IJMMS 31:11 (2002) 659–673
PII. S0161171202112300
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT
FUNCTIONS INVOLVING A LINEAR OPERATOR
NAK EUN CHO, J. PATEL, and G. P. MOHAPATRA
Received 28 December 2001
The purpose of this paper is to derive some argument properties of certain multivalent
functions in the open unit disk involving a linear operator. We also investigate their integral
preserving property in a sector.
2OOO Mathematics Subject Classification: 30C45.
1. Introduction. Let Ꮽp denote the class of functions of the form
f (z) = zp +
∞
an+p zn+p
p ∈ N = {1, 2, . . .}
(1.1)
n=1
which are analytic in the open unit disk ᐁ = {z : |z| < 1}. A function f ∈ Ꮽp is said to
be p-valently starlike of order α in ᐁ, if it satisfies
zf (z)
> α (0 ≤ α < p; z ∈ ᐁ).
Re
(1.2)
f (z)
We denote this class by ᏿p∗ (α). A function f ∈ Ꮽp is said to be p-valently convex of
order α in ᐁ, if it satisfies
zf (z)
(1.3)
Re 1 + > α (0 ≤ α < p; z ∈ ᐁ).
f (z)
The class of p-valently convex functions of order α is denoted by ᏷p (α). It follows
from (1.2) and (1.3) that
f ∈ ᏷p (α) ⇐⇒
zf ∈ ᏿p (α).
p
(1.4)
Further, a function f ∈ Ꮽp is said to be p-valently close-to-convex of order β and type
α, if there exists a function g ∈ ᏿p∗ (α) such that
zf (z)
(1.5)
Re
> β (0 ≤ α, β < p; z ∈ ᐁ).
g(z)
It is well known (see [10]) that every p-valently close-to-convex function is p-valent
in ᐁ.
For arbitrary fixed real numbers A and B (−1 ≤ B < A ≤ 1), let ᏼ(A, B) denote the
class of functions of the form
φ(z) = 1 + c1 z + c2 z2 + · · ·
(1.6)
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NAK EUN CHO ET AL.
which are analytic in ᐁ and satisfies the condition
φ(z) ≺
1 + Az
1 + Bz
(z ∈ ᐁ),
(1.7)
where the symbol ≺ stands for subordination. The class ᏼ(A, B) was introduced and
studied by Janowski [8].
We note that a function φ ∈ ᏼ(A, B), if and only if
φ(z) − 1 − AB < A − B
1 − B2 1 − B2
1−A
Re φ(z) >
2
(B ≠ −1, z ∈ ᐁ),
(1.8)
(B = −1, z ∈ ᐁ).
For a function f ∈ Ꮽ, given by (1.1), the generalized Bernardi-Libera-Livingston
integral operator F [1] is defined by
γ + p z γ−1
t
f (t) dt
zγ
0
∞
γ +p
an+p zn+p
= zp +
γ
+
p +n
n=1
F (z) =
(1.9)
(γ > −p; z ∈ ᐁ).
It readily follows from (1.9) that
f ∈ Ꮽp ⇒ F ∈ Ꮽp .
(1.10)
Let
φp (a, c; z) =
∞
(a)n n+p
z
(c)n
n=0
(c ≠ 0, −1, −2, . . . ; z ∈ ᐁ),
(1.11)
and we define a linear operator Lp (a, c) on Ꮽp by
Lp (a, c)f (z) = φp (a, c; z) ∗ f (z)
(z ∈ ᐁ),
(1.12)
where (x)n = Γ (n + x)/Γ (x) and the symbol ∗ is the Hadamard product or convolution. Clearly, Lp (a, c) maps Ꮽp into itself. Further, Lp (a, a) is the identity operator
and
Lp (a, c) = Lp (a, b)Lp (b, c) = Lp (b, c)Lp (a, b)
(b, c ≠ 0, −1, −2, . . .).
(1.13)
Thus, if a ≠ 0, −1, −2, . . . , then Lp (a, c) has an inverse Lp (c, a). We also observe that
for f ∈ Ꮽp ,
Lp (p + 1, p)f (z) =
zf (z)
,
p
Lp (µ + p, 1)f (z) = D µ+p−1 f (z),
(1.14)
where µ (µ > −p) is any real number. In case of p = 1 and µ ∈ N, D µ f (z) is the
Ruscheweyh derivative [14]. The operator Lp (a, c) was introduced and studied by
Saitoh and Nunokawa [15]. This operator is a generalization of the linear operator
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ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
L(a, c) introduced by Carlson and Shaffer [3] in their systemic investigation of certain
classes of starlike, convex, and prestarlike hypergeometric functions.
