10-21.2010. 111KB Jun 04 2011 12:07:21 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 21/2010
pp. 97-104
MAJORIZATION FOR CERTAIN CLASSES
OF ANALYTIC FUNCTIONS
Pranay Goswami and Zhi-Gang Wang
Abstract. In the present paper, we investigate the majorization properties
for certain classes of multivalent analytic functions defined by fractional derivatives.
Moreover, we point out some new or known consequences of our main result.
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: Analytic functions; Multivalent functions; Starlike functions; Subordination; Fractional calculus operators; Majorization property.
1. Introduction
Let f and g be analytic in the open unit disk
∆ = {z : z ∈ C , |z| < 1} .
(1)
We say that f is majorized by g in ∆ (see [3]) and write
f (z) ≪ g(z)
(z ∈ ∆),
(2)
if there exists a function ϕ, analytic in ∆ such that
|ϕ(z)| ≤ 1
and f (z) = ϕ(z)g(z)
(z ∈ ∆).
(3)
It may be noted here that (2) is closely related to the concept of quasi-subordination
between analytic functions.
For two functions f and g, analytic in ∆, we say that the function f is subordinate
to g in ∆, and write
f (z) ≺ g(z),
97
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
if there exists a Schwarz function ω, which is analytic in ∆ with
ω(0) = 0
and
|ω(z)| < 1
(z ∈ ∆)
such that
Indeed, it is known that
f (z) = g ω(z)
(z ∈ ∆).
f (z) ≺ g(z) =⇒ f (0) = g(0)
and f (∆) ⊂ g(∆).
Let Ap denote the class of functions of the form
f (z) = z p +
∞
X
ak z k
(p ∈ N := {1, 2, 3, ...}),
k=p+1
(4)
which are analytic in the open unit disk ∆. For simplicity, we write A1 =: A.
For two functions fj ∈ Ap (j = 1, 2) given by
fj (z) = z p +
∞
X
ak,j z k
k=p+1
(j = 1, 2; p ∈ N),
(5)
we define the Hadamard product (or convolution) of f1 and f2 by
(f1 ∗ f2 )(z) = z p +
∞
X
k=p+1
ak,1 ak,2 z k = (f2 ∗ f1 )(z).
In the following, we recall the definitions of fractional integral and fractional
derivative.
Definition 1. (See [6]; see also [8]) The fractional integral of order λ(λ > 0)
is defined, for a function f , analytic in a simply-connected of the complex plane
containing origin by
Zz
f (ξ)
1
−λ
dξ,
(6)
Dz f (z) =
Γ(λ)
(z − ξ)1−λ
0
where the multiplicity of
z − ξ > 0.
(z − ξ)λ−1
is removed by requiring log(z − ξ) to be real when
Definition 2. (See [6]; see also [8]) Under the hypothesis of Definition 1, the fractional derivative of f of order λ(λ ≥ 0)
Rz f (ξ)
1
dξ (0 ≤ λ < 1),
λ
Γ(1−λ)
(z−ξ)λ
Dz f (z) =
(7)
0
dn λ−n
f (z)
(n ≤ λ < n + 1; n ∈ N ∪ {0}),
dz n Dz
98
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
where the multiplicity of (z − ξ)λ−1 is removed by requiring log(z − ξ) to be real when
z − ξ > 0.
Recently, Patel and Mishra [7] defined the extended fractional differ-integral
operator
Ω(λ,p)
: A p → Ap
z
for a function f ∈ Ap and for a real number λ(−∞ < λ < p + 1) by
f (z) =
Ω(λ,p)
z
Γ(p + 1 − λ) λ λ
z Dz f (z),
Γ(p + 1)
(8)
where Dzλ f (z) is the fractional integral of f of order λ if 0 ≤ λ < p + 1. It is easy to
see from (8) that for a function f of the form (1), we have
Ω(λ,p)
f (z)
z
∞
X
Γ(n + p + 1)Γ(p + 1 − λ)
an z n
=z +
Γ(p + 1)Γ(n + p + 1 − λ)
p
n=p+1
(z ∈ ∆)
and
z(Ω(λ,p)
f (z))′ = (p − λ)Ω(λ+1,p)
f (z) + λΩ(λ,p)
f (z)
z
z
z
(−∞ < λ < p; z ∈ ∆). (9)
Definition 3. A function f (z) ∈ Ap is said to be in the class Spλ,j [A, B; γ] of
p-valent functions of complex order γ 6= 0 in ∆ if and only if
(j+1)
(λ,p)
1 + Az
z
Ω
f
(z)
z
1
(10)
1+
≺
−
p
+
j
(j)
γ
1 + Bz
(λ,p)
Ωz f (z)
(z ∈ ∆; −1 ≤ B < A ≤ 1; p ∈ N; j ∈ N0 := N ∪ {0}; γ ∈ C − {0}; −∞ < λ < p).
