MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 3 (2013), 403–411
September 2013
MAJORIZATION PROBLEM FOR A SUBCLASS OF p-VALENTLY
ANALYTIC FUNCTIONS DEFINED BY THE WRIGHT
GENERALIZED HYPERGEOMETRIC FUNCTION
S. P. Goyal and Sanjay Kumar Bansal
Abstract. In this paper we investigate the majorization problem for a subclass of p-valently
analytic functions involving the Wright generalized hypergeometric function. Some useful consequences of the main result are mentioned and relevance with some of the earlier results are also
pointed out.
1. Introduction
Let f and g be analytic functions in the open unit disk
U = {z : z ∈ C, 0 ≤ |z| < 1} .
We say that f (z) is majorized by g(z) in U [16] and 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)
Note that majorization is closely related to the concept of quasi-subordination
between analytic functions [22].
Further, f is said to be subordinate to g in U, if there exists a Schwarz function
w(z) which is analytic in U, with w(0) = 0 and |w(z)| < 1 (z ∈ U) such that
f (z) = g(w(z)) (z ∈ U). We denote this subordination by
f (z) ≺ g(z) (z ∈ U).
In particular, if f (z) is univalent in U, we have the following equivalence (see [18])
f (z) ≺ g(z) (z ∈ U) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).
2010 Mathematics Subject Classification: 30C45
Keywords and phrases: Analytic, p-valent, majorization, Wright generalized hypergeometric
function.
403
404
S.P. Goyal, S.K. Bansal
q
[α1 , A1 , A, B; γ]
2. The class Sp,l,s
Let Ap denote the class of functions of the form
∞
P
f (z) = z p +
ak z k , (p ∈ N),
(2.1)
k=p+1
which are analytic and multivalent in the open unit disk U. In particular if p = 1,
then A1 = A.
For the functions fj ∈ Ap given by
∞
P
fj (z) = z p +
ak,j z k , (j = 1, 2; p ∈ N),
k=p+1
we define the Hadamard product (or convolution) of f1 and f2 by
∞
P
(f1 ∗ f2 )(z) = z p +
ak,1 ak,2 z k = (f2 ∗ f1 )(z).
k=p+1
Let l, s ∈ N. For positive real parameters αi , Ai ; βj , Bj (i = 1, . . . , l; j = 1, . . . , s),
with
s
l
P
P
1+
Bj −
Ai ≥ 0,
j=1
i=1
the Fox-Wright function l ψs is defined by (see [24])
∞ Ql
n
X
j=1 Γ(αj + nAj )z
Q
ψ
[(α
,
A
)
;
(β
,
B
)
;
z]
=
l s
j
j 1,l
j
j 1,s
s
j=1 Γ(βj + nBj )n!
n=0
(z ∈ U).
In particular, when Ai = Bj = 1 (i = 1, . . . , l; j = 1, . . . , s), we have the following
relationship:
l Fs (α1 , . . . , αl ; β1 , . . . , βs ; z)
= Ω l ψs [(α1 , 1)1,l ; (βj , 1)1,s ; z] (l ≤ s + 1; z ∈ U),
where
Γ(β1 ) . . . Γ(βs )
.
Γ(α1 ) . . . (αl )
The Fox-Wright generalized hypergeometric function has been used in many papers
on geometric function theory [see e.g. [3–5,8,9]).
Corresponding to the function φp defined by
Ω :=
φp [(αj , Aj )1,l ; (βj , Bj )1,s ; z] = Ωz p l ψs [(αj , Aj )1,l ; (βj , Bj )1,s ; z]
(z ∈ U),
Dziok and Raina [8] considered a linear operator
θp [(α1 , A1 ), . . . , (αl , Al ); (β1 , B1 ), . . . , (βs , Bs )] : Ap −→ Ap
defined by the following Hadamard product
θp,l,s (α1 , A1 )f (z) := φp [(αj , Aj )1,l ; (bj , βj )1,s ; z] ∗ f (z).
