Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 104, 2005
ON CERTAIN CLASSES OF MULTIVALENT ANALYTIC FUNCTIONS
B.A. FRASIN
D EPARTMENT OF M ATHEMATICS
A L AL -BAYT U NIVERSITY
P.O. B OX : 130095
M AFRAQ , J ORDAN .
bafrasin@yahoo.com
Received 08 September, 2005; accepted 26 September, 2005
Communicated by A. Lupaş
A BSTRACT. In this paper we introduce the class B(p, n, µ, α) of analytic and p-valent functions
to obtain some sufficient conditions and some angular properties for functions belonging to this
class.
Key words and phrases: Analytic and p-valent functions, p-valent starlike functions and p-valent convex functions, Strongly
starlike and strongly convex functions.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION AND D EFINITIONS
Let A(p, n) denote the class of functions f (z) of the form
(1.1)
p
f (z) = z +
∞
X
ak z k ,
(p, n ∈ N := {1, 2, 3, . . .}),
k=p+n
which are analytic and p-valent in the open unit disk U = {z : z ∈ C and |z| < 1}. In
particular, we set A(1, 1) =: A. A function f (z) ∈ A(p, n) is said to be in the class S ∗ (p, n, α)
of p-valently starlike of order α in U if and only if it satisfies the inequality
0 zf (z)
(1.2)
Re
> α,
(z ∈ U; 0 ≤ α < p).
f (z)
On the other hand, a function f (z) ∈ A(p, n) is said to be in the class K(p, n, α) of p-valently
convex of order α in U if and only if it satisfies the inequality
zf 00 (z)
(1.3)
Re 1 + 0
> α,
(z ∈ U; 0 ≤ α < p).
f (z)
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
268-05
2
B.A. F RASIN
Furthermore, a function f (z) ∈ A(p, n) is said to be in the class C(p, n, α) of p-valently closeto-convex of order α in U if and only if it satisfies the inequality
Re
(1.4)
f 0 (z)
z p−1
>α
(z ∈ U; 0 ≤ α < p).
In particular, we write S ∗ (1, 1, 0) =: S ∗ , K(1, 1, 0) =: K and C(1, 1, 0) =: C, where S ∗ ,
K and C are the usual subclasses of A consisting of functions which are starlike, convex and
close-to-convex, respectively.
∗
Let S (p, n, α1 , α2 ) be the subclass of A(p, n) which satisfies
(1.5)
−
πα1
zf 0 (z)
πα2
< arg
<
,
2
f (z)
2
(z ∈ U; 0 < α1 , α2 ≤ p),
and let K(p, n, α1 , α2 ) be the subclass of A(p, n) which satisfies
πα1
zf 00 (z)
πα2
< arg 1 + 0
<
,
(z ∈ U; 0 < α1 , α2 ≤ p).
(1.6)
−
2
f (z)
2
∗
We note that S (1, 1, α1 , α2 ) =: S ∗ (α1 , α2 ), K(1, 1, α1 , α2 ) =: K(α1 , α2 ), where S ∗ (α1 , α2 )
and K(α1 , α2 ) are the subclasses of A introduced and studied by Takahashi and Nunokawa
∗
∗
∗
