Journal of Inequalities in Pure and
Applied Mathematics
CERTAIN SECOND ORDER LINEAR DIFFERENTIAL
SUBORDINATIONS
volume 5, issue 3, article 59,
2004.
V. RAVICHANDRAN
Department of Computer Applications
Sri Venkateswara College of Engineering
Pennalur, Sriperumbudur 602 105, India
Received 29 December, 2003;
accepted 03 April, 2004.
Communicated by: N.E. Cho
EMail: vravi@svce.ac.in
URL: http://www.svce.ac.in/∼vravi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
003-04
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Abstract
In this present investigation, we obtain some results for certain second order
linear differential subordination. We also discuss some applications of our results.
2000 Mathematics Subject Classification: Primary 30C80, Secondary 30C45.
Key words: Analytic functions, Hadamard product (or convolution), differential subordination, Ruscheweyh derivatives, univalent functions, convex functions.
The author is thankful to the referee for his comments.
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Differential Subordination with Convex Functions of Order α 8
Differential Subordination with Caratheodory Functions of
Order α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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1.
Introduction
Let H denote the class of all analytic functions in ∆ := {z ∈ C : |z| < 1}. For
a positive integer n and a ∈ C, let
)
(
∞
X
H[a, n] := f ∈ H : f (z) = a +
ak z k (n ∈ N := {1, 2, 3, . . .})
k=n
and
(
A(p, n) :=
f ∈ H : f (z) = z p +
∞
X
)
ak z k
(n, p ∈ N) .
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
k=n+p
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Set
Ap := A(p, 1), A := A1 .
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For two functions f, g ∈ H, we say that the function f (z) is subordinate to
g(z) in ∆ and write
f ≺g
f (z) ≺ g(z),
or
and
|w(z)| < 1
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if there exists a Schwarz function w(z) ∈ H with
w(0) = 0
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(z ∈ ∆),
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such that
J. Ineq. Pure and Appl. Math. 5(3) Art. 59, 2004
(1.1)
f (z) = g(w(z))
(z ∈ ∆).
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In particular, if the function g is univalent in ∆, the above subordination (1.1)
is equivalent to
f (0) = g(0) and f (∆) ⊂ g(∆).
Miller and Mocanu [2] considered the second order linear differential subordination
A(z)z 2 p00 (z) + B(z)zp0 (z) + C(z)p(z) + D(z) ≺ h(z),
where A, B, C and D are complex-valued functions defined on ∆ and h(z) is
any convex function and in particular h(z) = (1 + z)/(1 − z). In fact, they have
proved the following:
Theorem 1.1 (Miller and Mocanu [2, Theorem 4.1a, p.188]). Let n be a positive integer and A(z) = A ≥ 0. Suppose that the functions B(z), C(z), D(z) :
∆ → C satisfy <B(z) ≥ A and
(1.2)
[=C(z)]2 ≤ n[<B(z) − A]<(nB(z) − nA − 2D(z)).
If p ∈ H[1, n] and if
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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(1.3)
<{Az 2 p00 (z) + B(z)zp0 (z) + C(z)p(z) + D(z)} > 0,
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then
<p(z) > 0.
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Also Miller and Mocanu [2] have proved the following:
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Theorem 1.2 (Miller and Mocanu [2, Theorem 4.1e, p.195]). Let h be convex
univalent in ∆ with h(0) = 0 and let A ≥ 0. Suppose that k > 4/|h0 (0)| and
that B(z), C(z) and D(z) are analytic in ∆ and satisfy
< B(z) ≥ A + |C(z) − 1| − <(C(z) − 1) + k|D(z)|.
If p ∈ H[0, 1] satisfies the differential subordination
Az 2 p00 (z) + B(z)zp0 (z) + C(z)p(z) + D(z) ≺ h(z)
then p ≺ h.
Certain Second Order Linear
Differential Subordinations
In this paper, we extend Theorem 1.1 by assuming
2 00
0
<{Az p (z) + B(z)zp (z) + C(z)p(z) + D(z)} > α,
V. Ravichandran
(0 ≤ α < 1)
and Theorem 1.2 by assuming that the function h(z) is convex of order α. Certain results of Karunakaran and Ponnusamy [6], Juneja and Ponnusamy [7] and
Owa and Srivastava [8] are obtained as special cases. Also we give application
of our results to certain functions defined by the familiar Ruscheweyh derivatives.