In the present paper, we give some argument properties of certain class of analytic
functions in Ꮽp involving the linear operator Lp (a, c). An application of a certain
integral operator is also considered. The results obtained here, besides extending the
works of Bulboacă [2], Chichra [4], Cho et al. [5], Fukui et al. [6], Libera [9], Nunokawa
[13], and Sakaguchi [16], it yields a number of new results.
2. Main results. To establish our main results, we need the following lemmas.
Lemma 2.1 [11]. Let h(z) be convex (univalent) in ᐁ and let ψ(z) be analytic in ᐁ
with Re{ψ(z)} ≥ 0. If φ(z) is analytic in ᐁ and φ(0) = ψ(0), then
φ(z) + ψ(z)zφ (z) ≺ h(z) (z ∈ ᐁ)
(2.1)
φ(z) ≺ h(z) (z ∈ ᐁ).
(2.2)
implies
Lemma 2.2 [12]. Let φ(z) be analytic in ᐁ, φ(0) = 1, φ(z) ≠ 0 in ᐁ and suppose
that there exists a point z0 ∈ ᐁ such that
arg φ(z) < π η |z| < z0 ,
2
(2.3)
π
arg φ z0 = η,
2
where η > 0. Then
where
z 0 φ z 0
= ikη,
φ z0
(2.4)
π
1
1
d+
when arg φ z0 = η,
2
d
2
1
1
π
k≤−
d+
when arg φ z0 = − η,
2
d
2
(2.5)
k≥
where
1/η
= ±id
φ z0
(d > 0).
(2.6)
We now derive the following theorem.
Theorem 2.3. Let a > 0, −1 ≤ B < A ≤ 1, f ∈ Ꮽp , and suppose that g ∈ Ꮽp satisfies
Lp (a + 1, c)g(z)
1 + Az
≺
Lp (a, c)g(z)
1 + Bz
If
(z ∈ ᐁ).
Lp (a + 1, c)f (z)
Lp (a, c)f (z)
arg (1 − λ)
+λ
−β Lp (a, c)g(z)
Lp (a + 1, c)g(z)
π
< δ
2
(λ ≥ 0; 0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
(2.7)
(2.8)
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NAK EUN CHO ET AL.
then
π
Lp (a, c)f (z)
arg
−β < η
Lp (a, c)g(z)
2
(z ∈ ᐁ),
(2.9)
where η (0 < η ≤ 1) is the solution of the equation


λη sin(π /2) 1 − t(A, B)

η + 2 tan−1
, for B ≠ −1,
π
a(1 + A)/(1 + B) + λη cos(π /2) 1 − t(A, B)
δ=


η,
for B = −1,
(2.10)
when
t(A, B) =
2
A−B
sin−1
.
π
1 − AB
(2.11)
Proof. Let
Lp (a, c)f (z)
= β + (1 − β)φ(z).
Lp (a, c)g(z)
(2.12)
Then φ(z) is analytic in ᐁ with φ(0) = 1. On differentiating both sides of (2.12) and
using the identity
z Lp (a, c)f (z) = aLp (a + 1, c)f (z) − (a − p)Lp (a, c)f (z)
(2.13)
in the resulting equation, we deduce that
(1 − λ)
Lp (a + 1, c)f (z)
Lp (a, c)f (z)
λzφ (z)
+λ
− β = (1 − β) φ(z) +
,
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
ar (z)
(2.14)
where
r (z) =
Lp (a + 1, c)g(z)
.
Lp (a, c)g(z)
(2.15)
If we let
r (z) = ρe(π θ/2)i ,
(2.16)
then from (2.7) followed by (1.8), it follows that
1−A
1+A
<ρ<
,
1−B
1+B
−t(A, B) < θ < t(A, B)
(2.17)
for B ≠ −1,
when t(A, B) is given by (2.11), and
1−A
< ρ < ∞,
2
−1 < θ < 1
for B = −1.
