Journal of Inequalities in Pure and
Applied Mathematics
NON-AUTONOMOUS DIFFERENTIAL SUBORDINATIONS
RELATED TO A SECTOR
volume 4, issue 4, article 72,
2003.
SUKHJIT SINGH AND SUSHMA GUPTA
Sant Longowal Institute of Engineering and Technology,
Longowal-148106 (Punjab), India.
EMail: sushmagupta1@yahoo.com
Received 02 April, 2003;
accepted 11 September, 2003.
Communicated by: H.M. Srivastava
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
042-03
Quit
Abstract
Let λ(z) be a complex valued function defined in the unit disc E and let p(z)
be a function analytic in E with p(0) = 1 and p(z) 6= 0 in E. In this article, we
determine the largest constants γk , k = 1, 2, 3, . . . and conditions on λ(z) such
that for given α, β and δ, the non-autonomous differential subordination
zp0 (z) α
1 + z γk
β
(p(z)) 1 + λ(z) k
, z ∈ E,
≺
1−z
p (z)
implies
1+z δ
p(z) ≺
1−z
in E. Here the symbol ‘ ≺ ’ stands for subordination. Almost all the previously known results on differential subordination concerning a sector follow as
particular cases of our results.
2000 Mathematics Subject Classification: Primary 30C45, Secondary 30C50.
Key words: Univalent function, Starlike function, Subordination, Differential subordination.
The authors wish to thank Prof. S. Ponnusamy and Prof. A. Lecko for their help
during this work. They are also grateful to the refree for useful suggestions and
comments.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Proofs of Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4
Applications to Univalent Functions . . . . . . . . . . . . . . . . . . . . . 16
References
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
1.
Introduction
Let A denote the class of functions f which are analytic in the unit disc E =
{z : |z| < 1} and satisfy f (0) = 0, f 0 (0) = 1. Denote by A0 , the class of
functions p analytic in E for which p(0) = 1 and p(z) 6= 0 in E.
If f and g are analytic in E, we say that f is subordinate to g in E, written
as f (z) ≺ g(z) in E, if g is univalent in E, f (0) = g(0) and f (E) ⊂ g(E).
Let h be a univalent function in E and let ψ : C 2 → C, where C is the
complex plane. If an analytic function p satisfies the differential subordination
(1.1)
0
ψ(p(z), zp (z)) ≺ h(z), h(0) = ψ(p(0), 0), z ∈ E,
Non-autonomous Differential
Subordinations Related to a
Sector
then a univalent function q is said to be the dominant of the differential subordination (1.1), if p(0) = q(0) and p ≺ q for all p satisfying (1.1). Differential
subordination (1.1) is said to be non-autonomous if a function of z is allowed
to be present on the left hand side, in addition to the terms p(z) and zp0 (z).
Since 1981, when a formal study of differential subordination started with a
remarkable paper of Miller and Mocanu [5], several results concerning differential subordination in a sector have been proved (e.g. see [6], [9] and [3]).
In the present paper, we establish the following two theorems.
Sukhjit Singh and Sushma Gupta
Theorem 1.1. Let α ∈ [0, 1] be fixed and let δ ∈ (0, δ0 ], where δ0 is the solution
of the equation
π
βδπ = 2π − α
+ arctanη
2
for β ≥ 0 and for a suitable fixed η > 0 such that λ(z) : E → C satisfies
Go Back
(1.2)
δ Re λ(z)
≥ η, z ∈ E.
1 + δ|Imλ(z)|
Title Page
Contents
JJ
J
II
I
Close
Quit
Page 3 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
If p ∈ A0 satisfies the non-autonomous differential subordination
α γ
1+z 1
zp0 (z)
β
(1.3)
(p(z)) 1 + λ(z)
≺
, z ∈ E,
p(z)
1−z
then,
p(z) ≺
1+z
1−z
δ
in E,
where,
(1.4)
γ1 = βδ +
Non-autonomous Differential
Subordinations Related to a
Sector
2α
arctan η, 0 < δ ≤ δ0 .