If f ∈ Ap is given by the equation (2.1), then we have
Ql
∞
Γ(αj + nAj )
P
p
Qsj=1
θp,l,s (α1 , A1 )f (z) = z + Ω
an+p z n+p
Γ(β
j + nBj )n!
n=1
j=1
(z ∈ U).
Majorization problem p-valently analytic functions
405
In particular, for Ai = Bj = 1(i = 1, . . . , l; j = 1, . . . , s), we get the linear operator
Hp,l,s (α1 , . . . , αl ; β1 , . . . , βs )f (z) = Hp,l,s (α1 )f (z)
= zp +
∞
X
Πlj=1 (αj )n
an+p z n+p
s
Π
(β
)
n!
j
n
n=1 j=1
(z ∈ U),
studied by Dziok and Srivastava [10]. It should be remarked that the linear operator
Hp,l,s (α1 ) is a generalization of many other linear operators considered earlier. In
particular, for f (z) ∈ Ap , we have the following observations:
(i) Hp,2,1 (a, 1; c)f (z) = Lp (a; c)f (z) (a ∈ R; c ∈ R \ Z−
0 ), where Lp (a; c) is the
linear operator studied earlier by Saitoh [21]. It yields another operator L(a, c)f (z)
introduced by Carlson and Shaffer [6] for p=1.
(ii) Hp,2,1 (n + p, 1; 1)f (z) = Dn+p−1 f (z) (n ∈ N; n > −p), the linear operator
studied by Goel and Sohi [11]. In the case p = 1, we get Dn f (z), the well-known
Ruscheweyh derivative [20] of f (z) ∈ A.
λ
(iii) Hp,2,1 (c, λ + p; a)f (z) = Ipλ (a, c)f (z) (a, n ∈ N \ Z−
0 , λ > −p), where Ip is
the linear operator studied earlier by Cho, Kwon and Srivastava [7].
(iv) Hp,2,1 (1, p + 1; n + p)f (z) = In,p f (z) (n ∈ Z; n > −p), where In,p− is the
extended integral operator considerd by Liu and Noor [15].
λ,p
(v) Hp,2,1 (p + 1, 1; p + 1 − λ)f (z) = Ωλ,p
is
z f (z) (−∞ < λ < p + 1), where Ωz
the extended fractional differintegral operator studied by Patel and Mishra [19].
It is easy to verify the following three-term recurrence relation for the operator
θp,l,s :
α1
(q+1)
(q)
z (θp,l,s (α1 , A1 )f (z))
=
(θp,l,s (α1 + 1, A1 )f (z))
A1
µ
¶
α1
(q)
−
− p + q (θp,l,s (α1 , A1 )f (z))
(p ∈ N, q ∈ N ∪ {0}, p > q). (2.2)
A1
Using of the operator θp,l,s {(α1 , A1 )}, we now introduce the following subclass of
functions f ∈ Ap :
Definition 1. A function f (z) ∈ Ap is said to be in the class
q
Sp,l,s
[α1 , A1 , A, B; γ] of p-valently analytic functions of complex order γ 6= 0 in U
if and only if
z(θp,l,s (α1 , A1 )f (z))(q+1)
(A − B)z
−p+q ≺γ
,
(q)
1 + Bz
(θp,l,s (α1 , A1 )f (z))
(2.3)
(z ∈ U, −1 ≤ B < A ≤ 1, αi , Ai , Bj , βj > 0, (i = 1, . . . , l; j = 1, . . . , s), p ∈ N, q ∈
N0 , p > q and γ ∈ C∗ = C − {0}).
q
q
q
Also, Tp,l,s
(α1 , A1 ; γ) = Sp,l,s
(α1 , A1 , 1, −1; γ), where Tp,l,s
(α1 , A1 ; γ) denote
the class of functions f ∈ Ap satisfying the following inequality.
