URL:https://doi.org/10.28924/2291-8639 DOI:10.28924/2291-8639-17-2019-974
ON PROPERTIES OF CERTAIN ANALYTIC MULTIPLIER TRANSFORM OF COMPLEX ORDER
DEBORAH OLUFUNMILAYO MAKINDE1,∗, SHAHRAM NAJAFZADEH2
1Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
2Department of Mathematics, Payame Noor University, P. O. Box: 19395–3697, Tehran, Iran
∗Corresponding author: [email protected]
Abstract. The focus of this paper is to investigate the subclassesS∗C(γ, µ, α, λ;b),T S∗C(γ, µ, α, λ;b) =
T∩S∗C(γ, µ, α, λ;b) and obtain the coefficient bounds as well as establishing its relationship with certain existing results in the literature.
1. Introduction
LetA be the class of normalized analytic functions f in the open unit disc U ={z ∈C :|z|<1} with
f(0) =f0(0) = 0 and of the form
f(z) =z+
∞ X
n=2
anzn, an ∈C, (1.1)
andS the class of all functions inAthat are univalent in U. Also, the subclass of functions inAthat are
of the form
f(z) =z− ∞ X
n=2
anzn, an ≥0, (1.2)
is denoted byT and the subclassesS∗(α), C(γ) are given respectively by
S∗(α) =
f ∈S: Re
zf0(z)
f(z)
> γ z∈U
(1.3)
Received 2019-07-05; accepted 2019-08-12; published 2019-11-01. 2010Mathematics Subject Classification. 30C45.
Key words and phrases. analyticity; univalent; linear transformation; coefficient bounds.
c
2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
C(α) =
f ∈S: Re
1 +zf
00(z)
f0(z)
> γ z∈U,≥γ <1
. (1.4)
Moreover, the class T S∗(γ) denoted by T ∩S∗(γ) which is the subclass of function f ∈ T such that f is
starlike of orderγ and respectively, T C(γ) is the class of functionf ∈T such thatf is convex of orderγ.
An interesting unification of the classesS∗(α) andC(γ) denoted byS∗C(γ, β) which satisfies the condition
Re
zf0(z) +βz2f0(z)
βzf0(z) + (1−β)f(z)
> γ 0≥γ <1, z∈U. (1.5)
has been extensively studied by different researchers, for example, see [6] and [1,2,3]. The special cases for
β= 0,1 are given byS∗(γ) andC((γ)) respectively.
Furthermore, the class T S∗C(γ, β) which is the subclass of function f ∈ T such that f belongs the class
S∗C(γ, β), was studied by Altintas et al. and other researchers. For details see [ 3, 5, 6 ].
Using the unification in (5), Nizami Mustafa [6] introduced and investigated the class S∗C(γ, β;τ) and
T S∗C(γ, β;τ), 0≤α <1;β∈[0,1];τ∈C which he defined as follows
A functionf ∈Sgiven by (1.1) is said to belong to the classS∗C(γ, β;τ) if the following condition is satisfied
Re
1 + 1 τ
zf0(z) +βz2f0(z)
βzf0(z) + (1−β)f(z)−1
> γ 0≥γ <1;β∈[0,1];τ∈C− {0}, z∈U. (1.6)
Meanwhile, the author in [4] defined a linear transformationDm α,λf by
Dα,λm f(z) =z+
∞ X
n=2
α
1 +λ(n+α−2)
1 +λ(α−1) m
anzn, 0≤λ≤1;α≥1;m∈N∪0 (1.7)
Motivated by the work of Mustafa in [6], we study the effect of the application of the linear operatorDmα,λf
on the unification of the classes of the functionsS∗C(γ, β;τ).
Now, we define the classS∗C(γ, α, λ;b) to be class of functionsf ∈S which satisfies the condition
Re (
1 + 1 b
"
z(Dm
α,λf)0(z) +µz2(Dmα,λf)00(z) µz(Dm
α,λf)0(z) + (1−µ)(D m α,λf)(z)
−1 #)
> γ,0≥γ <1, z∈U; 0≤λ, µ≤1;α≥1;m∈N∪0
(1.8)
Also, we denote byDT the subclass of the class of functions in (7) which is of the form
Dα,λm f(z) =z− ∞ X
n=2
α
1 +λ(n+α−2)
1 +λ(α−1) m
anzn, 0≤λ, µ≤1;α≥1;m∈N∪0 (1.9)
and denote byT S∗C(γ, µ, α, λ;b) =T∩S∗C(γ, µ, α, λ;b) which is the class of functionsf in (1.9) such that
f belong to the classS∗C(γ, µ, α, λ;b) =T∩S∗C(γ, µ, α, λ;b).