π
Sukhjit Singh and Sushma Gupta
1
be fixed. Also let
Theorem 1.2. Let k = 2, 3, 4, . . . , α ∈ [0, 1], δ ∈ 0, k−1
β ≥ 0 be such that 0 ≤ βδ ≤ 2. For a suitable fixed η > 0, let λ(z) : E → C
be a function satisfying
(1.5)
δ|λ(z)| cos ψ
≥ η, z ∈ E,
x + δ|λ(z)|| sin ψ|
II
I
Go Back
|ψ − Argλ(z)| = (k − 1)
δπ
2
Close
Quit
and,
(1.7)
Contents
JJ
J
where,
(1.6)
Title Page
Page 4 of 23
x = (1 − (k − 1)δ)
1−(k−1)δ
2
(1 + (k − 1)δ)
1+(k−1)δ
2
.
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
If p ∈ A0 satisfies the non-autonomous differential subordination
α γ
1+z k
zp0 (z)
β
≺
, z ∈ E,
(1.8)
(p(z)) 1 + λ(z) k
p (z)
1−z
then,
p(z) ≺
1+z
1−z
δ
in E,
where,
2α
(1.9)
γk = βδ +
arctan η.
π
We claim that our results unify most of the previously known results related
to differential subordinations in a sector. Some special cases of Theorem 1.1
and Theorem 1.2 have been discussed in Section 3. In Section 4, we give some
applications of our results to the univalent functions.
We shall need the following lemma to prove our results.
Lemma 1.3. Let F be analytic in E and let G be analytic and univalent in E
except for points ζ such that limz→ζ F (z) = ∞, with F (0) = G(0). If F 6≺ G
in E, then there exist points z0 ∈ E, ζ0 ∈ ∂E (boundary of E) and an m ≥ 1
for which
(a) F (|z| < |z0 |) ⊂ G(E),
(b) F (z0 ) = G(ζ0 ) and
(c) z0 F 0 (z0 ) = mζ0 G0 (ζ0 ).
Lemma 1.3 is due to Miller and Mocanu [5].
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
2.
Proofs of Main Theorems
δ
Proof of Theorem 1.1. Let q(z) = 1+z
. Then we need to prove that (1.3)
1−z
implies p(z) ≺ q(z) in E. Suppose, on the contrary, that p 6≺ q in E. Then,
by Lemma 1.3, there exist points z0 ∈ E, ζ0 ∈ ∂E and an m ≥ 1 such that
p(z0 ) = q(ζ0 ) and z0 p0 (z0 ) = mζ0 q 0 (ζ0 ). Since p(z0 ) = q(ζ0 ) 6= 0, it follows
0
that ζ0 6= ±1. Thus 1+ζ
= ri for r 6= 0. Writing λ(z0 ) = Reiφ , |φ| < π/2, a
1−ζ0
simple calculation gives
α
α
mζ0 q 0 (ζ0 )
z0 p0 (z0 )
β
β
= (q(ζ0 )) 1 + λ(z0 )
(p(z0 )) 1 + λ(z0 )
p(z0 )
q(ζ0 )
2
α
iφ
r +1
imδRe
βδ
= (ri)
1+
2
r
= Ψ0 say.
Sukhjit Singh and Sushma Gupta
Title Page
Contents
Then,
(2.1)
Non-autonomous Differential
Subordinations Related to a
Sector
βδπ
+ α arctan
ArgΨ0 = ±
2
"
#
mδRcosφ
.
2r
− mδRsinφ
r 2 +1
Here a positive sign corresponds to r > 0 and a negative sign to r < 0.
mδR cos φ
, where x(r) = r22r+1 . Then, Max|x(r)| = 1
Let A = A(m, r) = x(r)−mδR
sin φ
for all values of r whether positive or negative.
If we define,
B = B(m, r) =
mδRcosφ
,
|x(r)| + mδR|sinφ|
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
then B is an increasing function of m for each fixed r, and therefore, for m ≥ 1,
we have
B≥
δRcosφ
δRcosφ
≥
≥ η, (using (1.2)).
|x(r)| + δR|sinφ|
1 + δR|sinφ|
Now, It is easy to check the following six cases:
(i) When r > 0 and sinφ ≤ 0, we have A = B.
(ii) When r > 0 and x(r) > mδRsinφ > 0, we get A > B.
Non-autonomous Differential
Subordinations Related to a
Sector
(iii) When r > 0 and x(r) < mδRsinφ, we get A < −B.
Sukhjit Singh and Sushma Gupta
(iv) When r < 0 and sinφ ≥ 0, we have A = −B.