(α) and K(1, 1, α, α) =: Kst (α) where Sst
(α) and
[7]. Also, we note that S (1, 1, α, α) =: Sst
Kst (α) are the familiar classes of strongly starlike functions of order α and strongly convex
functions of order α, respectively.
The object of the present paper is to investigate various properties of the following classes of
analytic and p-valent functions defined as follows.
Definition 1.1. A function f (z) ∈ A(p, n) is said to be a member of the class B(p, n, µ, α) if
and only if
z p µ−1
1−p 0
(1.7)
z f (z) − p < p − α,
(p ∈ N),
f (z)
for some α (0 ≤ α < p), µ ≥ 0 and for all z ∈ U.
Note that condition (1.7) implies that
!
p µ−1
z
(1.8)
Re
z 1−p f 0 (z) > α.
f (z)
We note that B(p, n, 2, α) ≡ S ∗ (p, n, α) , B(p, n, 1, α) ≡ C(p, n, α). The class B(1, 1, 3, α) ≡
B(α) is the class which has been introduced and studied by Frasin and Darus [3] (see also [1, 2]).
In order to derive our main results, we have to recall the following lemmas.
Lemma 1.1 ([4]). Let w(z) be analytic in U and such that w(0) = 0. Then if |w(z)| attains its
maximum value on circle |z| = r < 1 at a point zo ∈U, we have
zo w0 (z) = kw(zo ),
(1.9)
where
k ≥ 1 is a real number.
Lemma 1.2 ([6]). Let Ω be a set in the complex plane C and suppose that Φ(z) is a mapping
from C2 × U to C which satisfies Φ(ix, y; z) ∈
/ Ω for z ∈ U, and for all real x, y such that
y ≤ −n(1 + u22 )/2. If the function q(z) = 1 + qn z n + qn+1 z n+1 + · · · is analytic in U such that
Φ(q(z), zq 0 (z); z) ∈ Ω for all z ∈ U, then Re q(z) > 0.
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O N C ERTAIN C LASSES OF M ULTIVALENT A NALYTIC F UNCTIONS
3
Lemma 1.3 ([5]). Let q(z) be analytic in U with q(0) = 1 and q(z) 6= 0 for all z ∈ U. If there
exist two points z1 , z2 ∈ U such that
−
(1.10)
πα2
πα1
= arg q(z1 ) < arg q(z) < arg q(z2 ) =
2
2
for α1 > 0, α2 > 0, and for |z| < |z1 | = |z2 | , then we have
z1 q 0 (z1 )
= −i
q(z1 )
(1.11)
α1 + α2
2
z2 q 0 (z2 )
=i
q(z2 )
m
and
α1 + α2
2
m
where
1 − |a|
m≥
1 + |a|
(1.12)
and
π
a = i tan
4
α2 − α1
α1 + α2
.
2. S UFFICIENT C ONDITIONS FOR S TARLIKENESS AND C LOSE - TO - CONVEXITY
Making use of Lemma 1.1, we first prove
Theorem 2.1. If f (z) ∈ A(p, n) satisfies
(2.1)
zf 00 (z)
zf 0 (z)
+ (µ − 1) p −
+γ
1 + 0
f (z)
f (z)
<
zp
f (z)
µ−1
!
1−p 0
z f (z) − p (p − α)(γ(2p − α) + 1)
,
2p − α
(z ∈ U),
for some α (0 ≤ α < p) and µ, γ ≥ 0, then f (z) ∈B(p, n, µ, α).
Proof. Define the function w(z) by
(2.2)
zp
f (z)
µ−1