In this paper, we investigate the subclassesS∗C(γ, µ, α, λ;b) and
2. Coeffiecient bounds for the classes S Cα(γ, µ;b)and T S Cα(γ, µ;b)
Theorem 2.1. Let f be as defined in (1.1). Then the function Dmα,λf belongs to the classS∗C(γ, µ, α, λ;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;α≥1;m∈N∪0
if ∞ X n=2 α
1 +λ(n+α −2) 1 +λ(α−1)
m
[1 +µ(n−1)][n+|b|(1−γ)−1]
|an| ≤ |b|(1−γ)
The result is sharp for the function
Dmα,λf(z) =z+
|b|(1−γ)(1 +λ(α−1))m
α[1 +µ(n−1)][n+|b|(1−γ)](1 +λ(n+α−2))mz
n n≥2
Proof. By (1.8),f belong to the classS∗C(γ, µ, α, λ;b) if
Re (
1 + 1 b
"
z(Dm α,λf)
0(z) +µz2(Dm α,λf)
00(z)
µz(Dm
α,λf)0(z) + (1−µ)(Dmα,λf)(z) −1
#)
> γ
It suffices to show that:
1 b "
z(Dm α,λf)
0(z) +µz2(Dm α,λf)
00(z)
µz(Dm
α,λf)0(z) + (1−µ)(Dmα,λf)(z) −1 #
<1−γ (2.1)
Simple computation in (2.1), using (1.7), we have:
1 b "
z(Dmα,λf)0(z) +µz2(Dmα,λf)00(z)
µz(Dm
α,λf)0(z) + (1−µ)(D m α,λf)(z)
−1 # = 1 b
z+P∞ n=2nα
1+λ(n+ α−2) 1+λ(α−1)
m
anzn+µP ∞
n=2n(n−1)α
1+λ(n+ α−2) 1+λ(α−1)
m anzn
µz+P∞ n=2µnα
1+λ(n+α−2)
1+λ(α−1)
m
anzn+ (1−µ)
z+P∞ n=2α
1+λ(n+α−2)
1+λ(α−1)
m anzn
−1 = 1 b
z+P∞
n=2nα[1 +µ(n−1)]
1+λ(n+α−2)
1+λ(α−1)
m anzn
z+P∞
n=2α(1 +µ(n−1))
1+λ(n+α−2)
1+λ(α−1)
m anzn
−1 ≤1 b P∞
n=2α(n−1)[1 +µ(n−1)]
1+λ(n+α−2)
1+λ(α−1)
m |an|
1−P∞
n=2α(1 +µ(n−1))
1+λ(n+α−2)
1+λ(α−1)
m |an|
which is bounded by 1−γ if
P∞
n=2α(n−1)[1 +µ(n−1)]
1+λ(n+α−2)
1+λ(α−1)
m
|an| ≤ |b|(1−γ)1−P∞n=2α(1 +µ(n−1))
1+λ(n+α−2)
1+λ(α−1)
m |an|
which is equivalent to
P∞ n=2
h
α(n−1)[1 +µ(n−1)]1+1+λ(λn(+α−α−1)2)
m
+α|b|(1−γ)(1 +µ(n−1))1+1+λ(λn(+α−α−1)2)
mi
|an| ≤ |b|(1−γ)
Which implies that
∞ X
n=2
α
1 +λ(n+α−2)
1 +λ(α−1) m
[1 +µ(n−1)][n+|b|(1−γ)−1]
Thus, (2.1) is satisfied if (2.2) is satisfied.
Corollary 2.1. Let f be as defined in (1) and the functionDm
α,λf belongs to the class S∗C(γ, µ, α, λ;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;α≥1;m∈N∪0. Then
|an| ≤
|b|(1−γ)(1 +λ(α−1))m
α[1 +µ(n−1)][n+|b|(1−γ)−1](1 +λ(n+α−2))m
Corollary 2.2.Letf be as defined in (1.1). Then the functionDm
α,λf belongs to the classS∗C(γ, µ,1, λ, m;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
[(1 +λ(n−1))m[1 +µ(n−1)][n+|b|(1−γ)−1]]|an| ≤ |b|(1−γ) (2.3)
The result is sharp for the function
Dα,λm f(z) =z+ |b|(1−γ)
[1 +µ(n−1)][n+|b|(1−γ)−1](1 +λ(n−1))mz
n, n≥2
Corollary 2.3. Letf be as defined in (1.1). Then the functionDm
α,λfbelongs to the classS
∗C(γ, µ,1, λ,1;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
[(1 +λ(n−1)) [1 +µ(n−1)][n+|b|(1−γ)−1]]|an| ≤ |b|(1−γ) (2.4)
The result is sharp for the function
Dmα,λf(z) =z+ |b|(1−γ)
[1 +µ(n−1)][n+|b|(1−γ)−1](1 +λ(n−1))z
n, n ≥2
Corollary 2.4. Letf be as defined in (1.1). Then the functionDα,λm f belongs to the classS∗C(γ, µ,1,1,1;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
[n[1 +µ(n−1)][n+|b|(1−γ)−1]]|an| ≤ |b|(1−γ) (2.5)
The result is sharp for the function
Dmα,λf(z) =z+
|b|(1−γ)
n[1 +µ(n−1)][n+|b|(1−γ)−1]z
n, n≥2
Corollary 2.5. Letf be as defined in (1.1). Then the functionDm
α,λf belongs to the classS∗C(γ, µ,1,0,1;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
[[1 +µ(n−1)][n+|b|(1−γ)−1]]|an| ≤ |b|(1−γ) (2.6)
The result is sharp for the function
Dα,λm f(z) =z+ |b|(1−γ)
[1 +µ(n−1)][n+|b|(1−γ)−]z
n, n≥2
Corollary 2.6. Letf be as defined in (1.1). Then the functionDα,λf belongs to the classS C(γ,0,1, λ,0; 1),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
[[1 +µ(n−1)][n−γ]]|an| ≤1−γ (2.7)
The result is sharp for the function
Dα,λm f(z) =z+ 1−γ
[1 +µ(n−1)][n−γ]z
n, n≥2
This result agrees with the Corollary 2.1 in [6].