Clearly, we have the following relationships:
1. Spλ,j [1, −1; γ] = Spλ,j (γ);
2. S10,0 [1, −1; γ] = S(γ)
(γ ∈ C − {0});
3. S11,0 [1, −1; γ] = K(γ)
(γ ∈ C − {0});
4. S10,0 [1, −1; 1 − α] = S ∗ (α)
for 0 ≤ α < 1.
99
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
The class Spλ,j (γ) was introduced by Goyal and Goswami [2]. The classes S(γ) and
K(γ) are said to be classes of starlike and convex of complex order γ 6= 0 in ∆ which
were considered by Naser and Aouf [4] and Wiatrowski [9], and S ∗ (α) denote the
class of starlike functions of order α in ∆.
An majorization problem for the class S(γ) has recently been investigated by
Altinas et al. [1]. Also, majorization problems for the class S ∗ = S ∗ (0) have been
investigated by MacGregor [3]. In the present paper, we investigate a majorization
problem for the class Spλ,j [A, B; γ].
2. Majorization problem for the class Spλ,j [A, B; γ]
We begin by proving the following result.
Theorem 1. Let the function f ∈ Ap and suppose that g ∈ Spλ,j [A, B; γ]. If
(j) is majorized by (Ωλ,p g(z))(j) in ∆, then
(Ωλ,p
z f (z))
z
(j)
(j)
λ+1,p
λ+1,p
Ωz
(|z| ≤ r0 ),
(11)
g(z)
f (z)
≤ Ωz
where r0 = r0 (p, γ, λ, A, B) is smallest the positive root of the equation
r3 |γ(A−B)−(p−λ)B|−[(p−λ)+2|B|]r2 −[|γ(A−B)−(p−λ)B|+2]r +(p−λ) = 0,
(12)
(−1 ≤ B < A ≤ 1; p ∈ N; γ ∈ C − {0}).
Proof. Since g ∈ Spλ,j [A, B; γ], we find from (10) that
(j+1)
(λ,p)
1 z Ωz g(z)
1 + Aw(z)
1+
(j) − p + j = 1 + Bw(z) ,
γ
(λ,p)
Ωz g(z)
(13)
where ω is analytic in U with
ω(0) = 0
and
| ω(z) |< 1
(z ∈ U).
From (13), we get
(j+1)
(λ,p)
z Ωz g(z)
(p − j) + [(γ(A − B) + (p − j)B]w(z)
.
(j) =
1 + Bw(z)
(λ,p)
Ωz g(z)
100
(14)
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
By noting that
(j)
(j)
,
+ (λ − j) Ω(λ,p)
g(z)
g(z)
g(z))(j+1) = (p − λ) Ω(λ+1,p)
z(Ω(λ,p)
z
z
z
by virtue of (14) and (15), we get
(j)
(j)
(λ,p)
(λ+1,p)
(p
−
λ)[1
+
|B||z|]
Ωz g(z)
≤
.
g(z)
(p − λ) − |γ(A − B) − (p − λ)B||z| Ωz
(15)
(16)
(j)
(j)
(λ,p)
(λ,p)
in the unit disk ∆, from
is majorized by Ωz g(z)
Next, since Ωz f (z)
(3), we have
(j)
(j)
(λ,p)
=
ϕ(z)
Ω
g(z)
Ω(λ,p)
f
(z)
z
z
Differentiating it with respect to z and multiplying by z, we get
(j+1)
(j)
(j+1)
(λ,p)
′
(λ,p)
.
