In the present paper we investigate the majorization properties for certain classes of multivalent analytic functions defined by extended multiplier transformation. Moreover, we point out some new or known consequences of our main result

Jyoti Aggarwal

Rachna Mathur

CiteSeerX

Int. J. Open Problems Complex Analysis, Vol. 5, No. 2, July 2013
ISSN 2074-2827; Copyright c©ICSRS Publication, 2013
www.i-csrs.org
Majorization for Certain Classes
of Analytic Multivalent
Functions
Jyoti Aggarwal and Rachna Mathur
Department of Mathematics, Govt. Dugar (P.G.) College, Bikaner, (India)
e-mail: maths.jyoti86@gmail.com
Department of Mathematics, Govt. Dugar (P.G.) College, Bikaner, (India)
e-mail: rachnamathur@rediffmail.com
Abstract
In the present paper we investigate the majorization properties for
certain classes of multivalent analytic functions defined by extended
multiplier transformation. Moreover, we point out some new or known
consequences of our main result.
Keywords: Analytic functions, Multivalent functions, Starlike functions,
Subordination, Majorization property, Extended multiplier transformation.
2010 Mathematical Subject Classification: 30C45, 30C80.
1 Introduction
Let f and g be two analytic functions in the open unit disk U = {z ∈ C : |z| < 1}.
We say that f is majorized by g in U (see [20]), and we write
f(z) g(z) (z ∈ U),
if there exists a function ϕ, analytic in U such that
|ϕ(z)| ≤ 1 and f(z) = ϕ(z)g(z) (z ∈ U). (1.1)
42 Jyoti Aggarwal and Rachna Mathur
It may be noted that the notion of majorization is closely related to the concept
of quasi-subordination between analytic functions.
For two functions f and g analytic in U , we say that the function f is
subordinate to g in U , and we write
f(z) ≺ g(z),
if there exists a Schwarz function w, which (by definition) is analytic in U ,
with w(0) = 0 and |w(z)| < 1 for all z ∈ U , such that
f(z) = g (w(z)) (z ∈ U).
Let Ap denote the class of functions of the form
f(z) = zp +
∞∑
k=p+1
akz
k (p ∈ N := {1, 2, 3, . . . }) , (1.2)
which are analytic in the open unit disk U . For simplicity, we write A := A1.
Catas [1] extended the multiplier transformation and defined the operator
Iµp (n, λ) : Ap → Ap ,
on f ∈ Ap, defined by (1.2) by the following infinite series:
Iµp (n, λ)f(z) = zp +
∞∑
k=p+1
(
p+ λ+ µ(k − p)
p+ λ
)n
akz
k,
(λ ≥ 0; µ ≥ 0; p ∈ N; n ∈ N ∪ {0}) , (1.3)
where f ∈ Ap has the form (1.2). We note that:
I0p(0, 1)f(z) = f(z), I0p(1, 1) =
zf ′(z)
p
, and I0p(2, 1) =
zf ′(z)+z2f ′′(z)
p2
By specializing the parameters λ, µ and p, we obtain the following operators
studied by various authors:
(i) I1p(n, λ)f(z) = Ip(n, λ)f(z), see [2, 3, 4]
(ii) I1p(n, 0)f(z) = Dnpf(z) (see [5, 6, 7]
(iii) I11 (n, λ)f(z) = Inλf(z) (see [8, 9])
(iv) I11 (n, 0)f(z) = Dnf(z) (see [10, 11])
(v) Iµ1 (n, 0)f(z) = Dnµf(z) (see [12])
(vi) I11 (n, 1)f(z) = Inf(z) (see [13])
(vii) Iµp (n, 0)f(z) = Dnp,µf(z)
where Dnp,µf(z) is defined by
Dnp,µf(z) = z
p +
∞∑
k=p+1
(
p+µ(k−p)
p
)n
akz
k.
Majorization for Certain Classes of Analytic Multivalent Functions 43
Definition 1.1 A function f ∈ Ap is said to be in the class Sn,λp,q [A,B; γ], if
and only if
1 +
1
γ
(
z
(
Iµp (n, λ)f(z)
)(q+1)
(Iµp (n, λ)f(z))(q)
− p+ q + n
)
≺
1 + n
γ
+ Az
1 +Bz
, (1.4)
with −1 ≤ B < A ≤ 1, p ∈ N, n, q ∈ N0, λ ≥ 0; µ ≥ 0, |p+ λ| ≥
|γ(A−B) + (p− n+ λ)B|, γ ∈ C∗ := C \ {0}, where
(
Iµp (n, λ)f
)(q)
repre-
sents the q times derivative of Iµp (n, λ)f .
