General Mathematics Vol. 16, No. 2 (2008), 37-47
On a subclass of analytic functions1
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
Abstract
In the present paper, the authors study the differential inequality
·
³p
´′ ¸
p
m
m
>γ
q(z)
ℜ α q(z) + βz
where q(z) is an analytic function such that q(0) = 1 and α, β, γ, m
be real numbers and its applications to analytic functions in the open
unit disc E = {z : |z| < 1}.
Key Words: Univalent function, Starlike function, Convex function,
Multiplier transformation.
2000 Mathematical Subject Classification: 30C45.
1
Introduction
p
Let Ap denote the class of functions of the form f (z) = z +
∞
X
ak z k , p ∈
k=p+1
N = {1, 2, ....}, which are analytic in the open unit disc E = {z : |z| < 1}.
We write A1 = A. A function f ∈ Ap is said to be p-valent starlike of order
α(0 ≤ α < p) in E if
µ ′ ¶
zf (z)
> α, z ∈ E.
ℜ
f (z)
1
Received 3 July, 2007
Accepted for publication (in revised form) 25 September, 2007
37
38
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
We denote by Sp∗ (α), the class of all such functions. A function f ∈ Ap
is said to be p-valent convex of order α(0 ≤ α < p) in E if
µ
′′
zf (z)
ℜ 1+ ′
f (z)
¶
> α, z ∈ E.
Let Kp (α) denote the class of all those functions f ∈ Ap which are
multivalently convex of order α in E. Note that S1∗ (α) and K1 (α) are,
respectively, the usual classes of univalent starlike functions of order α and
univalent convex functions of order α, 0 ≤ α < 1, and will be denoted here
by S ∗ (α) and K(α), respectively. We shall use S ∗ and K to denote S ∗ (0)
and K(0), respectively which are the classes of univalent starlike (w.r.t. the
origin) and univalent convex functions.
For f ∈ Ap , we define the multiplier transformation Ip (n, λ) as
(1)
¶n
∞ µ
X
k+λ
ak z k , (λ ≥ 0, n ∈ Z).
Ip (n, λ)f (z) = z +
p
+
λ
k=p+1
p
The operator Ip (n, λ) has been recently studied by Aghalary et.al. [1].
Earlier, the operator I1 (n, λ) was investigated by Cho and Srivastava [3] and
Cho and Kim [4], whereas the operator I1 (n, 1) was studied by Uralegaddi
and Somanatha [10]. I1 (n, 0) is the well-known Sălăgean [9] derivative op∞
X
∗
n
k n ak z k , n ∈ N0 = N ∪ {0} and
erator D , defined as: D f (z) = z +
k=2
f ∈ A.
A function f ∈ A is said to belong to the class R if it satisfies the
condition
h ′
i
′′
ℜ f (z) + zf (z) > 0, z ∈ E
And it is well known that R ⊂ S ∗ .
A function f ∈ Ap is said to be in the class Sn (p, λ, α) for all z in E if it
satisfies
·
¸
α
Ip (n + 1, λ)f (z)
ℜ
> ,
Ip (n, λ)f (z)
p
On a subclass of analytic functions
39
for some α (0 ≤ α < p, p ∈ N). We note that S0 (1, 0, α) and S1 (1, 0, α) are
the usual classes S ∗ (α) and K(α) of starlike functions of order α and convex
functions of order α, respectively.
In 1989, Owa, Shen and Obradovic̆ [7] obtained a sufficient condition for
a function f ∈ A to belong to the class Sn (1, 0, α) = Sn (α).
Recently, Li and Owa [5] studied the operator I1 (n, 0).
In this paper, the authors study the differential inequality
·
³p
´′ ¸
p
m
m
q(z)
>γ
ℜ α q(z) + βz
where q(z) is an analytic function such that q(0) = 1 and α, β, γ, m be
real numbers. And the authors also discuss the applications of the above
mentioned differential inequality to multiplier transformation defined by (1)
in the open unit disc E = {z : |z| < 1}.
2
Preliminaries
We shall make use of the following lemma of Miller and Mocanu to prove
our result.
Lemma 2.1.([6])Let D be a subset of C × C (C is the complex plane) and
let Φ : D → C be a complex function. For u = u1 + iu2 , v = v1 + iv2
(u1 , u2 , v1 , v2 are reals), let Φ satisfy the following conditions:
(i) Φ is continuous in D;
(ii) (1, 0) ∈ D and ReΦ(1, 0) > 0; and
(iii) ℜΦ(iu2 , v1 ) ≤ 0 for all (iu2 , v1 ) ∈ D such that v1 ≤ −(1 + u22 )/2.