(2.18)
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
663
Let h(z) be the function which maps onto the angular domain {w : | arg{w}| < (π /2)δ}
with h(0) = 1. Applying Lemma 2.1 for this h(z) with ψ(z) = λ/(ar (z)), we see that
Re φ(z) > 0 in ᐁ and hence φ(z) ≠ 0 in ᐁ.
If there exists a point z0 ∈ ᐁ such that conditions (2.3) are satisfied, then by
Lemma 2.2 we obtain (2.4) under restrictions (2.5) and (2.6).
At first, suppose that p(z0 )1/η = id (d > 0). For the case B ≠ −1, we obtain
arg (1 − λ)
Lp (a, c)f z0
Lp (a + 1, c)f z0
+λ
−β
Lp (a, c)g z0
Lp (a + 1, c)g z0
z φ z
λ
0 0
= arg φ z0 + arg 1 +
ar z0
φ z0
e−(π θ/2)i
π
= η + arg 1 + iηkλ
2
ρa
ληk sin(π /2)(1 − θ)
π
−1
= η + tan
2
ρa + ληk cos(π /2)(1 − θ)
π
λη sin(π /2) 1 − t(A, B)
−1
≥ η + tan
2
a(1 + A)/(1 + B) + λη cos(π /2) 1 − t(A, B)
≥
(2.19)
π
δ,
2
where δ and t(A, B) are given by (2.10) and (2.11), respectively. Similarly, for the case
B = −1, we have
Lp (a + 1, c)f z0
Lp (a, c)f z0
π
+λ
− β ≥ η.
arg (1 − λ)
Lp (a, c)g z0
Lp (a + 1, c)g z0
2
(2.20)
This is a contradiction to the assumption of our theorem.
Next, suppose that φ(z0 )1/η = −id (d > 0). For the case B ≠ −1, applying the same
method as above, we have
Lp (a, c)f z0
Lp (a + 1, c)f z0
+λ
−β
Lp (a, c)g z0
Lp (a + 1, c)g z0
λη sin(π /2) 1 − t(A, B)
π
−1
≤ − η − tan
2
a(1 + A)/(1 + B) + λη cos(π /2) 1 − t(A, B)
π
≤ − δ,
2
arg (1 − λ)
(2.21)
where δ and t(A, B) are given by (2.10) and (2.11), respectively and for the case B = −1,
we have
L (a + 1, c)f z0
Lp (a, c)f z0
π
+λ p
−β ≤ − η
(2.22)
arg (1 − λ)
Lp (a, c)g z0
Lp (a + 1, c)g z0
2
which contradicts the assumption. Therefore we complete the proof of the theorem.
664
NAK EUN CHO ET AL.
Remark 2.4. For a = c = p, A = 1, B = −1, and λ = 1, Theorem 2.3 is the recent
result obtained by Nunokawa [13].
Taking a = µ + p (µ > −p), c = 1, A = 1, and B = 0 in Theorem 2.3, we have the
following corollary.
Corollary 2.5. If f ∈ Ꮽp satisfies
D µ+p f (z)
D µ+p−1 f (z)
arg (1 − λ) µ+p−1
+ λ µ+p
−β D
g(z)
D
g(z)
π
< δ
2
(2.23)
(λ ≥ 0; 0 < δ ≤ 1; 0 ≤ β < 1; z ∈ ᐁ)
for some g ∈ Ꮽp satisfying the condition
D µ+p g(z)
D µ+p−1 g(z) − 1 < α
(0 < α ≤ 1; z ∈ ᐁ),
(2.24)
then
D µ+p−1 f (z) π
< η (z ∈ ᐁ),
arg
D µ+p−1 g(z) 2
(2.25)
where η (0 < η ≤ 1) is the solution of the equation
2
δ = η + tan−1
π
λη sin π /2 − sin−1 α
.
(µ + p)(1 + α) + λη cos π /2 − sin−1 α
(2.26)
Letting B → A (A < 1) and g(z) = zp in Theorem 2.3, we get the following corollary.
Corollary 2.6. If f ∈ Ꮽp satisfies
Lp (a + 1, c)f (z)
Lp (a, c)f (z)
arg (1 − λ)
+
λ
−
β
p
p
z
z
π
< δ
2
(2.27)
(a > 0; λ ≥ 0; 0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
then
π
Lp (a, c)f (z)
arg
< η
−
β
p
z
2
(z ∈ ᐁ),
(2.28)
where η (0 < η ≤ 1) is the solution of the equation
δ = η+
2
λη
tan−1
.