Clearly, we have the following relationships:
(i) Sn,0p,q [1,−1; γ] = Snp,q(γ);
(ii) Sn,0p,0 [1,−1; γ] = Sn(p, γ);
(iii) S0,0p,q [1,−1; γ] = Sp,q(γ);
(iv) S1,01,0 [1,−1; γ] = S(γ);
(v) S0,01,0 [1,−1; 1− α] = S∗(α), for 0 ≤ α < 1.
The class Sp,q(γ) was introduced by O. Altintaş and H. M. Srivastava [15].
The class S(γ) is said to be class of starlike functions of complex order γ ∈ C∗
in U , which were considered by Nasr and Aouf [21] and Wiatrowski [22], while
S∗(α) denotes the class of starlike functions of order α in U .
A majorization problem for the class S(γ) has recently been investigated by
Altintaş et al. [16], and majorization problems for the class S∗ = S∗(0) have
been investigated by MacGregor [20]. Very recently, Goyal and Goswami [17]
generalized these results for the fractional derivative operator. In the present
paper we investigate a majorization problem for the class Sn,λp,q [A,B; γ], and we
give some special cases of our main result obtained for appropriate choices of
the parameters A, B and λ. Further results on majorization problems can be
found in [18], [19] etc.
2 Majorization problem for the class Sn,λp,q [A,B; γ]
We begin by stating and proving the following main result.
Theorem 2.1 Let the function f ∈ Ap, and suppose that g ∈ Sn,λp,q [A,B; γ]. If(
Iµp (n, λ)f
)(q)
is majorized by
(
Iµp (n, λ)g
)(q)
in U , then∣∣∣(Iµp (n+ 1, λ)f(z))(q)∣∣∣ ≤ ∣∣∣(Iµp (n+ 1, λ)g(z))(q)∣∣∣ for |z| ≤ R, (2.1)
44 Jyoti Aggarwal and Rachna Mathur
where R = R(p, γ, λ, µ, n, A,B) is the smallest positive root of the equation∣∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}∣∣∣r3
− |p+ λ| (1 + 2|B|) r2
−
[
|µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}|+ 2|p+ λ|
]
r
+ |p+ λ| = 0
(2.2)
and −1 ≤ B < A ≤ 1; p ∈ N; n, q ∈ N0; λ ≥ 0; |p+ λ| ≥ |µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}|;
γ ∈ C∗.
Proof.
Since g ∈ Sn,λp,q [A,B; γ], we find from (1.4) that
1 +
1
γ
(
z
(
Iµp (n, λ)g(z)
)(q+1)
(Iµp (n, λ)g(z))(q)
− p+ q + n
)
=
1 + n
γ
+ Aw(z)
1 +Bw(z)
, (2.3)
where ω is analytic in U , with ω(0) = 0 and |ω(z)| < 1 for all z ∈ U . From
(2.3), we get
z
(
Iµp (n, λ)g(z)
)(q+1)
(Iµp (n, λ)g(z))(q)
=
p− q + [γ(A−B) +B(p− q − n)]w(z)
1 +Bw(z)
. (2.4)
Now, making use of the relation
µz
(
Iµp (n, λ)g(z)
)(q+1)
= (p+ λ)
(
Iµp (n+ 1, λ)g(z)
)(q)
− [p(1− µ) + qµ+ λ]
(
Iµp (n, λ)g(z)
)(q)
,
(2.5)
and after some simple computations, we get from (2.4) the following relation.(
Iµp (n+ 1, λ)g(z)
)(q)
(Iµp (n, λ)g(z))(q)
=
p+ λ+ [µγ(A−B)+B{µ(p− q − n) + p(1− µ) + µq + λ}]w(z)
(p+ λ)(1 +Bw(z)
which, after some easy calculations gives∣∣∣(Iµp (n, λ)g(z))(q)∣∣∣ ≤
|p+ λ|[1 + |B||z|]
∣∣∣(Iµp (n+ 1, λ)g(z))(q)∣∣∣
|p+ λ| − |µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}| |z|
.
(2.6)
Majorization for Certain Classes of Analytic Multivalent Functions 45
Next, since
(
Iµp (n, λ)f
)(q)
is majorized by
(
Iµp (n, λ)g
)(q)
in the open unit disk
U , from (1.1) we have(
Iµp (n, λ)f(z)
)(q)
= ϕ(z)
(
Iµp (n, λ)g(z)
)(q)
.