z 1−p f 0 (z) = p + (p − α)w(z).
Then w(z) is analytic in U and w(0) = 0. It follows from (2.2) that
00
1+
0
zf (z)
zf (z)
− p + (µ − 1) p −
0
f (z)
f (z)
+γ
p
z
f (z)
µ−1
!
z 1−p f 0 (z) − p
= γ(p − α)w(z) +
(p − α)zw0 (z)
.
p + (p − α)w(z)
Suppose that there exists z0 ∈ U such that
(2.3)
max |w(z)| = |w(z0 )| = 1.
|z|<z0
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4
B.A. F RASIN
Then from Lemma 1.1, we have (1.9). Therefore, letting w(z0 ) = eiθ , with k ≥ 1, we obtain
that
!
p µ−1
0
00
z0 f (z0 )
z0
z0 f (z0 )
1−p 0
+ (µ − 1) p −
+γ
z0 f (z0 ) − p 1 + 0
f (z0 )
f (z0 )
f (z0 )
(p − α)zw0 (z0 ) = γ(p − α)w(z0 ) +
p + (p − α)w(z0 ) (p − α)k
≥ Re γ(p − α) +
p + (p − α)w(z0 )
p−α
> γ(p − α) +
2p − α
(p − α)(γ(2p − α) + 1)
=
,
2p − α
which contradicts our assumption (2.1). Therefore we have |w(z)| < 1 in U. Finally, we have
z p µ−1
(2.4)
z 1−p f 0 (z) − p = (p − α)|w(z)| < p − α (z ∈ U),
f (z)
that is, f (z) ∈ B(p, n, µ, α).
Letting µ = 1 in Theorem 2.1, we obtain
Corollary 2.2. If f (z) ∈ A(p, n) satisfies
(p − α)(γ(2p − α) + 1)
zf 00 (z)
1−p 0
<
(2.5)
1
+
+
γ(z
f
(z)
−
p)
, (z ∈ U),
f 0 (z)
2p − α
for some α (0 ≤ α < p) and γ ≥ 0,then f (z) ∈ C(p, n, α).
Letting p = n = 1 and γ = α = 0 in Corollary 2.2, we easily obtain
Corollary 2.3. If f (z) ∈A satisfies
zf 00 (z) 1
(2.6)
1 + f 0 (z) < 2 ,
(z ∈ U),
then f (z) ∈ C.
Letting µ = 2 and γ = 1 in Theorem 2.1, we obtain
Corollary 2.4. If f (z) ∈ A(p, n) satisfies
00
(p − α)(2p − α + 1)
zf
(z)
1 +
<
(2.7)
f 0 (z) 2p − α
(0 ≤ α < p; z ∈ U),
then f (z) ∈ S ∗ (p, n, α).
Letting p = n = 1 and α = 0 in Corollary 2.4, we easily obtain
Corollary 2.5. If f (z) ∈ A satisfies
00
3
zf
(z)
1 +
<
(2.8)
f 0 (z) 2
(z ∈ U),
then f (z) ∈ S ∗ .
Next we prove
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O N C ERTAIN C LASSES OF M ULTIVALENT A NALYTIC F UNCTIONS
5
Theorem 2.6. If f (z) ∈ A(p, n) satisfies
#
"
µ−1
zp
(2.9) Re
z 1−p f 0 (z)
f (z)
)
(
µ−1
00
0
zf
(z)
zp
zf
(z)
×
z 1−p f 0 (z) + 1 + 0
− (µ − 1)
f (z)
f (z)
f (z)
n
n
>δ δ+
+ p δ(2 − µ) −
,
2
2
then f (z) ∈ B(p, n, µ, δ), where 0 ≤ δ < p.
Proof. Define the function q(z) by
p µ−1
z
z 1−p f 0 (z) = δ + (p − δ)q(z).
(2.10)
f (z)
Then, we see that q(z) = 1 + qn z n + qn+1 z n+1 + · · · is analytic in U. A computation shows that
#2 "
µ−1
µ−1
zp
zf 00 (z)
zf 0 (z)
zp
1−p 0
1−p 0
z f (z) +
z f (z) 1 + 0
− (µ − 1)
f (z)
f (z)
f (z)
f (z)
= (p − δ)zq 0 (z) + (p − δ)2 q 2 (z) + (p − δ)[p(2 − µ) + 2δ]q(z) + δp(2 − µ) + δ 2