+
zϕ(z)
Ω
g(z)
=
zϕ
(z)
Ω
g(z)
z Ω(λ,p)
f
(z)
z
z
z
(17)
Using (15), in the above equation, it yields
(j)
(j)
(j)
zϕ′ (z) (λ,p)
(λ+1,p)
=
Ω(λ+1,p)
f
(z)
+
ϕ(z)
Ω
g(z)
Ω
g(z)
z
z
(p − λ) z
(18)
Thus, by noting that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [5])
|ϕ′ (z)| ≤
1 − |ϕ(z)|2
1 − |z|2
(z ∈ ∆)
(19)
and making use of (16) and (19) in (18), we get
(j)
(λ+1,p)
Ωz
f (z)
(j)
(λ+1,p)
|z|(1 + |B||z|)
1 − |ϕ(z)|2
,
Ω
g(z)
≤ |ϕ(z)| +
1 − |z|2 [(p − λ) − |γ(A − B) − (p − λ)B||z|] z
(20)
which upon setting
|z| = r
and |ϕ(z)| =ρ
(0 ≤ ρ ≤ 1)
leads us to the inequality
(j)
λ+1,p
Φ(ρ)
Ωz
≤
f (z)
(1 − r2 )(p − λ) − |γ(A − B) − (p − λ)B|r
101
(j)
λ+1,p
Ωz
,
g(z)
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
where
Φ(ρ) = −r(1+|B|r)ρ2 +(1−r2 )[(p−λ)−|γ(A−B)+(p−λ)B|r)]ρ+r(1+|B|r) (21)
takes its maximum value at ρ = 1, with r0 = r0 (p, γ, λ, A, B), where r0 (p, γ, λ, A, B)
is smallest the positive root of the equation (12). Furthermore, if 0 ≤ ρ ≤ r0 (p, γ, λ, A, B),
then the function ψ(ρ) defined by
ψ(ρ) = −σ(1+|B|σ)ρ2 +(1−σ 2 )[(p−λ)−|γ(A−B)+(p−λ)B|σ)]ρ+σ(1+|B|σ) (22)
is a increasing function on the interval 0 ≤ ρ ≤ 1, so that
ψ(ρ) ≤ ψ(1) = (1 − σ 2 )[(p − λ) − |γ(A − B) + (p − λ)B|σ)]
(0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r0 (p, γ, λ, A, B)).
(23)
Hence upon setting ρ = 1, in (22), we conclude that (11) of Theorem (1) holds true
for
|z| ≤ r0 = r0 (p, γ, λ, A, B),
where r0 (p, γ, λ, A, B) is the smallest positive root of the equation (12). This completes the proof of the Theorem 1.
Setting A = 1 and B = −1 in Theorem 1, we get the following result.
Corollary 1. (See [2]) Let the function f ∈ Ap and suppose that g ∈ Spλ,j (γ). If
(j)
(j)
λ,p
in ∆, then
is
majorized
by
Ω
g(z)
Ωλ,p
f
(z)
z
z
where
(j)
(j)
λ+1,p
λ+1,p
Ωz
g(z)
f (z)
≤ Ωz
r1 = r1 (p, γ, λ) =
k−
f or |z| ≤ r1 ,
p
k 2 − 4(p − λ)|2γ − p + λ|
,
2|2γ − p + λ|
(24)
(25)
(k = 2 + p − λ + |2γ − p + λ|; p ∈ N; γ ∈ C − {0}).
Putting A = 1, B = −1, and j = 0 in Theorem 1, we obtain the following result.
Corollary 2. Let the function f ∈ A and suppose that g ∈ Spλ,0 (γ). If Ωλ,p
z f (z) is
λ,p
majorized by (Ωz g(z)) in ∆, then
λ+1,p
g(z) f or |z| ≤ r1 ,
f (z) ≤ Ωλ+1,p
Ωz
z
where r1 is given by (25).