½ µ
¶¾
1 z(θp,l,s (α1 , A1 )f (z))(q+1)
Re
−
p
+
q
> −1.
γ
(θp,l,s (α1 , A1 )f (z))(q)
406
S.P. Goyal, S.K. Bansal
Obviously we have the following relationships:
q
(i) Tp,1,0
(1; γ) = Spq (γ);
0
(ii) T1,1,0
(1; γ) = S(γ)(γ ∈ C∗ ) (see [17] and [23]);
0
(iii) T1,1,0
(1; 1 − α) = S ∗ (α)(0 ≤ α < 1).
Further we observe that:
(i) For q = 0, l = s + 1, α1 = β1 = p, A1 = B1 = 1, αi = Ai = βj = Bj = 1
q
(i = 2, 3, . . . , s + 1; j = 2, 3, . . . , s), our class Tp,l,s
(α1 , A1 , ; γ) reduces to the class
∗
Sp (γ) (γ ∈ C ) of p-valently starlike functions of order γ in U, where
µ
¶¶
¾
½
µ
1 zf 0 (z)
−p
> 0, p ∈ N, γ ∈ C∗ .
Sp (γ) = f (z) ∈ Ap : Re 1 +
γ
f (z)
(ii) For q = 0, l = s+1, α1 = p+1, β1 = p, A1 = B1 = 1, αi = Ai = βj = Bj = 1
q
(i = 2, 3, . . . , s + 1; j = 2, 3, . . . , s), Tp,l,s
(α1 , A1 , ; γ) reduces to the class Kp (γ)
∗
(γ ∈ C ) of p-valently convex functions of order γ in U, where
½
µ
µ
¶¶
¾
1
zf 00 (z)
∗
Kp (γ) = f (z) ∈ Ap : Re 1 +
1+ 0
−p
> 0, p ∈ N, γ ∈ C .
γ
f (z)
We shall require the following lemma
Lemma 1. [1] Let γ ∈ C∗ and f ∈ Kpq (γ). Then
1
(γ ∈ C∗ ).
Kpq (γ) ⊂ Spq ( γ)
2
Altintas et al. [1] investigated the majorization problem for the class S(γ)(γ ∈
C∗ ). Macgregor [16] investigated the same problem for the class S ∗ ≡ S ∗ (0), while
Goyal and Goswami [14] and Goyal, Bansal and Goswami [13], Goswami and Wang
[12] have investigated the majorization problem for certain subclasses of analytic
functions defined by derivatives and Saitoh operators. In this paper we investigate
q
majorization problem for the class Sp,l,s
[α1 , A1 , A, B; γ] which is an extension of
all the aforementioned and related subclasses. We also give some special cases of
our main result.
q
3. Majorization problem for the class Sp,l,s
[α1 , A1 , A, B; γ]
We shall assume throughout the paper that −1 ≤ B < A ≤ 1, γ ∈ C∗ ; p ∈ N
and q ∈ N0 , αi , Ai , βj , Bj > 0, (i = 1, . . . , l; j = 1, . . . , s) and p > q.
q
Theorem 1. Let the function f ∈ Ap and suppose that g ∈ Sp,l,s
[α1 , A1 , A, B; γ]
α1
α1
and A1 > |γ(A − B) + A1 B|. If
(θp,l,s (α1 , A1 )f (z))(q) ¿ (θp,l,s (α1 , A1 )g(z))(q) (z ∈ U),
then
|(θp,l,s (α1 + 1, A1 )f (z))(q) | ≤ |(θp,l,s (α1 + 1, A1 )g(z))(q) | for |z| ≤ r0 ,
(3.1)
407
Majorization problem p-valently analytic functions
where r0 = r0 (γ, α1 , A1 , A, B) is the smallest positive root of the equation
·
¸
α1
α1
α1
3 α1
2
r | B + γ(A − B)| − (
+ 2|B|)r − |γ(A − B) +
B| + 2 r +
= 0. (3.2)
A1
A1
A1
A1
q
Proof. Since g ∈ Sp,l,s
[α1 , A1 , A, B; γ], we find from (2.3) that
z(θp,l,s (α1 , A1 )g(z))(q+1)
γ(A − B)ω(z)
−p+q =
,
1 + Bω(z)
(θp,l,s (α1 , A1 )g(z))(q)
(3.3)
where ω(z) = c1 z +c2 z 2 +. . . , ω ∈ P, P denotes the well known class of the bounded
analytic functions in U (see [18]) and satisfies the conditions
ω(0) = 0, and |ω(z)| ≤ |z| (z ∈ U).