= Φ(q(z), zq 0 (z); z),
where
Φ(r, s; t) = (p − δ)s + (p − δ)2 r2 + (p − δ)[p(2 − µ) + 2δ]r + δp(2 − µ) + δ 2 .
For all real x, y satisfying y ≤ −n(1 + x22 )/2, we have
Re Φ(ix, y; z) = (p − δ)y − (p − δ)2 x2 + δp(2 − µ) + δ 2
hn
i
n
≤ − (p − δ) − (p − δ)
+ p − δ x2 + δp(2 − µ) + δ 2
2
2
n
2
≤ δp(2 − µ) + δ − (p − δ)
2
n
n
=δ δ+
+ p δ(2 − µ) −
.
2
2
Let
n
n
n o
Ω = w : Re w > δ δ +
+ p δ(2 − µ) −
.
2
2
Then Φ(q(z), zq 0 (z); z) ∈ Ω and Φ(ix, y; z) ∈
/ Ω for all real x and y ≤ −n(1 + x22 )/2, z ∈ U.
By using Lemma 1.2, we have Re q(z) > 0, that is, f (z) ∈ B(p, n, µ, δ).
Letting µ = 1 in Theorem 2.6, we have the following:
Corollary 2.7. If f (z) ∈ A(p, n) satisfies
n
n
(2.11)
Re (z 1−p f 0 (z))2 + z 1−p f 0 (z) + z 2−p f 00 (z) > δ δ +
+p δ−
,
2
2
then f (z) ∈ C(p, n, δ),where 0 ≤ δ < p.
Letting p = n = 1 and δ = 0 in Corollary 2.7, we easily get
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6
B.A. F RASIN
Corollary 2.8. If f (z) ∈ A satisfies
1
(2.12)
Re (f 0 (z))2 + f 0 (z) + zf 00 (z) > − ,
2
then f (z) ∈ C.
Letting µ = 2 in Theorem 2.6, we have
Corollary 2.9. If f (z) ∈ A(p, n) satisfies
0
zf (z) z 2 f 00 (z)
n n
Re
+
>δ δ+
− p.
f (z)
f (z)
2
2
then f (z) ∈ S ∗ (p, n, δ),where 0 ≤ δ < p.
Letting p = n = 1 and δ = 0 in Corollary 2.9, we easily get
Corollary 2.10. If f (z) ∈ A satisfies
0
zf (z) z 2 f 00 (z)
1
Re
+
>− .
f (z)
f (z)
2
then f (z) ∈ S ∗ .
3. A RGUMENT P ROPERTIES
µ−1
zp
z 1−p f 0 (z) 6= δ for z ∈ U and 0 ≤ δ < p. If f (z) ∈
f (z)
Theorem 3.1. Suppose that
A(p, n) satisfies
1 − |a| (α1 + α2 )(p − δ)
π
−1
− α1 − tan
2
1 + |a|
2γ
( !
µ−1
zp
< arg
z 1−p f 0 (z)
f (z)
zf 0 (z)
γ
γδ
zf 00 (z)
− p + (µ − 1) p −
+
−
× 1+ 0
f (z)
f (z)
p−δ
p−δ
π
1 − |a| (α1 + α2 )(p − δ)
< α2 + tan−1
(3.1)
2
1 + |a|
2γ
for α1 , α2 , γ > 0, then
!
p µ−1
π
z
π
(3.2)
− α1 < arg
z 1−p f 0 (z) − δ < α2 .
2
f (z)
2
Proof. Define the function q(z) by
(3.3)
1
q(z) =
p−δ
zp
f (z)
µ−1
!
z 1−p f 0 (z) − δ .
Then we see that q(z) is analytic in U, q(0) = 1, and q(z) 6= 0 for all z ∈ U. It follows from
(3.3) that
!
p µ−1
z
zf 00 (z)
zf 0 (z)
γ
γδ
1−p 0
z f (z)
1+ 0
− p + (µ − 1) p −
+
−
f (z)
f (z)
f (z)
p−δ
p−δ
= (p − δ)zq 0 (z) + γq(z).
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O N C ERTAIN C LASSES OF M ULTIVALENT A NALYTIC F UNCTIONS
7
Suppose that there exists two points z1 , z2 ∈ U such that the condition (1.10) is satisfied, then