For two functions f (z) and g(z) given by
f (z) = z p +
∞
X
ak z k ,
g(z) = z p +
k=n+p
∞
X
bk z k
(n, p ∈ N) ,
k=n+p
the Hadamard product (or convolution) of f and g is defined by
(f ∗ g)(z) := z p +
∞
X
k=n+p
ak bk z k =: (g ∗ f )(z).
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The Ruscheweyh derivative of f (z) of order δ + p − 1 is defined by
(1.4) Dδ+p−1 f (z) :=
zp
(1 − z)δ+p
∗f (z)
(f ∈ A (p, n) ; δ ∈ R\ (−∞, −p])
or, equivalently, by
(1.5)
D
δ+p−1
∞ X
δ+k−1
f (z) := z +
ak z k
k
−
p
k=p+1
p
(f ∈ A (p, n) ; δ ∈ R\ (−∞, −p]) .
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
In particular, if δ = l (l + p ∈ N), we find from the definition (1.4) or (1.5) that
dl+p−1 l−1
zp
z
f
(z)
(l + p − 1)! dz l+p−1
(f ∈ A (p, n) ; l + p ∈ N) .
Dl+p−1 f (z) =
In our present investigation of the second order linear differential subordination, we need the following definitions and results:
Definition 1.1 (Miller and Mocanu [2, Definition 2.2b, p. 21]). Let Q be the
set of functions q that are analytic and univalent on ∆ \ E(q), where
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E(q) = {ζ ∈ ∂∆ : lim q(z) = ∞}
z→ζ
and are such that q 0 (ζ) 6= 0 for ζ ∈ ∂∆\E(q), where ∂∆ := {z ∈ C : |z| = 1},
∆ := ∆ ∪ ∂∆.
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Theorem 1.3 (Miller and Mocanu [2, Lemma 2.2d, p. 24]). Let q ∈ Q, with
q(0) = a. Let p(z) = a + pn z n + · · · be analytic in ∆ with p(z) 6≡ a and n ≥ 1.
If p(z) is not subordinate to q(z), then there exist points z0 = r0 eθ0 ∈ ∆ and
ζ0 ∈ ∂∆ − E(q), and an m ≥ n ≥ 1 for which p(∆r0 ) ⊂ q(∆),
(1.6)
(i)
(ii)
(iii)
p(z0 ) = q(ζ0 )
z0 p0 (z0 ) = mζ0 q 0 (ζ0 ), and
<[z0 p00 (z0 )/p0 (z0 ) + 1] ≥ m<[z0 q 00 (z0 )/q 0 (z0 ) + 1],
where ∆r := {z ∈ C : |z| < r}.
Theorem 1.4 (cf. Miller and Mocanu [2, Theorem 2.3i (i), p. 35]). Let Ω be
a simply connected domain and ψ : C3 × ∆ → C satisfies the condition
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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ψ(iσ, ζ, µ + iη; z) 6∈ Ω
for z ∈ ∆ and for real σ, ζ, µ, η satisfying ζ ≤ −n(1 + σ 2 )/2 and ζ + µ ≤ 0.
Let p(z) = 1 + pn z n + pn+1 z n+1 + · · · be analytic in ∆. If
ψ(p(z), zp0 (z), z 2 p00 (z); z) ∈ Ω,
then <p(z) > 0.
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2.
Differential Subordination with Convex
Functions of Order α
By appealing to Theorem 1.3, we first prove the following:
Theorem 2.1. Let h be a convex univalent function of order α, 0 ≤ α < 1, in
∆ with h(0) = 0 and let A ≥ 0. Suppose that
k > 22(1−α) |h0 (0)|
and that B(z), C(z) and D(z) are analytic in ∆ and satisfy
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
(2.1) n<B(z) ≥ n(1 − αn)A +
1
[|C(z) − 1| − <(C(z) − 1)] + k|D(z)|,
2β(α)
where
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 α
4 (1 − 2α)


α 6=

4 − 22α+1
β(α) :=


 (log 4)−1
α=
(2.2)
1
2
1
.
2
If p ∈ H[0, n] satisfies the differential subordination
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(2.3)
then p ≺ h.
2 00
0
Az p (z) + B(z)zp (z) + C(z)p(z) + D(z) ≺ h(z),
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Proof. Our proof of Theorem 2.1 is essentially similar to Theorem 1.2 of Miller
and Mocanu [2]. Let the subordination in (2.3) be satisfied so that D(0) = 0.