Corollary 2.7.Letf be as defined in (1.1). Then the functionDmα,λf belongs to the classS∗C(γ, µ,1, λ,0; 1),
0≥γ <1, z∈U; 0≤λ, µ≤1;m∈N∪0 if
∞ X
n=2
(n−γ)|an| ≤1−γ (2.8)
The result is sharp for the function
Dmα,λf(z) =z+ 1−γ n−γz
n, n≥2
This result agrees with the Corollary 2.2 in [6].
Theorem 2.2. Let f ∈DT. Then the functionDα,λm f belongs to the class DTS∗C(γ, µ, α, λ;b),
0≥γ <1, z∈U; 0≤λ, µ≤1;α≥1;m∈N∪0if and only if
∞ X
n=2
α(n−1)[1 +µ(n−1)][n+b(1−γ)] x
y m
|an| ≤ |b|(1−γ)
Proof. We shall prove only the necessity part of the Theorem as the sufficiency proof is similar to the proof
of Theorem 1.
Let f be as defined in (1.1) and Dm
α,λf belongs to the class T S∗C(γ, µ, α, λ;b), 0 ≥ γ < 1, z ∈ U; 0 ≤
λ, µ≤1;α≥1;m∈N∪0;b∈R− {0}, we have
Re (
1 + 1 b
"
z(Dm
α,λf)0(z) +µz2(Dmα,λf)00(z) µz(Dm
α,λf)0(z) + (1−µ)(D m α,λf)(z)
−1 #)
> γ (2.9)
Using (1.7) in (2.9) and by algebraic simplification, we have
Re
−P∞
n=2α(n−1)[1 +µ(n−1)]
1+λ(n+α−2)
1+λ(α−1)
m anzn
bnz−P∞
n=2α(1 +µ(n−1))
1+λ(n+α−2)
1+λ(α−1)
m anzn
o
≥γ−1
Choosingz to be real andz−→1, we have
−P∞
n=2α(n−1)[1 +µ(n−1)]
1+λ(n+α−2)
1+λ(α−1)
m an
bn1−P∞
n=2α(1 +µ(n−1))
1+λ(n+α−2)
1+λ(α−1)
m an
b∈R− {0}implies thatb could be greater or less than zero.
Letb >0 in (19), we have
− ∞ X
n=2
α(n−1)[1 +µ(n−1)] x
y m
an≥(γ−1)b (
1− ∞ X
n=2
α(1 +µ(n−1)) x
y m
an )
(2.11)
wherex= 1 +λ(n+α−2) andy= 1 +λ(α−1) From (20), we have
∞ X
n=2
α(n−1)[1 +µ(n−1)][n+b(1−γ)] x
y m
|an| ≤b(1−γ) (2.12)
Now supposeb <0, which implies thatb=−|b|and substitutingb=−|b| in (19), we have
P∞
n=2α(n−1)[1 +µ(n−1)]
x y
m an
|b|n1−P∞
n=2α(1 +µ(n−1))
x y
m an
o ≥ (2.13)
P∞
n=2α(n−1)[1 +µ(n−1)]
x y
m
|an| ≥(γ−1)|b| n
1−P∞
n=2α(1 +µ(n−1))
x y
m an
o
which implies
∞ X
n=2
α(n−1)[1 +µ(n−1)][n+b(1−γ)]
1 +λ(n+α−2)
1 +λ(α−1) m
|an| ≥ −b(1−γ) (2.14)
From (21) and (23), the proof of the necessity is completed.
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