by Lemma 1.3, we obtain (1.11) under the constraint (1.12). Therefore, we have
z1 q 0 (z1 )
0
arg(γq(z1 ) + (p − δ)zq (z1 )) = arg q(z1 ) + arg γ + (p − δ)
q(z1 )
π
(α1 + α2 )(p − δ)
= − α1 + arg γ − i
m
2
2
(α1 + α2 )(p − δ)
π
−1
= − α1 − tan
m
2
2γ
1 − |a| (α1 + α2 )(p − δ)
π
−1
≤ α1 − tan
2
1 + |a|
2γ
and
1 − |a| (α1 + α2 )(p − δ)
π
0
−1
,
arg(γq(z2 ) + (p − δ)zq (z2 )) ≥ α2 + tan
2
1 + |a|
2γ
which contradict the assumptions of the theorem. This completes the proof.
Letting µ = 1 in Theorem 3.1, we have
Corollary 3.2. Suppose that z 1−p f 0 (z) 6= δ for z ∈ U and 0 ≤ δ < p. If f (z) ∈ A(p, n)
satisfies
1 − |a| (α1 + α2 )(p − δ)
π
−1
− α1 − tan
2
1 + |a|
2γ
zf 00 (z)
γ
γδ
1−p 0
−p+
−
< arg z f (z) 1 + 0
f (z)
p−δ
p−δ
π
1 − |a| (α1 + α2 )(p − δ)
(3.4)
< α2 + tan−1
2
1 + |a|
2γ
for α1 , α2 , γ > 0, then
π
π
(3.5)
− α1 < arg z 1−p f 0 (z) − δ < α2 .
2
2
Letting α1 = α2 = 1 in Corollary 3.2, we have
Corollary 3.3. Suppose that z 1−p f 0 (z) 6= δ for z ∈ U and 0 ≤ δ < p. If f (z) ∈ A(p, n)
satisfies
00
π
zf
(z)
γ
γδ
p
−
δ
1−p
0
−1
arg z f (z) 1 +
< + tan
(3.6)
−p+
−
f 0 (z)
p−δ
p−δ 2
γ
for γ > 0, then f (z) ∈ C(p, n, δ).
Letting µ = 2 in Theorem 3.1, we have
Corollary 3.4. Suppose that zf 0 (z)/f (z) 6= δ for z ∈ U and 0 ≤ δ < p. If f (z) ∈ A(p, n)
satisfies
π
1 − |a| (α1 + α2 )(p − δ)
−1
− α1 − tan
2
1 + |a|
2γ
0
zf (z)
zf 00 (z) zf 0 (z)
γ
γδ
< arg
1+ 0
−
+
−
f (z)
f (z)
f (z)
p−δ
p−δ
π
1 − |a| (α1 + α2 )(p − δ)
(3.7)
< α2 + tan−1
2
1 + |a|
2γ
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8
B.A. F RASIN
for α1 , α2 , γ > 0, then
π
− α1 < arg
2
(3.8)
zf 0 (z)
−δ
f (z)
<
π
α2
2
Letting α1 = α2 = 1 in Corollary 3.4, we have
Corollary 3.5. Suppose that zf 0 (z)/f (z) 6= δ for z ∈ U and 0 ≤ δ < p. If f (z) ∈ A(p, n)
satisfies
0
zf 00 (z) zf 0 (z)
γ
γδ π
p−δ
zf (z)
−1
1+ 0
−
+
−
< + tan
(3.9) arg
f (z)
f (z)
f (z)
p−δ
p−δ 2
γ
for γ > 0, then f (z) ∈ S ∗ (p, n, δ).
Letting α1 = α2 , µ = p = n = 1 and δ = 0 in Theorem 3.1, we have
Corollary 3.6. If f (z) ∈ A satisfies
z(f 0 (z))2 π
−1 α
00
0
(3.10)
< 2 α + tan γ
arg zf (z) + f (z)(γ + 1) − f (z)
for γ > 0 , then
π
α,
(0 < α ≤ 1).
2
Taking α1 = α2 , p = n = 1, µ = 2 and δ = 0 in Theorem 3.1, we obtain
|arg f 0 (z)| <
(3.11)
Corollary 3.7. If f (z) ∈ A satisfies
!
0 2
π
0
2 00
zf
(z)
zf
(z)
α
z
f
(z)
(3.12)
−
+
(γ + 1) < α + tan−1
arg
2
f (z)
f (z)
f (z)
γ
∗
for γ > 0, then f (z) ∈ Sst
(α).
Finally, we prove
Theorem 3.8. Let q(z) analytic in U with q(0) = 1, and q(z) 6= 0. If