(v) When r < 0 and mδRsinφ < −|x(r)|, we have A > B.
(vi) When r < 0 and 0 > mδRsinφ > −|x(r)|, we get A < −B.
Since arctan is an increasing function of its argument, therefore, in view of
cases (i) and (ii), we get from (2.1),
βδπ
+ α arctanB
2
βδπ
≥
+ α arctan η
2
γ1 π
=
.
2
ArgΨ0 ≥
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
For the case (iii), we get
βδπ
+ α arctan(−B)
2
βδπ
≤
− α arctan η
2
γ1 π
γ1 π
= βδπ −
≤ 2π −
, since 0 ≤ βδ ≤ βδ0 < 2.
2
2
For cases (iv) and (vi), we get
ArgΨ0 <
βδπ
ArgΨ0 ≤ −
+ α arctan(−B)
2
βδπ
≤−
− α arctan η
2
γ1 π
=−
.
2
In case (v), we have
βδπ
+ α arctanB
2
βδπ
≥−
+ α arctan η
2
γ1 π = − βδπ −
2
γ1 π ≥ − 2π −
, since 0 ≤ βδ ≤ βδ0 < 2.
2
Combining all the above cases, we obtain
γ1 π
γ1 π
≤ | Arg Ψ0 | ≤ 2π −
,
2
2
ArgΨ0 > −
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
which is a contradiction to (1.3). Hence p(z) ≺
theorem is complete.
1+z δ
.
1−z
The proof of the
Proof of Theorem 1.2. Proceeding as in the proof of Theorem 1.1, we get
α
z0 p0 (z0 )
β
(p(z0 )) 1 + λ(z0 ) k
p (z0 )
α
imδR 2
−1−(k−1)δ i(φ−(k−1)δπ/2)
βδ
= (ri)
(r + 1)r
e
1+
2
= Θ0 say.
Now, if ψ − φ = −(k − 1) δπ
, then we get
2
"
#
βδπ
mδRcosψ
(2.2)
Arg Θ0 =
+ α arctan 2r1+(k−1)δ
.
2
− mδRsinψ
2
r +1
i
h
1+(k−1)δ
mδRcosψ
Let A = A(m, r) = x(r)−mδRsinψ
, where x(r) = 2r r2 +1 . Since (k − 1)δ <
1, it is easy to verify that for r > 0 (and, of course, also for r < 0),
max |x(r)| = (1 − (k − 1)δ)
1−(k−1)δ
2
(1 + (k − 1)δ)
1+(k−1)δ
2
Now, if we define,
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
= x, using (1.7).
Close
Quit
B = B(m, r) =
mδRcosψ
,
|x(r)| + mδR|sinψ|
Page 9 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
then, B is an increasing function of m and, therefore, for m ≥ 1, we have
B≥
δRcosψ
δRcosψ
≥
≥ η, using (1.5).
|x(r)| + δR|sinψ|
x + δR|sinψ|
Now, one can easily verify the following three cases:
Case (i). A = B, when sinψ ≤ 0.
Case(ii). A > B, when sinψ > 0 and x(r) > mδRsinψ.
Case (iii). A < −B, when sinψ > 0 and x(r) < mδRsinψ.
Since arctan is an increasing function of its argument, therefore, in view of
cases (i) and (ii), we get from (2.2)
βδπ
+ α arctan η
2
γk π
=
, using (1.9).
2
ArgΘ0 ≥
For the case (iii), we obtain
βδπ
+ α arctan(−B)
2
βδπ
≤
− α arctan η
2
γk π
≤ βδπ −
2
γk π
≤ 2π −
, since βδ ≤ 2.
2
ArgΘ0 <
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
Now, consider the case when r < 0. Writing r = −a, a > 0 we have
α
α
z0 p0 (z0 )
imδR (a2 + 1)ei(φ+(k−1)δπ/2)
β
βδ −iβδπ
(p(z0 )) 1 + λ(z0 ) k
=a e 2
1−
p (z0 )
2
a1+(k−1)δ
= Φ0 say.
If ψ − φ = (k − 1) δπ
, then
2
ArgΦ0 =
−βδπ
+ α arctan C,
2
mδRcosψ
where C = C(m, r) = −|x(r)|−mδRsinψ
.
It is now elementary to check the following three cases:
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Case (a). When sinψ ≥ 0, we have C = −B.