Since
k|h0 (0)| > 22(1−α) ,
there is an r0 , 0 < r0 < 1 such that
(1 + r0 )2(1−α)
(1 + r)2(1−α)
= k|h0 (0)| and 22(1−α) <
< k|h0 (0)|
r0
r
for r0 < r < 1. Since h is convex of order α in ∆, the function hr (z) = h(rz)
is convex of order α in ∆ (r0 < r < 1). By setting pr (z) = p(rz) for r0 <
r < 1, we see that the subordination (2.3) becomes
(2.4)
ur (z) := Az 2 p00r (z) + B(rz)zp0r (z) + C(rz)pr (z) + D(rz) ≺ hr (z)
(z ∈ ∆; r0 < r < 1).
Assume that pr is not subordinate to hr , for some r in (r0 , 1). Then by Theorem
1.3 there exist points z0 ∈ ∆, w0 ∈ ∂∆ and an m ≥ n ≥ 1 such that
pr (z0 ) = hr (w0 ), z0 p0r (z0 ) = mw0 h0r (w0 ),
(2.5)
(2.6)
<
z0 p00 (z0 )
1+ 0r
pr (z0 )
≥ m<
w0 h00 (w0 )
1+ 0 r
hr (w0 )
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Therefore we have
(2.7)
Certain Second Order Linear
Differential Subordinations
<
z02 p00r (z0 )
1+
mw0 h0 (w0 )
≥ mα.
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From Equations (2.5), (2.6) and (2.7), it follows that
2 00
z0 pr (z0 )
(2.8)
<
≥ m(mα − 1).
w0 h0r (w0 )
Since hr (z) is convex of order α or equivalently
zh00r (z)
< 1+ 0
> α (z ∈ ∆),
hr (z)
by [2, Theorem 3.3f, p.115], we have
zh0 (z)
< r
> β(α) (z ∈ ∆)
hr (z)
where β(α) is given by Equation (2.2) and this condition is equivalent to
hr (z)
1
1
zh0 (z) − 2β(α) ≤ 2β(α) (z ∈ ∆).
r
Therefore,
1
hr (w0 )
≥
{<[C(rz0 ) − 1] − |C(rz0 ) − 1|}.
(2.9) < (C(rz0 ) − 1)
w0 h0r (w0 )
2β
Since h is convex of order α, we have the following well-known estimate:
|h0 (z)| ≥
|h0 (0)|
(1 + r)2(1−α)
(|z| = r < 1).
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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By setting z = rw0 , we see that
(2.10)
|w0 h0r (w0 )| ≥
r|h0 (0)|
(1 + r)2(1−α)
(|w0 | = 1).
By setting
(2.11) V :=
Az02 p00r (z0 ) B(rz0 )z0 p0r (z0 )
+
w0 h0r (w0 )
w0 h0r (w0 )
+ (C(rz0 ) − 1)
pr (z0 )
D(rz0 )
+
,
w0 h0r (w0 ) w0 h0r (w0 )
V. Ravichandran
we see that
(2.12)
Certain Second Order Linear
Differential Subordinations
ur (z0 ) = hr (w0 ) + V w0 h0r (w0 ).
From (2.8), (2.9), (2.10) and (2.11), we have
1
<V ≥ m(mα − 1)A + m<B(rz0 ) +
[<(C(rz0 ) − 1) − |C(rz0 ) − 1|]
2β(α)
(1 + r)2(1−α)
−
|D(rz0 )|
r|h0 (0)|
≥ m[(nα − 1)A + <B(rz0 )]
1
[<(C(rz0 ) − 1) − |C(rz0 ) − 1|] − k|D(rz0 )|
+
2β(α)
≥ n[(nα − 1)A + <B(rz0 )]
1
−
[|C(rz0 ) − 1| − <(C(rz0 ) − 1)] − k|D(rz0 )| ≥ 0,
2β(α)
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it follows that ur (z0 ) 6∈ hr (∆), a contradiction. Therefore, pr ≺ hr for r ∈
(r0 , 1). By letting r → 1− , we obtain the desired conclusion p ≺ h.
Remark 2.1. When α = 0, n = 1, Theorem 2.1 reduces to Theorem 1.2 of
Miller and Mocanu [2].