"
#−1
µ−1

 π
p
π
z
z 1−p f 0 (z)
zq 0 (z) < η2
(3.13)
− η1 < arg q(z) +

 2
2
f (z)
for some f (z) ∈ B(p, n, µ, α) and (0 < η1 , η2 ≤ 1) then
π
π
(3.14)
− α1 < arg q(z) < α2
2
2
where α1 and α2 (0 < α1 , α2 ≤ 1) are the solutions of the following equations:
h
i

−1 p−α
π
2
(α
+
α
)m
sin
(1
−
sin
)
1
2
2
π
p
2
h
(3.15)
η1 = α1 + tan−1 
π
2(2p − α) + (α + α )m sin π (1 − 2 sin−1
1
2
2
π
2
π
sin−1 p−α
)
p

i
p−α
)
p
and

(3.16)
h
π
(1
2
−
2
h
η2 = α2 + tan−1 
π
2(2p − α) + (α1 + α2 )m sin π2 (1 −
J. Inequal. Pure and Appl. Math., 6(4) Art. 104, 2005
(α1 + α2 )m sin
i
2
π

i
sin−1 p−α
)
p
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O N C ERTAIN C LASSES OF M ULTIVALENT A NALYTIC F UNCTIONS
9
Proof. Suppose that there exists two points z1 , z2 ∈ U such that the condition (1.10) is satisfied,
then by Lemma 1.3, we obtain (1.11) under the constraint (1.12). Since f ∈ B(p, n, µ, α), we
have
p µ−1
z
iπφ
1−p 0
(3.17)
z f (z) = ρ exp
,
f (z)
2
where



(3.18)
α < ρ < 2p − α
2
p−α
2
p−α

 − sin−1
< φ < sin−1
.
π
p
π
p
Thus, we obtain


"
#−1
µ−1


p
z
arg q(z1 ) +
z 1−p f 0 (z)
zq 0 (z1 )


f (z)


#−1
"
µ−1
p
0
z
z1 q (z1 ) 
= arg q(z1 ) + arg 1 +
z 1−p f 0 (z)
f (z)
q(z1 )
−1 !
π
α1 + α2
iπφ
= − α1 + arg 1 − i
m ρ exp
2
2
2
π
!
(1
−
φ)
(α
+
α
)m
sin
π
1
2
2
≤ − α1 − tan−1
2
2ρ + (α1 + α2 )m sin π2 (1 − φ)
h
i

−1 p−α
π
2
(α
+
α
)m
sin
(1
−
sin
)
1
2
2
π
p
π
h
≤ − α1 − tan−1 
2
2(2p − α) + (α + α )m sin π (1 − 2 sin−1
1
2
2
π

i
p−α
)
p
π
= − η1
2
and
"
arg(q(z1 )+
zp
f (z)
µ−1
#−1
z 1−p f 0 (z)
zq 0 (z1 ))
h
i

−
π
h
i
≥ α2 + tan−1 
−1 p−α
π
2
2
2(2p − α) + (α1 + α2 )m sin 2 (1 − π sin
)
p
π
= η2 ,
2
where η1 and η2 being given by (3.15) and (3.16), respectively, which contradicts the assumption
(3.13). This completes the proof of Theorem 3.8.

(α1 + α2 )m sin
π
(1
2
2
π
sin−1 p−α
)
p
Letting q(z) = zf 0 (z)/f (z) in Theorem 3.8, we have
Corollary 3.9. Let 0 < η1 , η2 ≤ 1. If


"
#−1
µ−1

 π
p
π
z
(3.19)
− η1 < arg q(z) +
z 1−p f 0 (z)
zq 0 (z) < η2

 2
2
f (z)
J. Inequal. Pure and Appl. Math., 6(4) Art. 104, 2005
http://jipam.vu.edu.au/
10
B.A. F RASIN
∗
for some f (z) ∈ B(p, n, µ, α) then f (z) ∈ S (p, n, α1 , α2 ), where 0 < α1 , α2 ≤ 1.
Letting η1 = η2 in Corollary 3.9, we have
Corollary 3.10. Let 0 < η1 ≤ 1. If