Title Page
Case (b). When sinψ < 0 and mδR sinψ < −|x(r)|, we obtain
C > B.
Contents
Case (c). For sinψ < 0and mδRsinψ > −|x(r)|, we have C < −B.
For cases (a) and (c), we get
−βδπ
+ α arctan(−B)
2
−βδπ
≤
− α arctan η
2
γk π
=−
.
2
ArgΦ0 ≤
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
In case (b), we have
−βδπ
+ α arctan B
2
−βδπ
≥
+ α arctan η
2
γk π = − βδπ −
2
γk π ≥ − 2π −
, since βδ ≤ 2.
2
ArgΦ0 >
Combining the above six cases, three for r > 0 and three for r < 0, we have
α γk π z0 p0 (z0 )
β
≤ 2π − γk π ,
≤ Arg (p(z0 )) 1 + λ(z0 ) k
2
p (z0 )
2
which is a contradiction to (1.8). Hence p(z) ≺
proof.
1+z δ
1−z
. This completes the
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
3.
Special Cases
(i) Taking α = β = 1 in Theorem 1.1, we get the result of S. Ponnusamy [9,
p. 399, Lemma 1].
(ii) Setting α = β = 1 and λ(z) = λ, a positive real number, in Theorem 1.1,
we get Theorem 5 of Miller and Mocanu [6, p. 532].
(iii) Putting β = 1 and λ(z) = 1 in Theorem 1.1, we get the result of A. Lecko
et.al. [4, p. 198, Theorem 2.1].
(iv) In Theorem 1.1, taking α = 1, we get the following result:
Let β ≥ 0 and let δ0 be the solution of the equation
βδπ =
3π
− arctan η,
2
for a suitable fixed η > 0 such that λ(z) : E → C is a function satisfying
δ Re λ(z)
≥ η, z ∈ E.
1 + δ| Im λ(z)|
If p ∈ A0 satisfies the non-autonomous differential subordination
γ
1+z
β
β−1
0
p (z) + λ(z)p (z)zp (z) ≺
, z ∈ E,
1−z
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 23
then,
p(z) ≺
1+z
1−z
δ
in E,
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
where,
γ = βδ +
2
arctan η, 0 < δ ≤ δ0 .
π
(v) In Theorem 1.2, taking λ(z) = 1 and β = 1, we get the result of A. Lecko
[3, p. 344, Theorem 2.1].
(vi) Putting α = β = 1 and k = 2 in Theorem 1.2, we obtain the following
result:
For 0 < δ < 1 and η > 0, let λ(z) : E → C be a function satisfying
δ|λ(z)|cosψ
≥ η, z ∈ E,
x + δ|λ(z)||sinψ|
where |ψ − Argλ(z)| =
δπ
2
and x = (1 + δ)
1+δ
2
(1 − δ)
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
1−δ
2
.
Title Page
0
If p ∈ A satisfies,
Contents
0
p(z) + λ(z)
zp (z)
≺
p(z)
1+z
1−z
γ 2
in E, then,
p(z) ≺
1+z
1−z
δ
in E, where γ2 = δ + π2 arctan η.
Taking λ(z) = λ, a positive real number in the above result, we get the
well-known differential subordination (see [7, p. 268, 5.1-40]).
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
0
(z)
(vii) Taking β + α in place of β, λ(z) = 1, k = 2 and p(z) = zff (z)
, where
f ∈ A, in Theorem 1.2, we get the following (Also see Darus and Thomas
[2, p. 1050, Theorem 1]):
Let α ∈ [0, 1], δ ∈ (0, 1) and β ≥ 0 be such that 0 ≤ (β + α)δ ≤ 2. If
f ∈ A satisfies
zf 0 (z)
f (z)
β α γ
1+z 2
zf 00 (z)
≺
,
1+ 0
f (z)
1−z
Non-autonomous Differential
Subordinations Related to a
Sector
then,
zf 0 (z)
≺
f (z)
1+z
1−z
δ
in E,
Sukhjit Singh and Sushma Gupta
where,
Title Page
"
γ2 = βδ +
#
2α
δπ
δ
arctan tan
+
.
