EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 4, No. 1, 2011, 76-82
ISSN 1307-5543 – www.ejpam.com
On Certain Sufficient Conditions for Analytic Univalent
Functions
G. Murugusundaramoorthy1,∗ and N. Magesh2
1
2
School of Advanced Sciences, VIT University, Vellore - 632014, Tamilnadu, India.
Department of Mathematics, Government Arts College(Men), Krishnagiri-635001, Tamilnadu, India.
l
Abstract. In this paper, we introduce a new class Bm
(α, δ) of functions which is defined by hypergeometric function and obtain its relations with some well-known subclasses of analytic univalent functions. Furthermore, as a special case, we show that convex functions of order 1/2 are also members of
l
the family Bm
(α, δ).
2000 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Univalent functions, starlike functions, convex functions, Hadamard product,
generalized hypergeometric functions.
1. Introduction
Let A denote the class of functions of the form
f (z) = z +
∞
X
an z n
(1)
n=2
which are analytic and univalent in the open disc U = {z : |z| < 1} and normalized by
f (0) = 0 = f ′(0) − 1. We denote by S ∗ (α) and K(α) the subclasses of A consisting of all
functions which are, respectively starlike and convex of order α. Thus,
z f ′(z)
∗
S (α) = f ∈ A : Re
> α, 0 ≤ α < 1, z ∈ U
f (z)
and
K(α) =
∗
f ∈ A : Re 1 +
z f ′′(z)
f ′(z)
> α, 0 ≤ α < 1, z ∈ U .
Corresponding author.
Email addresses: gmsmoorthyyahoo.om (G. Murugusundaramoorthy), nmagi_2000yahoo.o.in
(N.Magesh)
http://www.ejpam.com
76
c 2010 EJPAM All rights reserved.
G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82
77
We notice that K(α) ⊂ S ∗ (α) ⊂ A . Further,
R(α) = f ∈ A : Re f ′(z) > α, 0 ≤ α < 1, z ∈ U .
If f and g are analytic functions in U, we say that f is subordinate to g, written f ≺ g,
if there is a function w analytic in U, with w(0) = 0, |w(z)| < 1, for all z ∈ U such that
f (z) = g(w(z)) for all z ∈ U. If g is univalent, then f ≺ g if and only if f (0) = g(0) and
f (U) ⊆ g(U).
∞
∞
P
P
For functions Φ ∈ A given by Φ(z) = z+
φn z n and Ψ ∈ A given by Ψ(z) = z+
ψn z n ,
n=2
n=2
we define the Hadamard product (or Convolution ) of Φ and Ψ by
(Φ ∗ Ψ)(z) = z +
∞
X
φn ψn z n , z ∈ U.
(2)
n=2
For complex parameters α1 , . . . , αl and β1 , . . . , βm (β j 6= 0, −1, . . . ; j = 1, 2, . . . , m) the
generalized hypergeometric function l Fm (z) is defined by
l Fm (z)
≡ l Fm (α1 , . . . αl ; β1 , . . . , βm ; z) :=
∞
X
(α1 )n . . . (αl )n z n
n=0
(β1 )n . . . (βm )n n!
(3)
(l ≤ m + 1; l, m ∈ N0 := N ∪ {0}; z ∈ U)
where N denotes the set of all positive integers and (α)n is the Pochhammer symbol defined
by
¨
1,
n=0
(α)n =
(4)
α(α + 1)(α + 2) . . . (α + n − 1), n ∈ N .
Let H(α1 , . . . αl ; β1 , . . . , βm ) : A → A be a linear operator defined by
[(H(α1 , . . . αl ; β1 , . . . , βm ))( f )](z) := z l Fm (α1 , α2 , . . . αl ; β1 , β2 . . . , βm ; z) ∗ f (z)
∞
X
Γn an z n
(5)
= z+
n=2
where
Γn =
(α1 )n−1 . . . (αl )n−1
(n − 1)!(β1)n−1 . . . (βm )n−1
.
(6)
l
For notational simplicity, we can use a shorter notation H m
[α1 ] for H(α1 , . . . αl ; β1 , . . . , βm )
l
in the sequel. The linear operator H m [α1 ] is called Dziok-Srivastava operator (see [3]), includes (as its special cases) various other linear operators introduced and studied by Bernardi
[1], Carlson and Shaffer [2], Libera [6], Livingston [7], Ruscheweyh [8] and Srivastava-Owa
[9].
l
For 0 ≤ α < 1 and δ ≥ 0, let Bm
(α, δ) consisting of functions of the form (1) and satisfying
the condition
Œδ
H l [α + 1] f (z) ‚
z
m 1
− 1 < 1 − α, z ∈ U.