"
#−1 µ−1
0 
 zf 0 (z)
p
0
π
zf (z)
z
1−p 0
arg
< η1
(3.20)
z
f
(z)
z
+
2


f
(z)
f
(z)
f
(z)
for some f (z) ∈ B(p, n, µ, α), then
0 arg zf (z) < π α1
(3.21)
f (z) 2
(0 < α1 ≤ 1),
∗
that is, f (z) ∈ Sst
(α1 ), where α1 is the solutions of the following equation:
i
h


−1 p−α
π
2
2α
m
sin
(1
−
sin
)
1
2
π
p
2
i .
h
(3.22)
η = α1 + tan−1 
−1 p−α
π
2
π
2(2p − α) + 2α m sin (1 − sin
)
1
2
π
p
Letting q(z) = q(z) = 1 + (zf 00 (z)/f 0 (z) in Theorem 3.8, we have
Corollary 3.11. Let 0 < η1 , η2 ≤ 1. If


"
#−1 µ−1
0 

p
00
00
zf (z)
z
zf (z)
π
(3.23)
− η1 < arg 1 + 0
+
z 1−p f 0 (z)
z 1+ 0

2
f (z)
f (z)
f (z) 
<
π
η1
2
for some f (z) ∈ B(p, n, µ, α) then f (z) ∈ K(p, n, α1 , α2 ), where 0 < α1 , α2 ≤ 1.
Letting η1 = η2 in Corollary 3.11, we have
Corollary 3.12. Let 0 < η1 ≤ 1. If


"
#−1 µ−1
0 

00
p
00
π
z
zf (z)
1−p 0
arg 1 + zf (z) +
< η1
(3.24)
z
f
(z)
z
1
+
2
0 (z)
0 (z)


f
f
(z)
f
for some f (z) ∈ B(p, n, µ, α) then
00
π
zf
(z)
arg 1 +
< α1
(3.25)
f 0 (z) 2
(0 < α1 ≤ 1),
that is, f (z) ∈ Kst (α1 ), where α1 is the solution of the following equation:
h i


−1 p−α
π
2
2α
m
sin
1
−
sin
1
2
π
p
2
h
i .
(3.26)
η1 = α1 + tan−1 
π
2(2p − α) + 2α m sin π (1 − 2 sin−1 p−α )
1
J. Inequal. Pure and Appl. Math., 6(4) Art. 104, 2005
2
π
p
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O N C ERTAIN C LASSES OF M ULTIVALENT A NALYTIC F UNCTIONS
11
R EFERENCES
[1] B.A. FRASIN, A note on certain analytic and univalent functions, Southeast Asian J. Math., 28
(2004), 829–836.
[2] B.A. FRASIN
ted.
AND
J. JAHANGIRI, A new and comprehensive class of analytic functions, submit-
[3] B.A. FRASIN AND M. DARUS, On certain analytic univalent function, Int. J. Math. and Math. Sci.,
25(5) (2001), 305–310.
[4] I.S. JACK, Functions starlike and convex of order α, J. London Math. Soc. (2)3 (1971), 469–474.
[5] M. NUNOKAWA, S. OWA, H. SAITOH, N.E. CHO AND N. TAKAHASHI, Some properties of
analytic functions at extermal points for arguments, preprint.
[6] S.S. MILLER AND P.T. MOCANU, Differential subordinations and inequalities in the complex
plane, J. Differ. Equations, 67 (1987), 199–211.
[7] N. TAKAHASHI AND M. NUNOKAWA, A certain connections between starlike and convex functions, Appl. Math. Lett., 16 (2003), 563–655.
J. Inequal. Pure and Appl. Math., 6(4) Art. 104, 2005
http://jipam.vu.edu.au/