102
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
Further putting λ = 0, j = 0 in Corollary 2, we get
Corollary 3. (See [1]) Let the function f ∈ A and suppose that g ∈ Spλ,0 (γ). If
λ,p
Ωλ,p
z f (z) is majorized by Ωz g(z) in ∆, then Let the function f (z) ∈ A be analytic
in th open unit disk ∆ and suppose that g ∈ S(γ). If f (z) is majorized by g(z) in
∆, then
|f ′ (z)| ≤ |g ′ (z)|
(|z| ≤ r2 )
where
r2 = r2 (γ) =
3 + |2γ − 1| −
p
9 + 2|2γ − 1| + |2γ − 1|2
.
2|2γ − 1|
For γ = 1, Corollary 3 reduces to the following result:
Corollary 4. (See [3]) Let the function f ∈ A be an analytic and univalent in the
open unit disk ∆ and suppose that g ∈ S ∗ = S ∗ (0). If f (z) is majorized by g(z) in
∆, then
√
|f ′ (z)| ≤ |g ′ (z)|
(|z| ≤ 2 − 3).
Acknowledgements. The first-named author is grateful to Prof. S. P. Goyal,
University of Rajasthan, for his guidance and encouragement.
References
[1] O. Altinas, O. Ozkan and H. M. Srivastava, Majorization by starlike functions
of complex order, Complex Var. 46 (2001), 207–218.
[2] S. P. Goyal and P. Goswami, Majorization for certain classes of analytic
functions defined by fractional derivatives, Appl. Math. Lett. (2009), in press.
[3] T. H. MacGreogor, Majorization by univalent functions, Duke Math. J. 34
(1967), 95–102.
[4] M. A. Naser and M. K. Aouf, Starlike function of complex order, J. Natur.
Sci. Math. 25 (1985), 1–12.
[5] Z. Nehari, Conformal mapping ,MacGra-Hill Book Company, New York,
Toronto and London (1955).
[6] S. Owa, On distortion theorems I, Kyunpook Math. J. 18 (1978), 53–59.
[7] J. Patel and A. K. Mishra, On certain subclasses of multivalent functions
associated with an extended fractional differintegral operator, J. Math. Anal. Appl.
332 (2007), 109–122.
[8] H. M. Srivastava, S. Owa (Eds.), Univalent Functions, Fractional Calculus,
and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John
Wiley and Sons, New York, 1989.
103
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
[9] P. Wiatrowski, On the coefficients of some family of holomorphic functions,
Zeszyry Nauk. Uniw. Lddz. Nauk. Mat.-Przyrod., 39 (1970), 75–85.
Pranay Goswami
Department of Mathematics,
Amity University Rajasthan,
Jaipur (India) - 302002
email: [email protected]
Zhi-Gang Wang
School of Mathematics and Computing Science,
Changsha University of Science and Technology,
Yuntang Campus, Changsha 410114, Hunan,
People’s Republic of China
email: [email protected]
104
ISSN: 1582-5329
No. 21/2010
pp. 97-104
MAJORIZATION FOR CERTAIN CLASSES
OF ANALYTIC FUNCTIONS
Pranay Goswami and Zhi-Gang Wang
Abstract. In the present paper, we investigate the majorization properties
for certain classes of multivalent analytic functions defined by fractional derivatives.
Moreover, we point out some new or known consequences of our main result.
2000 Mathematics Subject Classification: 30C45.
Keywords and phrases: Analytic functions; Multivalent functions; Starlike functions; Subordination; Fractional calculus operators; Majorization property.
1. Introduction
Let f and g be analytic in the open unit disk
∆ = {z : z ∈ C , |z| < 1} .
(1)
We say that f is majorized by g in ∆ (see [3]) and write
f (z) ≪ g(z)
(z ∈ ∆),
(2)
if there exists a function ϕ, analytic in ∆ such that
|ϕ(z)| ≤ 1
and f (z) = ϕ(z)g(z)
(z ∈ ∆).
(3)
It may be noted here that (2) is closely related to the concept of quasi-subordination
between analytic functions.
For two functions f and g, analytic in ∆, we say that the function f is subordinate
to g in ∆, and write
f (z) ≺ g(z),
97
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
if there exists a Schwarz function ω, which is analytic in ∆ with
ω(0) = 0
and
|ω(z)| < 1
(z ∈ ∆)
such that
Indeed, it is known that
f (z) = g ω(z)
(z ∈ ∆).
f (z) ≺ g(z) =⇒ f (0) = g(0)
and f (∆) ⊂ g(∆).