Differentiating the last equality with respect to z, and then multiplying by z,
we get
z
(
Iµp (n, λ)f(z)
)(q+1)
= zϕ′(z)
(
Iµp (n, λ)g(z)
)(q)
+ zϕ(z)
(
Iµp (n, λ)g(z)
)(q+1)
,
(2.7)
and using (2.5), we easily obtain
(p+ λ)
(
Iµp (n+ 1, λ)f(z)
)(q)
= zϕ′(z)
(
Iµ,(q)p (n, λ)g(z)
)(q)
+ (p+ λ)ϕ(z)
(
Iµp (n+ 1, λ)g(z)
)(q)
.
(2.8)
Thus, by noting that the Schwarz function ϕ satisfies the inequality (see, e.g.
Nehari [14])
|ϕ′(z)| ≤ 1− |ϕ(z)|
2
1− |z|2
, (z ∈ U) (2.9)
and using (2.6) and (2.9) in (2.8), we get∣∣∣(Iµp (n+ 1, λ)f(z))(q)∣∣∣ ≤(
|ϕ(z)|+ 1− |ϕ(z)|
2
1− |z|2
|p+ λ|(1 + |B||z|)|z|
|p+ λ| − |µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}| |z|
)
·
∣∣∣(Iµp (n+ 1, λ)g(z))(q)∣∣∣ ,
which upon setting
|z| = r and |ϕ(z)| = ρ, (0 ≤ ρ ≤ 1)
leads us to the inequality∣∣∣(Iµp (n+ 1, λ)f(z))(q)∣∣∣ ≤
Φ(r, ρ)
∣∣∣(Iµp (n+ 1, λ)g(z))(q)∣∣∣
(1− r2)
[
|p+ λ| −
∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}∣∣r] ,
(2.10)
where
Φ(r, ρ) = −|p+ λ|r(1 + |B|r)ρ2
+ (1− r2)
[
|p+ λ| −
∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}|r]ρ
+ |p+ λ|r(1 + |B|r).
46 Jyoti Aggarwal and Rachna Mathur
If we denote
Ψ(r, ρ) =
Φ(r, ρ)
(1− r2)
[
|p+ λ| −
∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}∣∣r] ,
then (2.10) becomes∣∣∣(Iµp (n+ 1, λ)f(z))(q)∣∣∣ ≤ Ψ(r, ρ) ∣∣∣(Iµp (n+ 1, λ)g(z))(q)∣∣∣ .
Now, in order to prove Theorem 2.1, we have to determine
R = max {r ∈ [0, 1] : Ψ(r, ρ) ≤ 1,∀ρ ∈ [0, 1]}
= max {r ∈ [0, 1] : χ(r, ρ) ≥ 0,∀ρ ∈ [0, 1]} ,
where
χ(r, ρ) = (1− ρ)(1− r2)
[
|p+ λ| −
∣∣µγ(A−B) +B{µ(p− q − n)
+ p(1− µ) + µq + λ}
∣∣r]
− (1− ρ2)|p+ λ|(1 + |B|r)r.
which shows that the inequality χ(r, ρ) ≥ 0 is equivalent to
ξ(r, ρ) =
[
|p+ λ| −
∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ)
+ µq + λ}
∣∣r] (1− r2)
− |p+ λ| (1 + |B|r) r(1 + ρ) ≥ 0,
while the function ξ(r, ρ) takes its minimum value at ρ = 1 withR = R(p, γ, λ, µ, n, A,B),
the smallest positive root of the equation (2.5), i.e.
min {ξ(r, ρ) : ρ ∈ [0, 1]} = ξ(r, 1) = φ(r),
where
φ(r) =
∣∣∣µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}∣∣∣r3
− |p+ λ| (1 + 2|B|) r2
−
[
|µγ(A−B) +B{µ(p− q − n) + p(1− µ) + µq + λ}|+ 2|p+ λ|
]
r
+ |p+ λ| = 0.
Thus it follows that φ(r) ≥ 0 for all r ∈ [0, R], which proves the desired result
(2.1).