(3.4)
Using (2.2) and (3.4) in (3.3), we get
(q)
| (θp,l,s (α1 , A1 )g(z))
provided that
α1
A1
|≤
α1
A1
−
α1
A1 [1
α1
|A
B
1
+ |B||z|]
+ (A − B)γ||z|
| (θp,l,s (α1 + 1, A1 )g(z))
(q)
Next, since (θp,l,s (α1 , A1 )f (z)) is majorized by (θp,l,s (α1 , A1 )g(z))
unit disk U, we have from (1.1) that
(q)
|,
(3.5)
α1
> |A
B + γ(A − B)| and z ∈ U.
1
(θp,l,s (α1 , A1 )f (z))
(q)
(q)
= ϕ(z) (θp,l,s (α1 , A1 )g(z))
(q)
,
in the
(3.6)
where |φ(z)| ≤ 1.
Differentiating (3.6) with respect to ‘z’ and multiplying by ‘z’, we get
z(θp,l,s (α1 , A1 )f (z))(q+1)
= zϕ0 (z) (θp,l,s (α1 , A1 )g(z))
(q)
+ zϕ(z)(θp,l,s (α1 , A1 )g(z))(q+1) ,
which on using (2.2) once again, yields
(q)
(θp,l,s (α1 + 1, A1 )f (z))
A1 0
(q)
(q)
=
zϕ (z) (θp,l,s (α1 , A1 )g(z)) + ϕ(z) (θp,l,s (α1 + 1, A1 )g(z)) .
α1
(3.7)
Thus, noting that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [18])
|ϕ0 (z)| ≤
1 − |ϕ(z)|2
1 − |z|2
(z ∈ U),
(3.8)
and making use of (3.5) and (3.8) in (3.7), we get
¶
·
µ
1 − |ϕ(z)|2
(q)
| (θp,l,s (α1 + 1, A1 )f (z)) | ≤ |ϕ(z)| +
1 − |z|2
Ã
!#
|z|(1 + |B||z|)
(q)
| (θp,l,s (α1 + 1, A1 )g(z)) |,
α1
α1
−
|
B
+
(A
−
B)γ||z|
A1
A1
408
S.P. Goyal, S.K. Bansal
which upon setting |z| = r and |ϕ(z)| = ρ (0 ≤ ρ ≤ 1) leads to the inequality
| (θp,l,s (α1 + 1, A1 )f (z))
≤
(1 −
(q)
|
α1
r2 )[ A
1
υ(ρ)
(q)
| (θp,l,s (α1 + 1, A1 )g(z)) |,
α1
− |A
B
+
(A
−
B)γ|r]
1
where
·
¸
α1
α1
− | B + (A − B)γ)|r + r(1 + |B|r).
υ(ρ) = −r(1 + |B|r)ρ + (1 − r )ρ
A1
A1
2
2
takes its maximum value at ρ = 1 with r0 = r0 (γ, α1 , A1 , A, B) is the smallest
positive root of the equation (3.2). Furthermore, if 0 ≤ σ ≤ r0 , then the function
χ(ρ) defined by
χ(ρ) = −σ(1 + |B|σ)ρ2
hα
i
α1
B + (A − B)γ)|σ + σ(1 + |B|σ) (3.9)
A1
A1
is an increasing function on the interval 0 ≤ ρ ≤ 1, so that
hα
i
α1
1
χ(ρ) ≤ χ(1) = (1 − σ 2 )
− | B + (A − B)γ)|σ (0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r0 ).