Let Ap denote the class of functions of the form
f (z) = z p +
∞
X
ak z k
(p ∈ N := {1, 2, 3, ...}),
k=p+1
(4)
which are analytic in the open unit disk ∆. For simplicity, we write A1 =: A.
For two functions fj ∈ Ap (j = 1, 2) given by
fj (z) = z p +
∞
X
ak,j z k
k=p+1
(j = 1, 2; p ∈ N),
(5)
we define the Hadamard product (or convolution) of f1 and f2 by
(f1 ∗ f2 )(z) = z p +
∞
X
k=p+1
ak,1 ak,2 z k = (f2 ∗ f1 )(z).
In the following, we recall the definitions of fractional integral and fractional
derivative.
Definition 1. (See [6]; see also [8]) The fractional integral of order λ(λ > 0)
is defined, for a function f , analytic in a simply-connected of the complex plane
containing origin by
Zz
f (ξ)
1
−λ
dξ,
(6)
Dz f (z) =
Γ(λ)
(z − ξ)1−λ
0
where the multiplicity of
z − ξ > 0.
(z − ξ)λ−1
is removed by requiring log(z − ξ) to be real when
Definition 2. (See [6]; see also [8]) Under the hypothesis of Definition 1, the fractional derivative of f of order λ(λ ≥ 0)
Rz f (ξ)
1
dξ (0 ≤ λ < 1),
λ
Γ(1−λ)
(z−ξ)λ
Dz f (z) =
(7)
0
dn λ−n
f (z)
(n ≤ λ < n + 1; n ∈ N ∪ {0}),
dz n Dz
98
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
where the multiplicity of (z − ξ)λ−1 is removed by requiring log(z − ξ) to be real when
z − ξ > 0.
Recently, Patel and Mishra [7] defined the extended fractional differ-integral
operator
Ω(λ,p)
: A p → Ap
z
for a function f ∈ Ap and for a real number λ(−∞ < λ < p + 1) by
f (z) =
Ω(λ,p)
z
Γ(p + 1 − λ) λ λ
z Dz f (z),
Γ(p + 1)
(8)
where Dzλ f (z) is the fractional integral of f of order λ if 0 ≤ λ < p + 1. It is easy to
see from (8) that for a function f of the form (1), we have
Ω(λ,p)
f (z)
z
∞
X
Γ(n + p + 1)Γ(p + 1 − λ)
an z n
=z +
Γ(p + 1)Γ(n + p + 1 − λ)
p
n=p+1
(z ∈ ∆)
and
z(Ω(λ,p)
f (z))′ = (p − λ)Ω(λ+1,p)
f (z) + λΩ(λ,p)
f (z)
z
z
z
(−∞ < λ < p; z ∈ ∆). (9)
Definition 3. A function f (z) ∈ Ap is said to be in the class Spλ,j [A, B; γ] of
p-valent functions of complex order γ 6= 0 in ∆ if and only if
(j+1)
(λ,p)
1 + Az
z
Ω
f
(z)
z
1
(10)
1+
≺
−
p
+
j
(j)
γ
1 + Bz
(λ,p)
Ωz f (z)
(z ∈ ∆; −1 ≤ B < A ≤ 1; p ∈ N; j ∈ N0 := N ∪ {0}; γ ∈ C − {0}; −∞ < λ < p).
Clearly, we have the following relationships:
1. Spλ,j [1, −1; γ] = Spλ,j (γ);
2. S10,0 [1, −1; γ] = S(γ)
(γ ∈ C − {0});
3. S11,0 [1, −1; γ] = K(γ)
(γ ∈ C − {0});
4. S10,0 [1, −1; 1 − α] = S ∗ (α)
for 0 ≤ α < 1.
99
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
The class Spλ,j (γ) was introduced by Goyal and Goswami [2]. The classes S(γ) and
K(γ) are said to be classes of starlike and convex of complex order γ 6= 0 in ∆ which
were considered by Naser and Aouf [4] and Wiatrowski [9], and S ∗ (α) denote the
class of starlike functions of order α in ∆.