Let r(z) = 1 + r1 z + r2 z 2 + . . . be regular in the unit disc E, such that
(r(z), zr′ (z)) ∈ D for all z ∈ E. If
ℜ[Φ(r(z), zr′ (z))] > 0, z ∈ E,
then ℜr(z) > 0, z ∈ E.
We, now, state and prove our main results.
40
3
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
Main Results
Lemma 3.1. Let q(z) = 1 + q1 z + q2 z 2 + .......... be an analytic function in
E which satisfies the condition
·
³p
´′ ¸
p
m
m
q(z)
(2)
>γ
ℜ α q(z) + βz
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
h p
i β + pβ 2 + 4γ(α + β)
2m
ℜ
q(z) >
2(α + β)
Proof. Let us define q(z) as
1
(3)
(q(z)) 2m = δ + (1 − δ)r(z),
z∈E
where δ is the nonnegative root of the quadratic equation
(α + β)δ 2 − βδ − γ = 0
given by
(4)
δ=
β+
p
β 2 + 4γ(α + β)
2(α + β)
which satisfies the condition 0 ≤ δ < 1.
Therefore r(z) is an analytic function in E and r(z) = 1+r1 z +r2 z 2 +....
In view of (2), we have
¸
·
´′
³p
p
1
1
m
m
q(z) − γ = ℜ
(5)ℜ
[α[δ + (1 − δ)r(z)]2
α q(z) + βz
1−γ
1−γ
+2β(1 − δ)(δ + (1 − δ)r(z))zr′ (z) − γ].
If D = C × C, define Φ(u, v) : D → C as under
Φ(u, v) =
¤
1 £
α [δ + (1 − δ)u]2 + 2β(1 − δ)(δ + (1 − δ)u)v − γ .
1−γ
Then Φ(u, v) is continuous in D, (1, 0) ∈ D and ℜ Φ(1, 0) =
α−γ
1−γ
> 0.
On a subclass of analytic functions
41
Further, in view of (5), we get ℜΦ(r(z), zr′ (z)) > 0, z ∈ E.
Let u = u1 + iu2 , v = v1 + iv2 where u1 , u2 , v1 and v2 are all reals. Then,
1+u2
for (iu2 , v1 ) ∈ D, with v1 ≤ − 2 2 , we have
¤
1 £
α [δ + (1 − δ)iu2 ]2 + 2β(1 − δ)(δ + (1 − δ)iu2 )v1 − γ
ℜΦ(iu2 , v1 ) = ℜ
1−γ
¤
1 £
(α + β)δ 2 − 2βδ − γ
1−γ
≤ 0.
=
In view of (3), (4) and Lemma 2.1, proof now follows.
Theorem 3.1. If f ∈ Ap satisfies
 r
!′ 
Ãr
m Ip (n, λ)f (z)
m Ip (n, λ)f (z)
>γ
ℜ α
+ βz
p
z
zp
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
r
p
β + β 2 + 4γ(α + β)
2m Ip (n, λ)f (z)
>
.
ℜ
zp
2(α + β)
Proof. Let us write
Ip (n, λ)f (z)
q(z) =
zp
In view of Lemma 3.1, the proof follows.
Theorem 3.2. If f ∈ Ap satisfies
 s
Ãs
!′ 
Ip (n + 1, λ)f (z)
Ip (n + 1, λ)f (z) 
+ βz m
>γ
ℜ α m
Ip (n, λ)f (z)
Ip (n, λ)f (z)
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
s
p
β 2 + 4γ(α + β)
β
+
I
(n
+
1,
λ)f
(z)
p
>
.
ℜ 2m
Ip (n, λ)f (z)
2(α + β)
Proof. Let us write
Ip (n + 1, λ)f (z)
Ip (n, λ)f (z)
In view of Lemma 3.1, the proof follows.
q(z) =
42
4
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
Corollaries
By taking p = 1 and λ = 0 in Theorem 3.1. We have the following
Corollary 4.1. If f ∈ A satisfies
 r
!′ 
Ãr
n
n
m D f (z)
m D f (z)
>γ
+ βz
ℜ α
z
z
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
r
p
n
β + β 2 + 4γ(α + β)
2m D f (z)
>
.