1+δ
1−δ
π
2
(1 + δ) 2 (1 − δ) 2 cos δπ
2
(viii) Writing β = 1, α = 1, λ(z) = 1, k = 2 and p(z) = zf 0 (z)/f (z),
where f ∈ A, in Theorem 1.2, we get well-known result of Nunokawa
and Thomas [8, p. 364, Theorem1].
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
4.
Applications to Univalent Functions
In this section, we give some applications of our results to univalent functions
and obtain some new conditions for univalence, starlikeness and strongly starlikeness.
A function f ∈ A is said to be strongly starlike of order α, 0 < α ≤ 1, if
0
απ
zf
(z)
<
arg
in E.
f (z) 2
We denote the set of all such functions by S ∗ (α). The class S ∗ (α) was introduced and studied independently by Brannan and Kirwan [1] and Stankiewicz
[10]. Note that S ∗ (1) is the usual class of starlike functions f in A satisfying
0 zf (z)
Re
> 0, z ∈ E.
f (z)
We denote this class by St.
First of all, we note that if f ∈ A, then the functionals
are all members of the class A0 .
f (z)
, f 0 (z)
z
and
zf 0 (z)
f (z)
Theorem 4.1. Let α ∈ [0, 1] and let δ0 = 0.6165 . . . be the unique root of the
equation
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
2 arctan(1 − δ) + π(1 − 2δ) = 0.
Close
Further, let β ≥ 0 be such that βδ ≤ βδ0 ≤ 2 for 0 < δ ≤ δ0 . If a function
f ∈ A, f 0 (z) 6= 0, z ∈ E, satisfies
α γ
zf 00 (z)
1+z
0
β
(4.2)
(f (z)) 1 + 0
≺
, z ∈ E,
f (z)
1−z
Quit
(4.1)
Page 16 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
then f ∈ St where γ = βδ +
2α
arctan
π
δ.
Proof. In Theorem 1.1, taking λ(z) = 1 and p(z) = f 0 (z), the subordination
(4.2) implies
0
f (z) ≺
(4.3)
1+z
1−z
δ
, z ∈ E,
where γ = βδ + 2α
arctan δ. Again using Theorem 1.1 with α = β = λ(z) = 1
π
f (z)
and p(z) = z , we get from (4.3),
µ
f (z)
1+z
≺
, z ∈ E,
z
1−z
where δ = µ +
2
π
arctanµ which is equivalent to (4.1) with µ = 1 − δ. Now
zf 0 (z) f (z) 0
arg
≤ |argf (z)| + arg
f (z) z π
π
≤ (δ + µ) = ,
2
2
and the desired result follows.
Writing p(z) =
f (z)
z
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
and λ(z) = 1 in Theorem 1.1, we get:
Lemma 4.2. Let α ∈ [0, 1]be fixed and let δ ∈ (0, δ0 ], where δ0 is the solution
of the equation
π
βδπ = 2π − α
+ arctan δ
2
Quit
Page 17 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
for fixed β ≥ 0. If f ∈ A,
f (z)
z
f (z)
z
6= 0, z ∈ E, satisfies
β−α
0
(f (z)) ≺
then,
f (z)
≺
z
where γ = βδ +
2α
π
α
1+z
1−z
1+z
1−z
γ
in E,
δ
, z ∈ E,
arctan δ.
Theorem 4.3. Let α ∈ [0, 1], β ≥ 0 be fixed. Suppose that γ0 , α/2 < γ0 ≤ α,
is the unique root of the equation
2γ − α π
(4.4)
β = (α − γ)cot
.
α
2
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
Let f ∈ A,
f (z)
z
6= 0, z ∈ E, satisfy
(4.5)
f (z)
z
β−α
0
α
(f (z)) ≺
1+z
1−z
γ
in E.
JJ
J
II
I
Go Back
Close
Then f ∈ St.
Proof. In view of (4.5) and Lemma 4.2, we have
f (z)
≺
z
1+z
1−z
Quit
Page 18 of 23
δ
, z ∈ E,
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
where δ is given by the equation
γ = βδ +
(4.6)
2α
arctan δ.
π
We observe that (4.6) is equivalent to (4.4) with δβ = α − γ. Now
β−α β zf 0 (z) f
(z)
f
(z)
≤ arg(f 0 (z))α
α arg
+ arg
f (z) z
z
απ
γπ βδπ
≤
+
=
,
2
2
2
and the conclusion follows.