A1
A1
Hence, upon setting ρ = 1 in (3.9), we conclude that (3.1) of Theorem 1 holds
true for |z| ≤ r0 = r0 (γ, α1 , A1 , A, B) where r0 (γ, α1 , A1 , A, B) is the smallest
positive¯ root of the equation
(3.2). In fact, as one can see easily, in any case,
¯
¯ α1
¯
either ¯ A1 B + (A − B)γ ¯ 6= 0, or if it is equal to zero, (3.2) has a unique root in
+ (1 − σ 2 )
1
−|
the interval (0, 1) and this is the smallest positive root of equation (3.2). This
completes the proof of the theorem.
4. Special cases
Setting Ai = Bj = 1, (i = 1, . . . , l; j = 1, . . . , s) in Theorem 1, we get the
following result:
q
Corollary 1. Let the function f ∈ Ap and suppose that g ∈ Sp,l,s
(α1 , A, B; γ)
(q)
and α1 > |γ(A − B) − α1 B|. If (Hp,l,s (α1 )f (z)) ¿ (Hp,l,s (α1 )g(z))(q) , z ∈ U,
then
|(Hp,l,s (α1 + 1)f (z))(q) | ≤ |(Hp,l,s (α1 + 1)g(z))(q) | for |z| ≤ r1 ,
where r1 = r1 (γ, α1 , A, B) is the smallest positive root of the equation
h
i
r13 |γ(A − B) + α1 B| − (α1 + 2|B|)r12 − |γ(A − B) + α1 B| + 2 r1 + α1 = 0.
Setting A = 1 and B = −1, in Theorem 1, we get
q
Corollary 2. Let the function f ∈ Ap and suppose that g ∈ Tp,l,s
(α1 , A1 , γ)
¯
¯
¯
α1
α1 ¯
(q)
(q)
and A1 > ¯2γ − A1 ¯. If (θp,l,s (α1 , A1 )f (z)) ¿ (θp,l,s (α1 , A1 )g(z)) in U, then
|(θp,l,s (α1 + 1, A1 )f (z))(q) | ≤ |(θp,l,s (α1 + 1, A1 )g(z))(q) |f or|z| ≤ r2 ,
(4.1)
409
Majorization problem p-valently analytic functions
where



r2 = r2 (γ, α1 , A1 ) =
(k = 2 +
α1
A1
+ |2γ −
α1
A1 |;


γ ∈ C∗ ).
q
k−
¯
¯
k2 −4 A1 ¯2γ− A1 ¯
α
¯
α
¯
1
1
2¯2γ− A1 ¯
α
,
if 2γ 6=
α1
A1 ,
if 2γ =
α1
A1
1
α1
α1 +2A1 ,
Remark 1. The expression under the square root in (4.1) is positive, since
¯
¯ µ
¯
¯¶2
¯
¯
¯
α1 ¯¯
α1 ¯¯
α1 ¯¯
α1 ¯¯
4α1
α1 ¯¯
2
¯
k −4
− 2γ −
+4+
+ 4 ¯2γ −
2γ −
=
A1 ¯
A1 ¯
A1 ¯
A1 ¯
A1
A1 ¯
¯
¯
¯¶2
¯
µ
¯
α1 ¯¯
α1 ¯¯
α1 ¯¯
+ 4 + 8 ¯¯2γ −
> 0.