Setting µ = 1 in (2.1) and (2.2), we get the following results of [2]:
Majorization for Certain Classes of Analytic Multivalent Functions 47
Corollary 2.1 Let the function f ∈ Ap and suppose that g ∈ Sn,λp,q (γ). If
(Ip(n, λ)f)(q) is majorized by (Ip(n, λ)g)(q) in U , then∣∣∣(Ip(n+ 1, λ)f(z))(q)∣∣∣ ≤ ∣∣∣(Ip(n+ 1, λ)g(z))(q)∣∣∣ for |z| ≤ r1,
where r1 is the smallest positive root of the following cubic equation∣∣∣γ(A−B) +B(p− n+ λ)∣∣∣r3 − |p+ λ| (1 + 2|B|) r2
−
[
|γ(A−B) +B(p− n+ λ)|+ 2|p+ λ|
]
r + |p+ λ| = 0
(2.11)
and −1 ≤ B < A ≤ 1; p ∈ N; n, q ∈ N0; λ ≥ 0; |p+ λ| ≥ |γ(A−B) +B(p− n+ λ)|;
γ ∈ C∗.
Setting A = 1 and B = −1 in Corollary 2.1, the equation (2.2) becomes∣∣∣2γ−(p−n+λ)∣∣∣r3−3|p+λ|r2−[|2γ−(p−n+λ)|+2|p+λ|]r+|p+λ| = 0. (2.12)
We see that r = −1 is one of the roots of this equation, and the other two
roots are given by
|2γ − p+ n− λ|r2 −
[
|2γ − p+ n− λ|+ 3|p+ λ|
]
r + |p+ λ| = 0,
so we may easily find the smallest positive root of (2.12). Hence, we have the
following result:
Corollary 2.2 Let the function f ∈ Ap and suppose that g ∈ Sn,λp,q (γ). If
(Ip(n, λ)f)(q) is majorized by (Ip(n, λ)g)(q) in U , then∣∣∣(Ip(n+ 1, λ)f(z))(q)∣∣∣ ≤ ∣∣∣(Ip(n+ 1, λ)g(z))(q)∣∣∣ for |z| ≤ r1,
where
r1 = r1(p, γ, λ, n) =
η −
√
η2 − 4|2γ − p+ n− λ||p+ λ|
2|2γ − p+ n− λ|
,
with η = 3|p + λ| + |2γ − p + n − λ|, and p ∈ N, n ∈ N0, λ ≥ 0, |p + λ| ≥
|2γ − p+ n− λ|, γ ∈ C∗.
Substituting λ = 0 in (2.11), we obtain a Sălăgean-type operator for mul-
tivalent function defined by
Dnpf(z) = Ip(n, 0) = zp +
∞∑
k=p+1
(
k
p
)n
akz
k, (n ∈ N0) ,
where f ∈ Ap is given by (1.2).
Putting A = 1, B = −1, and λ = 0 in Corollary 2.2, we obtain the following
result for the operator Dnp :
48 Jyoti Aggarwal and Rachna Mathur
Corollary 2.3 Let the function f ∈ Ap and suppose that g ∈ Snp,q(γ). If(
Dnpf
)(q)
is majorized by
(
Dnp g
)(q)
, then∣∣∣(Dn+1p f(z))(q)∣∣∣ ≤ ∣∣∣(Dn+1p g(z))(q)∣∣∣ for |z| ≤ r2,
where
r2 = r2(p, γ, n) =
η1 −
√
η21 − 4p|2γ − p+ n|
2|2γ − p+ n|
,
with η1 = 3p+ |2γ − p+ n|, and p ∈ N, n ∈ N0, γ ∈ C∗.
Further, putting n = q = 0, p = 1 in Corollary 2.3, we obtain the result of
Altinas et. al. [16]:
Corollary 2.4 Let the function f ∈ A be univalent in the open unit disk ∆,
and suppose that g ∈ S(γ). If f is majorized by g in ∆, then
|f ′(z)| ≤ |g′(z)| for |z| ≤ r3,
where
r3 = r3(γ) =
3 + |2γ − 1| −
√
9 + 2|2γ − 1|+ |2γ − 1|2
2|2γ − 1|
,
with γ ∈ C∗.
Also, putting γ = 1 in Corollary 2.4, we obtain the well-known result given
by MacGregor [20]:
Corollary 2.5 Let the function f ∈ A be univalent in the open unit disk ∆,
and suppose that g ∈ S∗. If f is majorized by g in ∆, then
|f ′(z)| ≤ |g′(z)| for |z| ≤ 2−
√
3.
3 Open Problem
In this paper we have solved majorization problem for the class of analytic
functions. Are these results can be obtained for meromorphic multivalent
function?
Majorization for Certain Classes of Analytic Multivalent Functions 49
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Majorization for Certain Classes of Analytic Multivalent Functions

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