ℜ
z
2(α + β)
By taking p = 1, n = 0 and λ = 0 in Theorem 3.1. We have the following
Corollary 4.2. If f ∈ A satisfies
 r
Ãr
!′ 
m f (z)
m f (z)
>γ
ℜ α
+ βz
z
z
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
r
p
β + β 2 + 4γ(α + β)
2m f (z)
>
.
ℜ
z
2(α + β)
By taking p = 1, n = 0, λ = 0, α = 1 and m = 2 in Theorem 3.1. We
have the following result of Aouf, M.K. and Hossen, H.M. [2] for n = 1 .
Corollary 4.3.If f ∈ A satisfies
r
Ãr
!′ 
2 f (z)
2 f (z)
>γ
ℜ
+ βz
z
z
where β and γ be real numbers such that β ≥ 0 and γ < 1
then
r
p
β + β 2 + 4γ(1 + β)
4 f (z)
>
.
ℜ
z
2(1 + β)
On a subclass of analytic functions
43
By taking p = 1, n = 1 and λ = 0 in Theorem 3.1. We have the following
Corollary 4.4. If f ∈ A satisfies
·
³p
´′ ¸
p
′
′
m
m
ℜ α f (z) + βz
>γ
f (z)
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
p ′
β + β 2 + 4γ(α + β)
2m
f (z) >
.
ℜ
2(α + β)
By taking p = 1, n = 1, λ = 0, α = 1 and m = 2 in Theorem 3.1. We
have the following result of Owa, S. and Wu, Z. [8].
Corollary 4.5. If f ∈ A satisfies
·
³p
´′ ¸
p
′
′
2
2
f (z) + βz
f (z)
>γ
ℜ
where β and γ be real numbers such that β ≥ 0 and γ < 1
then
p
p
β 2 + 4γ(1 + β)
β
+
ℜ 4 f ′ (z) >
.
2(1 + β)
1
By taking p = 1, n = 1, λ = 0 and m = in Theorem 3.1. We have the
2
following
Corollary 4.6. If f ∈ A satisfies
#
"
µh
i 2 ¶′
h ′ i2
′
>γ
ℜ α f (z) + βz f (z)
where α, β and γ be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
β + β 2 + 4γ(α + β)
′
ℜf (z) >
2(α + β)
i.e. f is close-to-convex and hence univalent.
By taking p = 1, n = 2 and λ = 0 in Theorem 3.1. We have the following
Corollary 4.7. If f ∈ A satisfies
·
³p
´′ ¸
p
′
′′
′
′′
m
m
ℜ α f (z) + zf (z) + βz
f (z) + zf (z)
>γ
44
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
p ′
β + β 2 + 4γ(α + β)
′′
2m
f (z) + zf (z) >
.
ℜ
2(α + β)
1
By taking p = 1, n = 2, λ = 0 and m = in Theorem 3.1. We have the
2
following
Corollary 4.8. If f ∈ A satisfies
"
·³
¸′ #
³ ′
´
´
2
2
′′
′
′′
ℜ α f (z) + zf (z) + βz f (z) + zf (z)
>γ
where α, β and γ be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
³ ′
´ β + pβ 2 + 4γ(α + β)
′′
ℜ f (z) + zf (z) >
2(α + β)
i.e. f ∈ R and hence f ∈ S ∗ .
By taking p = 1 and λ = 0 in Theorem 3.2. We have the following
Corollary 4.9. If f ∈ A satisfies
 s
!′ 
Ãs
n+1
n+1
D f (z)
D f (z) 
ℜ α m
+ βz m
>γ
n
D f (z)
Dn f (z)
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
s
p
n+1 f (z)
β + β 2 + 4γ(α + β)
2m D
ℜ
>
.
Dn f (z)
2(α + β)
1
By taking p = 1, λ = 0 and m =
in Theorem 3.2. We have the
2
following
Corollary 4.10. If f ∈ A satisfies


õ
µ n+1
¶2
¶2 !′
n+1
D f (z)
D f (z)
>γ
ℜ α
+ βz
n
D f (z)
Dn f (z)
On a subclass of analytic functions
45
where α, β and γ be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
β + β 2 + 4γ(α + β)
Dn+1 f (z)
>
ℜ n
D f (z)
2(α + β)
p
β + β 2 + 4γ(α + β)
, 0 ≤ δ < 1.
i.e. f ∈ S(δ), where δ =
2(α + β)
By taking p = 1, n = 0 and λ = 0 in Theorem 3.2. We have the following
Corollary 4.11. If f ∈ A satisfies
 s
Ãs
!′ 
′
′
zf (z)
zf (z) 
ℜ α m
+ βz m
>γ
f (z)
f (z)
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
#
" s
p
′
β + β 2 + 4γ(α + β)
zf
(z)
2m
>
.