π
a
(2.29)
Corollary 2.7. Under the hypothesis of Corollary 2.6, we have
arg H (z) − β < π η (z ∈ ᐁ),
2
(2.30)
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
where the function H(z) is defined in ᐁ by
z
Lp (a, c)f (t)
dt
H(z) =
tp
0
665
(2.31)
and η (0 < η ≤ 1) is the solution of (2.29).
Remark 2.8. Taking a = c = p, λ = 1, and β = 0 in Corollary 2.6, a = c = p and
β = 0 in Corollary 2.7, we get the corresponding results obtained by Cho et al. [5].
Setting A = 1 − (2α/p) (0 ≤ α < p), B = −1, and δ = 1 in Theorem 2.3, we have the
following corollary.
Corollary 2.9. Let a > 0, f ∈ Ꮽp , and g ∈ ᏿p∗ (α). If
Lp (a, c)f (z)
Lp (a + 1, c)f (z)
Re (1 − λ)
+λ
> β (λ ≥ 0; 0 ≤ β < 1; z ∈ ᐁ), (2.32)
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
then
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
>β
(z ∈ ᐁ).
(2.33)
Remark 2.10. For a = c = p = 1 and α = 0, Corollary 2.9 is the result by Bulboacă
[2]. If we put a = c = p = 1, β = 0, and g(z) = z in Corollary 2.9, then we have the result
due to Chichra [4]. Further, taking a = c = p, λ = 1, and α = β = 0 in Corollary 2.9, we
get the corresponding results of Libera [9] and Sakaguchi [16].
Theorem 2.11. If f ∈ Ꮽp satisfies
π
Lp (a, c)f (z)
< δ (0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
arg
−
β
zp
2
then
z
π
(γ + p) 0 t γ−1 Lp (a, c)f (t) dt
arg
< η
−
β
zγ+p
2
(0 < γ + p; z ∈ ᐁ),
where η (0 < η ≤ 1) is the solution of the equation
η
2
−1
δ = η + tan
.
π
γ +p
Proof. Consider the function φ(z) defined in ᐁ by
z
(γ + p) 0 t γ−1 Lp (a, c)f (t) dt
= β + (1 − β)φ(z).
zγ+p
(2.34)
(2.35)
(2.36)
(2.37)
Then φ(z) is analytic in ᐁ with φ(0) = 1. Differentiating both sides of (2.37) and
simplifying, we get
Lp (a, c)f (z)
zφ (z)
−
β
=
(1
−
β)
φ(z)
+
.
(2.38)
zp
γ +p
Now, by using Lemma 2.1 and a similar method in the proof of Theorem 2.3, we get
(2.35).
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NAK EUN CHO ET AL.
Taking a = p + 1, c = p, β = ρ/p, and δ = 1 in Theorem 2.11, we have the following
corollary.
Corollary 2.12. If f ∈ Ꮽp satisfies
Re
f (z)
zp−1
>ρ
(0 ≤ ρ < p; z ∈ ᐁ),
(2.39)
then
z
π
(γ + p) 0 t γ−1 f (t) dt
< η (z ∈ ᐁ),
arg
−
ρ
zγ+p
2
where η (0 < η ≤ 1) is the solution of the equation
η
2
η + tan−1
= 1.
π
γ +p
Theorem 2.13. If f ∈ Ꮽp satisfies
Lp (a + 1, c)f (z) a − p − γ π
arg
< δ
−
Lp (a, c)f (z)
a
2
(2.40)
(2.41)
(a > 0; p + γ > 0; 0 < δ ≤ 1; z ∈ ᐁ),
(2.42)
then
π
zγ Lp (a, c)f (z)
< η
arg z
γ−1
2
t
L
(a,
c)f
(t)
dt
p
0
(z ∈ ᐁ),
(2.43)
where η (0 < η ≤ 1) is the solution of (2.36).
Proof. Our proof of Theorem 2.13 is much akin to that of Theorem 2.3. Indeed,
in place of (2.37), we define the function φ(z) by
φ(z) =
zγ Lp (a, c)f (z)
z
(γ + p) 0 t γ−1 Lp (a, c)f (t) dt
(z ∈ ᐁ),
(2.44)
and apply Lemma 2.1 (with ψ(z) = 1/(γ +p)) as before. We choose to skip the details
involved.