From the proof of Theorem 2.1, it is clear that the condition h(0) = 0 in not
necessary when C(z) = 1 and hence the following:
Corollary 2.2. Let h be a convex univalent function of order α, 0 ≤ α < 1, in
∆, h(0) = a and let A ≥ 0. Suppose that
Certain Second Order Linear
Differential Subordinations
k > 22(1−α) /|h0 (0)|
V. Ravichandran
and that B(z) and D(z) are analytic in ∆ with D(0) = 0 and
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(2.13)
n < B(z) ≥ n(1 − αn)A + k|D(z)|
for all z ∈ ∆. If p ∈ H[a, n], p(0) = h(0), satisfies the differential subordination
(2.14)
Az 2 p00 (z) + B(z)zp0 (z) + p(z) + D(z) ≺ h(z),
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then p ≺ h.
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By taking A = 0 and D(z) = 0 in Theorem 2.1, we obtain the following:
Corollary 2.3. Let h be a convex univalent function of order α, 0 ≤ α < 1, in
∆ with h(0) = 0. Let B(z) and C(z) be analytic functions on ∆ satisfying
<B(z) ≥
1
[|C(z) − 1| − <(C(z) − 1)],
2nβ(α)
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where β(α) is as given in Theorem 2.1. If p ∈ H[0, n] satisfies the subordination
B(z)zp0 (z) + C(z)p(z) ≺ h(z),
then p(z) ≺ h(z).
By taking B(z) = 1, α = 0, n = 1, in Corollary 2.3, we have the following:
Corollary 2.4. Let h be a convex univalent function in ∆ with h(0) = 0. Let
C(z) be analytic functions on ∆ satisfying
<C(z) > |C(z) − 1|.
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
If the analytic function p(z) satisfies the subordination
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zp0 (z) + C(z)p(z) ≺ h(z),
then p(z) ≺ h(z).
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3.
Differential Subordination with Caratheodory
Functions of Order α
By appealing to Theorem 1.4, we now prove the following:
Theorem 3.1. Let n be a positive integer and A(z) = A ≥ 0. Suppose that the
functions B(z), C(z), D(z) : ∆ → C satisfy <B(z) ≥ A and
(3.1) [= C(z)]2 ≤ n [<B(z) − A]
2+δ
δ + 2α
× n(<B(z) − A) −
< C(z) −
<(D(z) − α) .
1−α
1−α
If p ∈ H[1, n] and
(3.2)
then
2 00
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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0
<{Az p (z) + B(z)zp (z) + C(z)p(z) + D(z)} > α
δ + 2α
<p(z) >
.
δ+2
(α < 1),
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Proof. Define the function P (z) by
Close
p(z) − γ
P (z) :=
1−γ
where
δ + 2α
γ :=
.
δ+2
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Then inequality (3.2) can be written as
<{ψ(P (z), zP 0 (z), z 2 P 00 (z); z)} > 0,
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where
ψ(r, s, t; z) = At + B(z)s + C(z)r +
γC(z) + D(z) − α
.
1−γ
In view of Theorem 1.4, it is enough to show that
<ψ(iσ, ζ, µ + iη; z) ≤ 0
for all real numbers σ, ζ, µ and η with ζ ≤
z ∈ ∆. Now,
−n(1+σ 2 )
,
2
ζ + µ ≤ 0 and for all
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
<ψ(iσ, ζ, µ + iη; z)
γC(z) + D(z) − α
= µA + ζ<B(z) − σ=C(z) + <
1−γ
γC(z) + D(z) − α
≤ ζ(<B(z) − A) − σ=C(z) + <
1−γ
1
≤ − n[<B(z) − A]σ 2 + 2=C(z)σ
2
γC(z) + D(z) − α
+n[<B(z) − A] − 2<
≤ 0,
1−γ
provided (3.1) holds. This completes the proof of our Theorem 3.1.
For α = δ = 0, Theorem 3.1 reduces to Theorem 1.1.
By taking D = 0 and C(z) = 1 in Theorem 3.1, we have the following:
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Corollary 3.2. Let A ≥ 0 and <B(z) − A > δ > 0. If p ∈ H[1, n] satisfies
<{Az 2 p00 (z) + B(z)zp0 (z) + p(z)} > α
(α < 1)
then
nδ + 2α
.
nδ + 2
Corollary 3.3. Let λ(z) and R(z) be functions defined on ∆ and
<p(z) >
<λ(z) > δ +
2+δ
<R(z) ≥ 0.
(1 − α)n
If p ∈ H[1, n] satisfies
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
<{λ(z)zp0 (z) + p(z) + R(z)} > α
(α < 1),
then
2α + δn
.
2 + δn
A special case of Corollary 3.3 is obtained by Owa and Srivastava [8, Lemma
2, p. 254].
The proof of the following theorem is similar and hence it is omitted.