>
− ¯2γ −
¯
A1
A1
A1 ¯
Further, putting l = s + 1, α1 = β1 = p, A1 = B1 = 1, αi = Ai = βj = Bj = 1,
(i = 2, . . . , s + 1; j = 2, . . . , s), in Corollary 2, we get
Corollary 3. [1] Let the function f ∈ Ap and suppose that g ∈ Spq . If
(f (z))(q) ¿ (g(z))(q) in U, then
|(f (z))(q+1) | ≤ |(g(z))(q+1) | for |z| ≤ r3 ,
where
p
k 2 − 4p|2γ − p + q|
2|2γ − p + q|
(k = 2 + p − q + |2γ − p + q|; and p ∈ N, q ∈ N0 , γ ∈ C∗ ).
r3 = r3 (γ, p, q) =
k−
(4.2)
Putting q = 0 in Corollary 3, we obtain
Corollary 4. Let the function f ∈ Ap and suppose that g ∈ Sp (γ). If
f (z) ¿ g(z) in U, then
|f 0 (z)| ≤ |g 0 (z)| for |z| ≤ r4 ,
where
r4 = r4 (γ, p) =
k−
p
k 2 − 4p|2γ − p|
2|2γ − p|
(4.3)
(k = 2 + p + |2γ − p|; and p ∈ N, γ ∈ C∗ ).
Putting q = 0, l = s + 1, β1 = p, α1 = p + 1, A1 = B1 = 1, αi = Ai = Bj =
βj = 1 (i = 2, 3, . . . , s + 1; j = 2, 3, . . . , s) in Corollary 2, with the aid of Lemma 1,
we get the following result.
Corollary 5. Let the function f ∈ Ap and suppose that g ∈ Kp (γ). If
f (z) ¿ g(z) in U, then
|f 0 (z)| ≤ |g 0 (z)|f or|z| ≤ r5 ,
where
r5 = r5 (γ, p) =
k−
(k = 2 + p + |γ − p|; and p ∈ N, γ ∈ C∗ ).
p
k 2 − 4p|γ − p|
2|γ − p|
(4.4)
410
S.P. Goyal, S.K. Bansal
Further, putting l = 2, s = 1, α1 = α, α2 = 1, β1 = β, A1 = A2 = B1 = 1 in
Corollary 2, we get
q
Corollary 6. Let the function f ∈ Ap and suppose that g ∈ Tp,2,1
(α1 , 1, β; γ)
and α ≥ |2γ − α|. If (Lp (α, β)f (z))(q) ¿ (Lp (α, β)g(z))(q) in U, then
|(Lp (α + 1, β)f (z))(q) | ≤ |(Lp (α + 1, β)g(z))(q) |f or|z| ≤ r6 ,
where
(
r6 = r6 (γ, α) =
√
k−
α
α+2
k2 −4α|2γ−α|
,
2|2γ−α|
if 2γ 6= α,
, if 2γ = α
(4.5)
(k = 2 + α + |2γ − α| and γ ∈ C \ {0}).
This is a known result obtained recently by Goyal, Bansal and Goswami [13].
Further, putting α = 1, we get a known result obtained by Altinas et al. [2],
which contains another known result obtained by MacGregor [16], when γ = 1.
Also, putting α = p + 1 and β = p − λ + 1 in Corollary 6, we get a known result
obtained by Goyal and Goswami [14].
Remark 2. In view of Remark 1 mentioned with Corollary 1, it can be
proved easily that the expressions under the square roots occurring in (4.2)–(4.5)
are positive.
Acknowledgement. The authors are thankful to the worthy referee for his
useful suggestions for the improvement of the paper. The first author (S P G) is
also thankful to CSIR, New Delhi, India for awarding Emeritius Scientist under
scheme No. 21(084)/10/EMR-II.
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(received 05.09.2011; in revised form 24.09.2012; available online 01.11.2012)
Department of Mathematics, University of Rajasthan, Jaipur-302004, India
E-mail: somprg@gmail.com
Department of Mathematics, Global Institute of Technology, Sitapura, Jaipur 302022, India
E-mail: bansalindian@gmail.com