Setting a = c = p and δ = 1 in Theorem 2.13, we obtain the following corollary.
Corollary 2.14. If f ∈ Ꮽp satisfies
Re
zf (z)
f (z)
> −γ
(γ + p > 0; z ∈ ᐁ),
(2.45)
then
π
zγ f (z)
< η
arg z
γ−1
2
f (t) dt 0 t
(z ∈ ᐁ),
(2.46)
where η (0 < η ≤ 1) is the solution of (2.41).
Replacing f (z) by zf (z)/p in Corollary 2.14, we deduce the following corollary.
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
667
Corollary 2.15. If f ∈ Ꮽp satisfies
zf (z)
Re 1 + f (z)
> −γ
(γ + p > 0; z ∈ ᐁ),
(2.47)
then
π
zf (z)
z
< η (z ∈ ᐁ),
arg
2
f (z) − (γ/zγ ) 0 t γ−1 f (t) dt (2.48)
where η (0 < η ≤ 1) is the solution of (2.41).
By setting γ = 0 in Corollary 2.15, we have the following corollary.
Corollary 2.16. If f ∈ ᏷p (0), then
zf (z) π
< η (z ∈ ᐁ),
arg
f (z) 2
(2.49)
where η (0 < η ≤ 1) is the solution of the equation:
η+
2
η
tan−1
= 1.
π
p
(2.50)
Similarly, we have the following theorem.
Theorem 2.17. If f ∈ Ꮽp satisfies
π
Lp (a + 1, c)f (z)
arg
−β < δ
Lp (a, c)f (z)
2
(a > 0; 0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
(2.51)
then
Lp (a, c)f (z) π
< η (z ∈ ᐁ),
arg
zp
2
(2.52)
where η (0 < η ≤ 1) is the solution of the equation
δ=
η
2
tan−1
.
π
(1 − β)a
(2.53)
Theorem 2.18. Let f ∈ Ꮽp and suppose that
p(1 − B)
a
(a > 0; −1 ≤ B < A ≤ 1).
(2.54)
Lp (a + 1, c)f (z)
Lp (a + 1, c)f (z)
arg (1 − λ)
+λ − β Lp (a, c)g(z)
Lp (a, c)g(z)
(2.55)
B < A ≤ B+
If
π
< δ
2
(λ ≥ 0; 0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
668
NAK EUN CHO ET AL.
for some g ∈ Ꮽp satisfying
Lp (a + 1, c)g(z)
1 + Az
≺
Lp (a, c)g(z)
1 + Bz
(z ∈ ᐁ),
(2.56)
then
π
Lp (a + 1, c)f (z)
arg
−β < η
Lp (a, c)g(z)
2
(z ∈ ᐁ),
(2.57)
where η (0 < η ≤ 1) is the solution of the equation


λη sin(π /2) 1 − t(A, B)

η + 2 tan−1 , for B ≠ −1,
π
p(1+B)+a(A−B) /(1+B)+λη cos(π /2) 1−t(A, B)
δ=


η,
for B = −1,
(2.58)
when
t(A, B) =
2
sin−1
π
a(A − B)
.
p 1 − B 2 − aB(A − B)
(2.59)
Proof. Let
Lp (a + 1, c)f (z)
= β + (1 − β)φ(z),
Lp (a, c)g(z)
r (z) =
Lp (a + 1, c)g(z)
,
Lp (a, c)g(z)
(2.60)
we have
(1 − λ)
Lp (a + 1, c)f (z)
Lp (a + 1, c)f (z)
λzφ (z)
+λ .
− β = (1 − β) φ(z) +
Lp (a, c)g(z)
ar (z) + p − a
Lp (a + 1, c)g(z)
(2.61)
The remaining part of the proof of Theorem 2.18 is similar to that of Theorem 2.3. So
we omit the details.
Put a = c = p, λ = 1, A = α/p, and B = 0 in Theorem 2.18, we have the following
corollary.