<p(z) >
Theorem 3.4. Let n be a positive integer and A(z) = A ≥ 0. Suppose that the
functions B(z), C(z), D(z) : ∆ → C satisfy <B(z) ≥ A and
(3.3) [=C(z)]2
δ + 2α
2+δ
≤ n[<B(z)−A] n(<B(z) − A) −
< C(z) −
<(D(z) − α) .
1−α
1−α
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If p ∈ H[1, n] satisfies
(3.4)
<{Az 2 p00 (z) + B(z)zp0 (z) + C(z)p(z) + D(z)} < α
then
<p(z) <
(α > 1),
δ + 2α
.
δ+2
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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4.
Applications
We now give certain applications of our results obtained in Section 2 and 3.
Theorem 4.1. Let γ ∈ C with γ 6= −1, −2, −3, . . . and let φ, Φ be analytic
functions on ∆ with φ(z)Φ(z) 6= 0 for z ∈ ∆. If
<C(z) − |C(z) − 1| > 1 − 2nβ(α)<B(z),
where
γΦ(z) + zΦ0 (z)
Φ(z)
and C(z) :=
,
φ(z)
φ(z)
then the integral operator defined by
Z z
1
I(f )(z) := γ
tγ−1 f (t)φ(t)dt
z Φ(z) 0
B(z) :=
satisfies I(f )(z) ≺ h(z) for every function f (z) ≺ h(z) where h(z) is a convex
function of order α.
Proof. The result follows immediately from Corollary 2.3.
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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Theorem 4.2. Let h be a convex univalent function of order α in ∆, 0 ≤ α < 1
and h(0) = 1. Let M, N, R be analytic in ∆ with R(0) = 0 and
M (z) = z n + . . . , and N (z) = z n + . . . .
Let
βN (z)
< 0
> k|R(z)|
zN (z)
22(1−α)
k> 0
|h (0)|
.
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If
(4.1)
β
M 0 (z)
M (z)
+ (1 − β)
+ R(z) ≺ h(z),
0
N (z)
N (z)
then
M (z)
≺ h(z).
N (z)
Proof. Let the function p(z) be defined by
Certain Second Order Linear
Differential Subordinations
p(z) = M (z)/N (z).
V. Ravichandran
Then p(0) = 1 = h(0) and it follows that
M 0 (z)
N (z) 0
p(z) +
zp
(z)
=
.
zN 0 (z)
N 0 (z)
Also, a computation shows that the subordination in (4.1) is equivalent to
p(z) +
βN (z) 0
zp (z) + R(z) ≺ h(z).
zN 0 (z)
The result now follows by an application of Corollary 2.2
Remark 4.1. When β = 1, α = 0, Theorem 4.2 reduces to [2, Theorem 4.1h, p.
199] of Miller and Mocanu. If α = 0 and R(z) = 0, then Theorem 4.2 reduces
to a result of Juneja and Ponnusamy [7, Corollary 1, p. 290].
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J. Ineq. Pure and Appl. Math. 5(3) Art. 59, 2004
More generally, we have the following:
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Theorem 4.3. Let δ > −p be any real number, λ ∈ C with <λ ≥ 0. Let R(z)
be a function defined on ∆ with R(0) = 0 and h(z) a convex function of order
α, 0 ≤ α < 1, h(0) = 1. Let g ∈ Ap satisfy
Dδ+p−1 g(z)
22(1−α)
< λ δ+p
≥ µ(δ + p)|R(z)|,
k> 0
.
D g(z)
|h (0)|
If f ∈ Ap satisfies
δ+p−1
µ
µ−1
Dδ+p f (z) Dδ+p−1 f (z)
D
f (z)
+ λ δ+p
+ R(z) ≺ h(z),
(1 − λ)
Dδ+p−1 g(z)
D g(z) Dδ+p−1 g(z)
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
then
Dδ+p−1 f (z)
Dδ+p−1 g(z)
µ
≺ h(z).
Proof. Let the function p(z) be defined by
δ+p−1
µ
D
f (z)
p(z) :=
.
Dδ+p−1 g(z)
Then a computation shows that the following subordination holds:
B(z)zp0 (z) + p(z) + R(z) ≺ h(z),
where
Dδ+p−1 g(z)
λ
B(z) :=
.
µ(δ + p) Dδ+p g(z)
The result follows by an application of Corollary 2.2.
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When R(z) = 0 and µ = 1, the Theorem 4.3 reduces to Juneja and Ponnusamy [7, Theorem 1, p. 289].