An majorization problem for the class S(γ) has recently been investigated by
Altinas et al. [1]. Also, majorization problems for the class S ∗ = S ∗ (0) have been
investigated by MacGregor [3]. In the present paper, we investigate a majorization
problem for the class Spλ,j [A, B; γ].
2. Majorization problem for the class Spλ,j [A, B; γ]
We begin by proving the following result.
Theorem 1. Let the function f ∈ Ap and suppose that g ∈ Spλ,j [A, B; γ]. If
(j) is majorized by (Ωλ,p g(z))(j) in ∆, then
(Ωλ,p
z f (z))
z
(j)
(j)
λ+1,p
λ+1,p
Ωz
(|z| ≤ r0 ),
(11)
g(z)
f (z)
≤ Ωz
where r0 = r0 (p, γ, λ, A, B) is smallest the positive root of the equation
r3 |γ(A−B)−(p−λ)B|−[(p−λ)+2|B|]r2 −[|γ(A−B)−(p−λ)B|+2]r +(p−λ) = 0,
(12)
(−1 ≤ B < A ≤ 1; p ∈ N; γ ∈ C − {0}).
Proof. Since g ∈ Spλ,j [A, B; γ], we find from (10) that
(j+1)
(λ,p)
1 z Ωz g(z)
1 + Aw(z)
1+
(j) − p + j = 1 + Bw(z) ,
γ
(λ,p)
Ωz g(z)
(13)
where ω is analytic in U with
ω(0) = 0
and
| ω(z) |< 1
(z ∈ U).
From (13), we get
(j+1)
(λ,p)
z Ωz g(z)
(p − j) + [(γ(A − B) + (p − j)B]w(z)
.
(j) =
1 + Bw(z)
(λ,p)
Ωz g(z)
100
(14)
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
By noting that
(j)
(j)
,
+ (λ − j) Ω(λ,p)
g(z)
g(z)
g(z))(j+1) = (p − λ) Ω(λ+1,p)
z(Ω(λ,p)
z
z
z
by virtue of (14) and (15), we get
(j)
(j)
(λ,p)
(λ+1,p)
(p
−
λ)[1
+
|B||z|]
Ωz g(z)
≤
.
g(z)
(p − λ) − |γ(A − B) − (p − λ)B||z| Ωz
(15)
(16)
(j)
(j)
(λ,p)
(λ,p)
in the unit disk ∆, from
is majorized by Ωz g(z)
Next, since Ωz f (z)
(3), we have
(j)
(j)
(λ,p)
=
ϕ(z)
Ω
g(z)
Ω(λ,p)
f
(z)
z
z
Differentiating it with respect to z and multiplying by z, we get
(j+1)
(j)
(j+1)
(λ,p)
′
(λ,p)
.
+
zϕ(z)
Ω
g(z)
=
zϕ
(z)
Ω
g(z)
z Ω(λ,p)
f
(z)
z
z
z
(17)
Using (15), in the above equation, it yields
(j)
(j)
(j)
zϕ′ (z) (λ,p)
(λ+1,p)
=
Ω(λ+1,p)
f
(z)
+
ϕ(z)
Ω
g(z)
Ω
g(z)
z
z
(p − λ) z
(18)
Thus, by noting that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [5])
|ϕ′ (z)| ≤
1 − |ϕ(z)|2
1 − |z|2
(z ∈ ∆)
(19)
and making use of (16) and (19) in (18), we get
(j)
(λ+1,p)
Ωz
f (z)
(j)
(λ+1,p)
|z|(1 + |B||z|)
1 − |ϕ(z)|2
,
Ω
g(z)
≤ |ϕ(z)| +
1 − |z|2 [(p − λ) − |γ(A − B) − (p − λ)B||z|] z
(20)
which upon setting
|z| = r
and |ϕ(z)| =ρ
(0 ≤ ρ ≤ 1)
leads us to the inequality
(j)
λ+1,p
Φ(ρ)
Ωz
≤
f (z)
(1 − r2 )(p − λ) − |γ(A − B) − (p − λ)B|r
101
(j)
λ+1,p
Ωz
,
g(z)
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
where
Φ(ρ) = −r(1+|B|r)ρ2 +(1−r2 )[(p−λ)−|γ(A−B)+(p−λ)B|r)]ρ+r(1+|B|r) (21)
takes its maximum value at ρ = 1, with r0 = r0 (p, γ, λ, A, B), where r0 (p, γ, λ, A, B)
is smallest the positive root of the equation (12). Furthermore, if 0 ≤ ρ ≤ r0 (p, γ, λ, A, B),
then the function ψ(ρ) defined by
ψ(ρ) = −σ(1+|B|σ)ρ2 +(1−σ 2 )[(p−λ)−|γ(A−B)+(p−λ)B|σ)]ρ+σ(1+|B|σ) (22)
is a increasing function on the interval 0 ≤ ρ ≤ 1, so that
ψ(ρ) ≤ ψ(1) = (1 − σ 2 )[(p − λ) − |γ(A − B) + (p − λ)B|σ)]
(0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r0 (p, γ, λ, A, B)).