Corollary 2.19. If f ∈ Ꮽp satisfies
π
zf (z)
arg
< δ
−
β
g (z)
2
(0 ≤ β < p; 0 < δ ≤ 1; z ∈ ᐁ),
(2.62)
for some g ∈ Ꮽp satisfying the condition
zg (z)
<α
−
p
g(z)
(0 < α ≤ p; z ∈ ᐁ),
(2.63)
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
669
then
π
zf (z)
arg
− β < η (z ∈ ᐁ),
g(z)
2
(2.64)
where η (0 < η ≤ 1) is the solution of the equation
2
δ = η + tan−1
π
η sin π /2 − sin−1 (α/p)
.
p + α + η cos π /2 − sin−1 (α/p)
(2.65)
Lemma 2.20. Let
α = ξ+
0 ≤ (a − 1)/a < ξ < α < 1
ξ
γ + p + aξ
(2.66)
and the function G(z) be defined by
G(z) =
γ +p
zγ
z
t γ−1 g(t) dt
0
g ∈ Ꮽp
(2.67)
for γ > (aξ 2 + (p + 1 − a)ξ − p)/(1 − ξ). If g ∈ Ꮽp satisfies
Lp (a + 1, c)g(z)
<α
−
1
L (a, c)g(z)
p
(z ∈ ᐁ),
(2.68)
Lp (a + 1, c)G(z)
L (a, c)G(z) − 1 < ξ
p
(z ∈ ᐁ).
(2.69)
then
Proof. Defining the function w(z) by
Lp (a + 1, c)G(z)
= 1 + ξw(z),
Lp (a, c)G(z)
(2.70)
we see that w(z) is analytic in ᐁ with w(0) = 0. Now, using the identities
z Lp (a, c)G(z) = aLp (a + 1, c)G(z) − (a − p)Lp (a, c)G(z),
z Lp (a, c)G(z) = (γ + p)Lp (a, c)g(z) − γLp (a, c)G(z)
(2.71)
(2.72)
in (2.70), we get
Lp (a, c)G(z)
γ +p
=
.
Lp (a, c)g(z)
γ + p + aξw(z)
(2.73)
Making use of the logarithmic differentiation of both sides of (2.73) and using identity
(2.71) for both g(z) and f (z) in the resulting equation, we deduce that
Lp (a + 1, c)g(z)
zw (z)
= ξ w(z) +
.
−
1
L (a, c)g(z)
γ + p + aξw(z) p
(2.74)
670
NAK EUN CHO ET AL.
We assume that there exists a point z0 ∈ ᐁ such that max|z|<|z0 | |w(z)| = |w(z0 )| = 1.
Then by Jack’s lemma [7], we have z0 w (z0 ) = kw(z0 ) (k ≥ 1). Let w(z0 ) = eiθ , and
apply this result to w(z) at z0 ∈ ᐁ, we get
L (a + 1, c)g z k
0
p
− 1 = ξ 1 +
Lp (a, c)g z0
γ + p + aξeiθ 1/2
(γ + p + k)2 + 2aξ(γ + p + k) cos θ + (aξ)2
.
=ξ
(γ + p)2 + 2aξ(γ + p) cos θ + (aξ)2
(2.75)
Since the right side of (2.75) is decreasing for 0 ≤ θ < 2π and γ > {aξ 2 +(p +1−a)ξ −
p}/(1 − ξ), we obtain
ξ(γ + p + 1 + aξ)
L (a + 1, c)g z p
0 − 1
≤
,
Lp (a, c)g z0
γ + p + aξ
(2.76)
which contradicts our hypothesis and hence we get
w(z) = 1 Lp (a + 1, c)G(z) − 1 < 1
ξ Lp (a, c)G(z)
(z ∈ ᐁ).
(2.77)
This completes the proof of Lemma 2.20.
Remark 2.21. We note that for a = c = p = 1, Lemma 2.20 yields the corresponding result obtained by Fukui et al. [6].
Theorem 2.22. Let α be as given in (2.66) and γ ∗ > max{(aξ 2 + (p + 1 − a)ξ −
p)/(1 − ξ), aξ − p}. If f ∈ Ꮽp satisfies
π
Lp (a + 1, c)f (z)
arg
−β < δ
Lp (a, c)g(z)
2
(0 ≤ β < 1; 0 < δ ≤ 1; z ∈ ᐁ),
(2.78)
for some f ∈ Ꮽp satisfying condition (2.68), then
π
Lp (a + 1, c)F (z)
arg
− β < η (z ∈ ᐁ),
Lp (a, c)G(z)
2
(2.79)
where the function F (z) and G(z) are defined for γ ∗ by (1.9) and (2.67), respectively
and η (0 < η ≤ 1) is the solution of the equation
2
δ = η + tan−1
π
η sin π /2 − sin−1 aξ/ γ ∗ + p
.