Theorem 4.4. Let α be a complex number <α > 0 and β < 1. Let M, N, R be
analytic in ∆ with R(0) = 0 and
M (z) := z n + c1 z n+k + · · · ,
Let
<
N (z) := z n + d1 z n+k + · · · .
αN (z)
2 + δk
>δ+
<R(z).
0
zN (z)
(1 − β)k
If
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
(4.2)
then
<[α
M (z)
M 0 (z)
+ (1 − α)
+ R(z)] > β,
0
N (z)
N (z)
Contents
M (z)
2β + kδ
<
>
.
N (z)
2 + kδ
Proof. Let p(z) := M (z)/N (z). Then p(0) = 1 = h(0). It follows that
N (z) 0
M 0 (z)
zp (z) = 0 .
p(z) +
zN 0 (z)
N (z)
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Then
<p(z) +
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αN (z) 0
M 0 (z)
M (z)
zp
(z)
+
R(z)
=
<[α
+ (1 − α)
+ R(z)]
0
0
zN (z)
N (z)
N (z)
> β.
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J. Ineq. Pure and Appl. Math. 5(3) Art. 59, 2004
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If B(z) is defined by B(z) := αN (z)/[zN 0 (z)], then it follows that
<B(z) > δ +
2 + δk
<R(z).
(1 − β)k
The result now follows by an application of Corollary 3.3
Remark 4.2. For R(z) = 0, β = 0, Theorem 4.4 is due to Karunakaran and
Ponnusamy [6, Theorem B, p. 562].
Theorem 4.5. Let δ > −p be any real number, λ ∈ C with <λ ≥ 0. Let R(z)
be a function defined on ∆ with R(0) = 0, 0 ≤ α < 1. Let g ∈ Ap satisfies
Dδ+p−1 g(z)
µ(δ + p)(2 + δ)
> µ(δ + p)δ +
<R(z) ≥ 0.
< λ δ+p
D g(z)
1−α
If f ∈ Ap satisfies
(
)
δ+p−1
µ
µ−1
D
f (z)
Dδ+p f (z) Dδ+p−1 f (z)
< (1 − λ)
+ λ δ+p
+ R(z) > α,
Dδ+p−1 g(z)
D g(z) Dδ+p−1 g(z)
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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then
Dδ+p−1 f (z)
Dδ+p−1 g(z)
µ
≥
2α + δ
.
2+δ
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Proof. Let
p(z) :=
δ+p−1
D
f (z)
δ+p−1
D
g(z)
µ
.
J. Ineq. Pure and Appl. Math. 5(3) Art. 59, 2004
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Then a computation shows that
<{B(z)zp0 (z) + p(z) + R(z)} > α,
where
B(z) :=
λ
Dδ+p−1 g(z)
.
µ(δ + p) Dδ+p g(z)
The result follows easily.
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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References
[1] B.C. CARLSON AND S.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[2] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory
and Applications, Series in Pure and Applied Mathematics, No. 225, Marcel
Dekker, New York, (2000).
[3] S.S. MILLER, P.T. MOCANU AND M.O. READE, Differential subordinations and univalent functions, Michigan Math. J., 28 (1981), 157–171.
Certain Second Order Linear
Differential Subordinations
[4] S.S. MILLER AND P.T. MOCANU, The theory and applications of second
order differential subordinations, Studia Univ. Babes-Bolyai, Math., 34(4)
(1989), 3–33.
V. Ravichandran
[5] S.S. MILLER AND P.T. MOCANU, Classes of univalent integral operators,
J. Math. Anal. Appl., 157 (1991), 147–165.
Contents
[6] V. KARUNAKARAN AND S. PONNUSAMY, Differential subordinations
and conformal mappings I, Indian J. Pure Appl. Math., 20 (1989), 560–
565.
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[7] S. PONNUSAMY AND O.P. JUNEJA, Some applications of first order differential subordinations, Glasnik Math., 25(45) (1990), 287–296.
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[8] S. OWA AND H.M. SRIVASTAVA, Analytic solutions of a class of BriotBouqet differential equations, in Current Topics in Analytic Function Theory (H.M. Srivastava and S. Owa, Ed.), World Scientific, Singapore, 252–
259 (1992).
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J. Ineq. Pure and Appl. Math. 5(3) Art. 59, 2004
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[9] H. SAITOH, A linear operator and its applications of first order differential
subordinations, Math. Japonica, 44 (1996), 31–38.
Certain Second Order Linear
Differential Subordinations
V. Ravichandran
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