ℜ
f (z)
2(α + β)
By taking p = 1, n = 0, λ = 0 and m = 12 in Theorem 3.2. We have the
following
Corollary 4.12. If f ∈ A satisfies

õ ′ ¶2 !′ 
µ ′ ¶2
zf (z)
zf (z)
>γ
+ βz
ℜ α
f (z)
f (z)
where α, β and γ be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
′
β + β 2 + 4γ(α + β)
zf (z)
>
ℜ
f (z)
2(α + β)
p
β + β 2 + 4γ(α + β)
i.e. f ∈ S ∗ (δ), where δ =
, 0 ≤ δ < 1.
2(α + β)
By taking p = 1, n = 1 and λ = 0 in Theorem 3.2. We have the following
Corollary 4.13. If f ∈ A satisfies
 s
Ãs
!′ 
′′
′′
zf (z)
zf (z) 
+ βz m 1 + ′
>γ
ℜ α m 1 + ′
f (z)
f (z)
46
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
where α, β, γ and m be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
#
" s
p
′′
β + β 2 + 4γ(α + β)
2m 1 + zf (z)
.
>
ℜ
f ′ (z)
2(α + β)
By taking p = 1, n = 1, λ = 0 and m = 21 in Theorem 3.2. We have the
following
Corollary 4.14. If f ∈ A satisfies


õ
¶2
¶2 !′
µ
′′
′′
zf (z)
zf (z)
>γ
1+ ′
+ βz
ℜ α 1 + ′
f (z)
f (z)
where α, β and γ be real numbers such that α ≥ 0, β ≥ 0 and α > γ
then
p
¶
µ
′′
β + β 2 + 4γ(α + β)
zf (z)
>
ℜ 1+ ′
f (z)
2(α + β)
p
β + β 2 + 4γ(α + β)
i.e. f ∈ K(δ), where δ =
, 0 ≤ δ < 1.
2(α + β)
References
[1] R. Aghalary, Ali,M. Rosihan, S. B. Joshi and V. Ravichandran,Inequalities for analytic functions defined by certain linear operators, International J. of Math. Sci., to appear.
[2] M. K. Aouf and H. M. Hossen, A note on certain subclass of analytic
functions, Mathematica (Cluj), 39(62),N o 1,1997,pp.3-5.
[3] N. E. Cho and H. M. Srivastava,Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math.
Comput. Modelling, 37(2003), 39-49.
[4] N. E. Cho and T. H. Kim,Multiplier transformations and strongly
close-to-convex functions, Bull. Korean Math. Soc., 40(2003), 399410.
On a subclass of analytic functions
47
[5] J. Li and S. Owa,Properties of the Sălăgean operator, Georgian Math.
J., 5, 4(1998), 361-366.
[6] S. S. Miller and P. T. Mocanu,Differential subordinations and inequalities in the complex plane, J. Diff. Eqns., 67(1987), 199-211.
[7] S. Owa, C.Y. Shen and M. Obradović,Certain subclasses of analytic
functions, Tamkang J. Math., 20(1989), 105-115.
[8] S. Owa, Z. Wu,A note on certain subclass of analytic functions, Math.
Japon, 34(1989), 413-416.
[9] G. S. Sălăgean,Subclasses of univalent functions, Lecture Notes in
Math., 1013, 362-372, Springer-Verlag, Heideberg,1983.
[10] B. A. Uralegaddi and C. Somanatha,Certain classes of univalent functions, in Current Topics in Analytic Function Theory, H. M. Srivastava
and S. Owa (ed.), World Scientific, Singapore, (1992), 371-374.
Sukhwinder Singh
Department of Applied Sciences
Baba Banda Singh Bahadur Engineering College
Fatehgarh Sahib -140407 (Punjab)
INDIA.
E-mail: ss billing@yahoo.co.in
Sushma Gupta, Sukhjit Singh
Department of Mathematics
Sant Longowal Institute of Engineering & Technology
Longowal-148106 (Punjab)
INDIA.
E-mail: sushmagupta1@yahoo.com
E-mail: sukhjit d@yahoo.com