γ ∗ + p + aξ + η cos π /2 − sin−1 aξ/ γ ∗ + p
(2.80)
Proof. Consider the function φ(z) defined in ᐁ by
Lp (a + 1, c)F (z)
= β + (1 − β)φ(z).
Lp (a, c)G(z)
(2.81)
ARGUMENT ESTIMATES OF CERTAIN MULTIVALENT FUNCTIONS . . .
671
Then φ(z) is analytic in ᐁ with φ(0) = 1. Taking logarithmic differentiation on both
sides of (2.81) and using identity (2.71) in the resulting equation, we get
z Lp (a + 1, c)F (z)
Lp (a + 1, c)G(z)
zφ (z)
= p −a+a
+ (1 − β)
.
Lp (a + 1, c)F (z)
Lp (a, c)G(z)
β + (1 − β)φ(z)
(2.82)
From the definition of F (z), we have
∗
γ + p Lp (a, c)f (z) = a Lp (a + 1, c)F (z) + γ ∗ Lp (a + 1, c)F (z).
(2.83)
Again, from (2.71) and (2.72), it follows that
∗
γ + p Lp (a + 1, c)g(z) = zLp (a + 1, c)G(z) + p + γ ∗ − a Lp (a, c)G(z).
(2.84)
Thus, by using (2.83) and (2.84) followed by (2.82), we obtain
Lp (a + 1, c)f (z)
zφ (z)
− β = (1 − β) φ(z) +
,
Lp (a, c)g(z)
ar (z) + γ ∗ + p − a
(2.85)
where r (z) = Lp (a + 1, c)G(z)/Lp (a, c)G(z). By using Lemma 2.20, we have
r (z) ≺ 1 + ξz
(z ∈ ᐁ),
(2.86)
where ξ is given by (2.66). Letting
ar (z) + γ ∗ + p − a = ρeiπ θ/2
(2.87)
and using the techniques of Theorem 2.3, the remaining part of the proof of Theorem
2.22 follows.
Remark 2.23. We easily find the following:



aξ − p,






 2(a − p) − 1
,
γ>

2






aξ 2 + (p + 1 − a)ξ − p



,
1−ξ
if
2a − 1
a−1
<ξ<
,
a
2a
if ξ =
if
2a − 1
,
2a
(2.88)
2a − 1
< ξ < 1.
2a
Taking a = c = p in Theorem 2.22, we get the following corollary.
Corollary 2.24. Let
α = ξ+
ξ
γ ∗ + p(1 + ξ)
(p − 1)/p < ξ < α < 1 ,
(2.89)
672
NAK EUN CHO ET AL.
where γ ∗ > max{(pξ 2 + ξ − p)/(1 − ξ), p(ξ − 1)}. If f ∈ Ꮽp satisfies
π
zf (z)
arg
−β < δ
g(z)
2
(0 ≤ β < p; 0 < δ ≤ 1; z ∈ ᐁ)
(2.90)
for some g ∈ Ꮽp satisfying the condition
zg (z)
< pα
−
p
g(z)
(z ∈ ᐁ),
(2.91)
then
π
zF (z)
arg
−β <
G(z)
2
(z ∈ ᐁ),
(2.92)
where η (0 < η ≤ 1) is the solution of the equation
δ = η+
2
tan−1
π
η sin π /2 − sin−1 pξ/ γ ∗ + p
.
γ ∗ + p(1 + ξ) + η cos π /2 − sin−1 pξ/ γ ∗ + p
(2.93)
Acknowledgment. This work was supported by Korea Research Foundation
Grant KRF-2001-015-DP0013.
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Nak Eun Cho: Division of Mathematical Sciences, Pukyong National University,
Pusan 608-737, Korea
E-mail address: necho@pknu.ac.kr
J. Patel: Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar
751 004, India
E-mail address: jpatelmath@sify.com
G. P. Mohapatra: Silicon Institute
Bhubaneswar 751035, India
E-mail address: girijamo@hotmail.com
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