and this minimal value
is
p3 + p(n + p − 1)(n + 2p − 1)r + (n + p − 1)3 r2
.
|p + (n + p − 1)reiθ |2
Now, by computation we see that
00
1 + zh (z) − p = (n + p − 1)(n − 1)r .
|p + (n + p − 1)reiθ |
h0 (z)
Then
H.A. Al-Kharsani and S.S.
Al-Hajiry
Title Page
Contents
p3 + p(n + p − 1)(n + 2p − 1)r + (n + p − 1)3 r2
(n + p − 1)(n − 1)r ≤
,
p − (n + p − 1)r
which leads to
(n + p − 1)(n − 1)r ≤ p2 − (n + p − 1)2 r.
Hence,
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
p2
r≤
.
(n + p − 1)(2n + p − 2)
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Theorem 5.3. Let f (z) ∈ U CV, then (f (z))p ∈ U CVp .
Proof. Let w(z) = (f (z))p , then
1+z
f 00 (z)
f 0 (z)
w00 (z)
=
1
+
z
+
(p
−
1)z
.
w0 (z)
f 0 (z)
f (z)
Then we find
00
w (z)
w00 (z)
Re 1 + z 0
− z 0
− (p − 1)
w (z)
w (z)
f 00 (z)
f 0 (z)
= Re 1 + z 0
+ (p − 1)z
f (z)
f (z)
00
0
f (z)
f
(z)
− z 0
+ (p − 1)z
− (p − 1) .
f (z)
f (z)
Since f (z) ∈ U CV , therefore we have
f 00 (z)
f 0 (z)
+ (p − 1)z
Re 1 + z 0
f (z)
f (z)
00
f (z)
f 0 (z)
+ (p − 1)z
− (p − 1)
− z 0
f (z)
f (z)
0 0
f (z)
f (z)
≥ (p − 1)Re z
− z
− 1
f (z)
f (z)
f (z) ∈ U CV , then f (z) ∈ SP [7] which means that
0 0
f (z)
f (z)
Re z
− z
− 1 ≥ 0.
f (z)
f (z)
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
H.A. Al-Kharsani and S.S.
Al-Hajiry
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Then
00
w (z)
w00 (z)
Re 1 + z 0
− z 0
− (p − 1) ≥ 0.
w (z)
w (z)
The following diagram shows the extermal function k(z) of the class U CV
when (k(z))p , p = 2:
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
H.A. Al-Kharsani and S.S.
Al-Hajiry
Title Page
The following diagram shows the extermal function K(z) of the class U CVp
when p = 2:
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And the following diagram shows that (k(z))p ≺ K(z):
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
H.A. Al-Kharsani and S.S.
Al-Hajiry
5.1.
Remarks
Title Page
Taking p = 1 in Theorem 3.1, we obtain the corresponding Theorem 1 of [7].
Taking p = 1 in Theorem 4.1, we obtain the corresponding Theorem 3 of
[3].
Taking p = 1 in inequality (4.3), we obtain Theorem 6 of [7], and in inequalities (4.5), (4.6), we obtain Theorem 5 of [7].
Taking p = 1 in Theorem 5.2, we obtain Theorem 2 of [4].
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References
[1] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math.,
56(1) (1991), 87–92.
[2] S. KANAS AND A. WISNIOWSKA, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105(1-2) (1999), 327–336.
[3] W.C. MA AND D. MINDA, Uniformly convex functions, Ann. Polon.
Math., 57(2) (1992), 165–175.
[4] S. OWA, On uniformly convex functions, Math. Japonica, 48(3) (1998),
377–384.
[5] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math.,
2(37) (1936), 347–408.
[6] F. RONNING, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 47 (1993), 123–134.
[7] F. RONNING, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118(1) (1993), 189–196.
[8] F. RONNING, On uniform starlikeness and related properties of univalent
functions, Complex Variables Theory Appl., 24(3-4) (1994), 233–239.
Subordination Results for the
Family of Uniformly Convex
p−valent Functions
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Al-Hajiry
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