(7)
l
z
H m [α1 ] f (z)
G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82
78
l
The class Bm
(α, δ) is a unified class of analytic functions which includes various new
subclasses of analytic univalent functions. We observe that
Example 1. If l = 2 and m = 1 with α1 = 1, α2 = 1, β1 = 1 then
(
)
δ
z
B12 (α, δ) := f ∈ A : f ′(z)
− 1 < 1 − α, δ ≥ 0, 0 ≤ α < 1, z ∈ U.
f (z)
The class B12 (α, δ) has been studied by Frasin and Jahangiri [5]. Further B12 (α, 2) has been
studied by Frasin and Darus [4]. Also we note that B12 (α, 1) ≡ S ∗ (α) and B12 (α, 0) ≡ R(α).
Example 2. If l = 2 and m = 1 with α1 = η + 1 (η > −1), α2 = 1, β1 = 1, then
)
(
δ
Dη+1 f (z) z
B(η, α, δ) := f ∈ A : − 1 < 1 − α, η > −1, δ ≥ 0, 0 ≤ α < 1, z ∈ U. ,
z
Dη f (z)
where Dη f (z) is called Ruscheweyh derivative operator [8] defined by
Dη f (z) :=
z
(1 − z)η+1
∗ f (z) ≡ H12 (η + 1, 1; 1) f (z).
Also we observe that B(0, α, 1) ≡ K(α).
Example 3. If l = 2 and m = 1 with α1 = µ + 1(µ > −1), α2 = 1, β1 = µ + 2, then
(
)
‚
Œδ
J
z
µ+1 f (z)
B(µ, α, δ) := f ∈ A : − 1 < 1 − α, µ > −1, δ ≥ 0, 0 ≤ α < 1, z ∈ U ,
z
Jµ f (z)
where Jµ is a Bernardi operator [1] defined by
Zz
µ+1
t µ−1 f (t)d t ≡ H12 (µ + 1, 1; µ + 2) f (z).
Jµ f (z) := µ
z
0
Note that the operator J1 was studied earlier by Libera [6] and Livingston [7].
Example 4. If l = 2 and m = 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), then
(
)
δ
L(a + 1, c) f (z) z
− 1 < 1 − α, δ ≥ 0, 0 ≤ α < 1, z ∈ U ,
B(a, c, α, δ) := f ∈ A : z
L(a, c) f (z)
where L(a, c) is a well-known Carlson-Shaffer linear operator [2] defined by
!
∞
X
(a)k k+1
L(a, c) f (z) :=
z
∗ f (z) ≡ H12 (a, 1; c) f (z).
(c)
k
k=0
The object of the present paper is to investigate the sufficient condition for functions to be
l
in the class Bm
(α, δ). Furthermore, as a special case, we show that convex functions of order
l
1/2 are also members of the family Bm
(α, δ).
G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82
79
2. Main Results
To prove our results we need the following lemma.
Lemma 1. [5] Let p be analytic in U with p(0) = 1 and suppose that
zp′(z)
3α − 1
Re 1 +
>
.
p(z)
2α
Then Re {p(z)} > α for z ∈ U and
1
2
(8)
≤ α < 1.
Using Lemma 1, we first prove the following theorem.
Theorem 1. Let f (z) be the functions of the form (1), δ ≥ 0 and
(α1 + 1)
where β =
3α−1
,
2α
l
Hm
[α1 + 2] f (z)
l [α + 1] f (z)
Hm
1
− δα1
l
Hm
[α1 + 1] f (z)
l [α ] f (z)
Hm
1
1
2
≤ α < 1. If
+ α1 (δ − 1) ≺ 1 + βz,
(9)
l
then f (z) ∈ Bm
(α, δ).
Proof. Define the function p(z) by
p(z) :=
l
[α1 + 1] f (z)
Hm
‚
z
Œδ
(10)
l [α ] f (z)
Hm
1
z
Then the function p(z) is analytic in U and p(0) = 1. Therefore, differentiating (10) logarithmically and the simple computation yields
zp′(z)
p(z)
= (α1 + 1)
l
Hm
[α1 + 2] f (z)
l [α + 1] f (z)
Hm
1
− δα1
l
Hm
[α1 + 1] f (z)
l [α ] f (z)
Hm
1
+ α1 (δ − 1) − 1.
By the hypothesis of the theorem, we have
zp′(z)
3α − 1
Re 1 +
>
.
p(z)
2α
Hence by Lemma 1, we have
(
‚
Œδ )
l
Hm
[α1 + 1] f (z)
z
Re
> α, z ∈ U.
l [α ] f (z)
z
Hm
1
l
Therefore in view of definition f (z) ∈ Bm
(α, δ).