Remark 4.1. Taking α = 1 in Theorem 4.3, we get the Theorem 1 of Ponnusamy
[9, p. 403].
Taking α = β =
1
2
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
in Theorem 4.3, we get
Example 4.1. For f ∈ A, the differential subordination
γ
p
1
+
z
f 0 (z) ≺
in E,
1−z
implies that f is starlike in E where γ = 0.3082 . . . is given by 2 arctan (1 −
2γ) + π(1 − 4γ) = 0.
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 23
Setting p(z) = f 0 (z) and α = λ(z) = 1 in Theorem 1.1, we obtain
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
Corollary 4.4. For β ≥ 0, if an analytic function f in A satisfies
2β+1
2
1+z
0
β
0
β−1
00
(f (z)) + (f (z)) zf (z) ≺
in E,
1−z
then
f 0 (z) ≺
1+z
in E,
1−z
and, hence, f is univalent in E.
(z)
Writing p(z) = zff (z)
, k = 2 and (β + α) in place of β in Theorem 1.2, we
get the following result:
Non-autonomous Differential
Subordinations Related to a
Sector
Theorem 4.5. Let α ∈ [0, 1] and β ≥ 0 be fixed. Let δ ∈ (0, 1) be such that
0 ≤ (β + α)δ ≤ 2. For a fixed η > 0, let λ(z) : E → C be a function satisfying
Sukhjit Singh and Sushma Gupta
0
Title Page
δ|λ(z)|cosψ
≥ η, z ∈ E,
x + δ|λ(z)||sinψ|
1+δ
Contents
1−δ
where |ψ − Arg λ(z)| = δπ
and x = (1 + δ) 2 (1 − δ) 2 .
2
If f ∈ A satisfies the differential subordination
0 β α γ
zf (z)
zf 0 (z)
zf 00 (z)
1+z 2
(1 − λ(z))
+ λ(z) 1 + 0
≺
f (z)
f (z)
f (z)
1−z
in E, then
JJ
J
II
I
Go Back
Close
Quit
zf 0 (z)
≺
f (z)
1+z
1−z
δ
in E i.e. f ∈ S ∗ (δ) in E, where γ2 = (β + α)δ +
Page 20 of 23
2α
arctan
π
η.
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
Taking β = 0, α = 1, λ(z) = λ, λ real, in Theorem 4.5, we obtain the
well-known result [7, p. 266, Cor. 5. 1i. 1].
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
References
[1] D.A. BRANNAN AND W.E. KIRWAN, On some classes of bounded univalent functions, J. London Math. Soc., 1(2) (1969), 431–443.
[2] M. DARUS AND D.K. THOMAS, α-logarithmically convex functions, Indian J. Pure Appl. Math., 29(10) (1998), 1049–1059.
[3] A. LECKO, On differential subordinations connected with convex functions related to a sector, Kyungpook Math. J., 39 (1999), 343–350.
[4] A. LECKO, M. LECKO, M. NUNOKAWA AND S. OWA, Differential
subordinations and convex functions related to a sector, Mathematica,
40(63)(2) (1998), 197–205.
[5] S.S. MILLER AND P.T. MOCANU, Differential subordinations and univalent functions, Michigan Math. J., 28 (1981), 157–171.
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
[6] S.S. MILLER AND P.T. MOCANU, Marx- Strohhäcker differential subordination systems, Proc. Amer. Math. Soc., 99(3) (1987), 527–534.
[7] S.S. MILLER AND P.T. MOCANU, Differential Subordinations-Theory
and Applications, Marcel Dekker Inc., New York, Basel.
JJ
J
II
I
Go Back
[8] M. NUNOKAWA AND D.K. THOMAS, On convex and starlike functions
in a sector, J. Austral. Math. Soc. (series A), 60 (1996), 363–368.
Close
[9] S. PONNUSAMY, Differential subordinations concerning starlike functions, Proc. Indian Acad. Sci. (Math. Sci.), 104(2) (1994), 397–411.
Page 22 of 23
Quit
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au
[10] J. STANKIEWICZ, Some remarks concerning starlike functions, Bull.
Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phy., 18 (1970), 143–146.
Non-autonomous Differential
Subordinations Related to a
Sector
Sukhjit Singh and Sushma Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 23
J. Ineq. Pure and Appl. Math. 4(4) Art. 72, 2003
http://jipam.vu.edu.au