(23)
Hence upon setting ρ = 1, in (22), we conclude that (11) of Theorem (1) holds true
for
|z| ≤ r0 = r0 (p, γ, λ, A, B),
where r0 (p, γ, λ, A, B) is the smallest positive root of the equation (12). This completes the proof of the Theorem 1.
Setting A = 1 and B = −1 in Theorem 1, we get the following result.
Corollary 1. (See [2]) Let the function f ∈ Ap and suppose that g ∈ Spλ,j (γ). If
(j)
(j)
λ,p
in ∆, then
is
majorized
by
Ω
g(z)
Ωλ,p
f
(z)
z
z
where
(j)
(j)
λ+1,p
λ+1,p
Ωz
g(z)
f (z)
≤ Ωz
r1 = r1 (p, γ, λ) =
k−
f or |z| ≤ r1 ,
p
k 2 − 4(p − λ)|2γ − p + λ|
,
2|2γ − p + λ|
(24)
(25)
(k = 2 + p − λ + |2γ − p + λ|; p ∈ N; γ ∈ C − {0}).
Putting A = 1, B = −1, and j = 0 in Theorem 1, we obtain the following result.
Corollary 2. Let the function f ∈ A and suppose that g ∈ Spλ,0 (γ). If Ωλ,p
z f (z) is
λ,p
majorized by (Ωz g(z)) in ∆, then
λ+1,p
g(z) f or |z| ≤ r1 ,
f (z) ≤ Ωλ+1,p
Ωz
z
where r1 is given by (25).
102
P. Goswami, Z. Wang - Majorization for certain classes of analytic functions
Further putting λ = 0, j = 0 in Corollary 2, we get
Corollary 3. (See [1]) Let the function f ∈ A and suppose that g ∈ Spλ,0 (γ). If
λ,p
Ωλ,p
z f (z) is majorized by Ωz g(z) in ∆, then Let the function f (z) ∈ A be analytic
in th open unit disk ∆ and suppose that g ∈ S(γ). If f (z) is majorized by g(z) in
∆, then
|f ′ (z)| ≤ |g ′ (z)|
(|z| ≤ r2 )
where
r2 = r2 (γ) =
3 + |2γ − 1| −
p
9 + 2|2γ − 1| + |2γ − 1|2
.
2|2γ − 1|
For γ = 1, Corollary 3 reduces to the following result:
Corollary 4. (See [3]) Let the function f ∈ A be an analytic and univalent in the
open unit disk ∆ and suppose that g ∈ S ∗ = S ∗ (0). If f (z) is majorized by g(z) in
∆, then
√
|f ′ (z)| ≤ |g ′ (z)|
(|z| ≤ 2 − 3).
Acknowledgements. The first-named author is grateful to Prof. S. P. Goyal,
University of Rajasthan, for his guidance and encouragement.
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Pranay Goswami
Department of Mathematics,
Amity University Rajasthan,
Jaipur (India) - 302002
email: [email protected]
Zhi-Gang Wang
School of Mathematics and Computing Science,
Changsha University of Science and Technology,
Yuntang Campus, Changsha 410114, Hunan,
People’s Republic of China
email: [email protected]
104