For l = 2 and m = 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), we obtain the following
corollary.
G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82
Corollary 1. Let
1
2
≤ α < 1. If f ∈ A and
L[a + 2, c] f (z)
(a + 1)
Re
80
L[a + 1, c] f (z)
then
¨
Re
L[a + 1, c] f (z)
− δa
L[a, c] f (z)
L[a + 1, c] f (z)
L[a, c] f (z)
z
+ a(δ − 1) >
3α − 1
2α
, z ∈ U,
δ «
z
> α, z ∈ U.
Therefore f (z) ∈ B(a, c, α, δ).
Taking l = 2 and m = 1 with α1 = µ + 1(µ > −1), α2 = 1, β1 = µ + 2, we get
Corollary 2. Let
1
2
≤ α < 1. If f ∈ A and
¨
(µ + 2)
Re
Jµ+2 f (z)
Jµ+1 f (z)
then
− δ(µ + 1)
(
Re
Jµ+1 f (z)
Jµ f (z)
‚
Jµ+1 f (z)
z
«
+ (µ + 1)(δ − 1) >
2α
, z ∈ U,
Œδ )
> α, z ∈ U.
Jµ f (z)
z
3α − 1
Therefore f (z) ∈ B(µ, α, δ).
Choosing l = 2 and m = 1 with α1 = η + 1 (η > −1), α2 = 1, β1 = 1, we have
Corollary 3. Let
1
2
¨
Re
(η + 2)
≤ α < 1. If f ∈ A and
Dη+2 f (z)
Dη+1 f
(z)
then
− δ(η + 1)
¨
Re
Dη+1 f (z)
Dη+1 f (z)
Dη f
(z)
z
«
+ (η + 1)(δ − 1) >
δ «
Dη f (z)
z
3α − 1
> α, z ∈ U.
Therefore f (z) ∈ B(η, α, δ).
Choosing l = 2 and m = 1 with α1 = 1, α2 = 1 and β1 = 1, we have
Corollary 4. [5] If f ∈ A and
z f ′′(z)
z f ′(z)
3α − 1
Re 1 +
+δ 1−
>
, z ∈ U,
f ′(z)
f (z)
2α
then
¨
Re
Therefore f (z) ∈ B12 (α, δ).
f ′(z)
z
f (z)
δ «
> α, z ∈ U.
2α
, z ∈ U,
G. Murugusundaramoorthy and N. Magesh / Eur. J. Pure Appl. Math, 4 (2011), 76-82
Choosing l = 2 and m = 1 with α1 = 2, α2 = 1, β1 = 1, δ = 1 and α =
81
1
2
we have
1
2
we have
1
2
we have
1
2
we have
Corollary 5. If f ∈ A and
¨ 2
«
z f ′′′ + 6z f ′′(z) + 6 f ′(z) z f ′′(z)
3
Re
−
> , z ∈ U,
2 f ′(z) + z f ′′
f ′(z)
2
then
1+
Re
z f ′′(z)
> 0, z ∈ U.
f ′(z)
That is, f (z) ∈ K.
Choosing l = 2 and m = 1 with α1 = 2, α2 = 1, β1 = 1, δ = 0 and α =
Corollary 6. If f ∈ A and
¨
Re
z 2 f ′′′ + 4z f ′′(z) + 2 f ′(z)
«
>
2 f ′(z) + z f ′′
then
z f ′′(z)
f ′(z) +
Re
>
2
1
2
1
2
, z ∈ U,
, z ∈ U.
Choosing l = 2 and m = 1 with α1 = 1, α2 = 1, β1 = 1, δ = 1 and α =
Corollary 7. If f ∈ A and
Re
z f ′′(z)
f ′(z)
then
Re
−
z f ′(z)
>
f (z)
z f ′(z)
f (z)
>
1
2
−3
2
, z ∈ U,
, z ∈ U.
That is, f (z) is starlike of order 1/2 .
Choosing l = 2 and m = 1 with α1 = 1, α2 = 1, β1 = 1, δ = 0 and α =
Corollary 8. If f ∈ A and
Re
1+
then
Re
z f ′′(z)
f ′(z)
>
1
2
, z∈U
1
f ′(z) > , z ∈ U.
2
That is f (z) ∈ B(0, 1/2) = R1/2 .
ACKNOWLEDGEMENTS The authors would like to thank the referee(s) for their insightful
comments and suggestions.
REFERENCES
82
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