AbstractIn this paper we investigate a majorization problem for a subclass of p-valently analytic function involving a generalized fractional differintegral operator. Some useful consequences of the main result are mentioned and relevance with some of the earlier results are also pointed out

Aouf, M.K.

Prajapat, J.K.

Elsevier - Publisher Connector 

Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator 

Computers and Mathematics with Applications 63 (2012) 42–47
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Majorization problem for certain class of p-valently analytic function
defined by generalized fractional differintegral operator
J.K. Prajapat a,∗, M.K. Aouf b
a Department of Mathematics, Central University of Rajasthan, Kishangarh, Distt.-Ajmer, Rajasthan, India
b Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e i n f o
Article history:
Received 13 June 2011
Received in revised form 24 October 2011
Accepted 24 October 2011
Keywords:
Analytic function
Differential subordination
Majorization
Fractional differintegral operator
a b s t r a c t
In this paper we investigate a majorization problem for a subclass of p-valently
analytic function involving a generalized fractional differintegral operator. Some useful
consequences of the main result are mentioned and relevance with some of the earlier
results are also pointed out.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Let functions f and g are analytic in the open unit disk
U = {z : z ∈ C, |z| < 1}. (1.1)
We say that f is majorized by g in U and write
f (z)≪ g(z) (z ∈ U), (1.2)
if there exist a function ϕ(z), analytic in U such that
|ϕ(z)| ≤ 1 and f (z) = ϕ(z)g(z) (z ∈ U). (1.3)
Note that majorization is closely related to quasi-subordination [1].
Further, we say that the function f is subordinated to g , and write f (z) ≺ g(z), z ∈ U, if there exists a functionw analytic
in U, with
|w(z)| < 1 and w(0) = 0 (z ∈ U), (1.4)
such that
f (z) = g(w(z)) (z ∈ U). (1.5)
In particular, if f (z) is univalent in U, we have the following equivalence:
f (z) ≺ g(z) (z ∈ U)⇐⇒ f (0) = g(0) and f (U) ⊂ g(U).
We recall here the following generalized fractional integral and generalized fractional derivative operators due to
Srivastava et al. [2] (see also [3]).
∗ Corresponding author.
E-mail addresses: jkp_0007@rediffmail.com (J.K. Prajapat), mkaouf127@yahoo.com (M.K. Aouf).
0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2011.10.065
J.K. Prajapat, M.K. Aouf / Computers and Mathematics with Applications 63 (2012) 42–47 43
Definition 1. For real numbers λ > 0, µ and η, Saigo hypergeometrc fractional integral operator Iλ,µ,η0,z is defined by
Iλ,µ,η0,z f (z) =
z−λ−µ
Γ (λ)
∫ z
0
(z − t)λ−12F1

λ+ µ,−η; λ; 1− t
z

f (t) dt, (1.6)
where the function f (z) is analytic in a simply-connected region of the complex z-plane containing the origin, with the order
f (z) = O(|z|ε) (z → 0; ε > max{0, µ− η} − 1),
and the multiplicity of (z − t)λ−1 is removed by requiring log(z − t) to be real when (z − t) > 0.
Definition 2. Under the hypotheses of Definition 1, Saigo hypergeometric fractional derivative operator Jλ,µ,η0,z is defined by
Jλ,µ,η0,z f (z) =

1
Γ (1− λ)
d
dz

zλ−µ
∫ z
0
(z − t)−λ2F1

µ− λ, 1− η; 1− λ; 1− t
z

f (t) dt

(0 ≤ λ < 1);
dn
dzn
Jλ−n,µ,η0,z f (z) (n ≤ λ < n+ 1; n ∈ N),
(1.7)
where the multiplicity of (z − t)−λ is removed as in Definition 1.
It may be remarked that
Iλ,−λ,η0,z f (z) = D−λz f (z) (λ > 0)
and
Jλ,λ,η0,z f (z) = Dλz f (z) (0 ≤ λ < 1),
where D−λz denotes fractional integral operator and Dλz denotes fractional derivative operator considered by Owa [4].
LetAp denote the class of functions of the form
f (z) = zp +
∞−
n=1
an+pzn+p (p ∈ N = {1, 2, 3, . . .}), (1.8)
which are analytic and p-valent in the open unit disk U. Recently Goyal and Prajapat [5], introduced generalized fractional
differintegral operator Sλ,µ,η0,z : Ap → Ap, by
S
λ,µ,η
0,z f (z) =

Γ (1+ p− µ)Γ (1+ p+ η − λ)
Γ (1+ p)Γ (1+ p+ η − µ) z
µJλ,µ,η0,z f (z) (0 ≤ λ < η + p+ 1, z ∈ U);
Γ (1+ p− µ)Γ (1+ p+ η − λ)
Γ (1+ p)Γ (1+ p+ η − µ) z
µI−λ,µ,η0,z f (z) (−∞ < λ < 0, z ∈ U).
(1.9)
It is easily seen from (1.9) that for a function f of the form (1.8), we have
S
λ,µ,η
0,z f (z) = zp +
∞−
n=1
(1+ p)n(1+ p+ η − µ)n
(1+ p− µ)n(1+ p+ η − λ)n ap+nz
p+n
= zp3F2(1, 1+ p, 1+ p+ η − µ; 1+ p− µ, 1+ p+ η − λ; z) ∗ f (z)
(z ∈ U; p ∈ N; µ, η ∈ R; µ < p+ 1;−∞ < λ < η + p+ 1) (1.10)
where ∗ denotes usual Hadamard product of analytic functions and pFq is well known generalized hypergeometric function.
The operator Sλ,µ,η0,z satisfies the following three-term recurrence relation:
z(Sλ,µ,η0,z f (z))
(j+1) = (p+ η − λ)(Sλ+1,µ,η0,z f (z))(j) − (η + j− λ)(Sλ,µ,η0,z f (z))(j). (1.11)
Note that
S0,0,00,z f (z) = f (z), S1,1,10,z f (z) = S1,0,00,z f (z) =
zf ′(z)
p
and
S2,1,10,z f (z) =
zf ′(z)+ z2f ′′(z)
p2
.
We also note that
S
λ,λ,η
0,z f (z) = Sλ,µ,00,z f (z) = Ωλ,pz f (z),
44 J.K. Prajapat, M.K. Aouf / Computers and Mathematics with Applications 63 (2012) 42–47
whereΩλ,pz is an extended fractional differintegral operator studied very recently by [6] (see also [7]). On the other hand, if
we set
λ = −α, µ = 0 and η = β − 1,
in (1.10) and using
Iα,0,β−10,z f (z) =
1
zβΓ (α)
∫ z
0
tβ−1

1− t
z
α−1
f (t)dt,
we obtain following p-valent generalization of multiplier transformation operator [8,9]:
Qαβ,pf (z) =

p+ α + β − 1
p+ β − 1

α
zβ
∫ z
0
tβ−1

1− t
z
α−1
f (t)dt
= zp +
∞−
n=1
Γ (p+ β + n)Γ (p+ α + β)
Γ (p+ α + β + n)Γ (p+ β)an+pz
n+p (β > −p;α + β > −p). (1.12)
On the other hand, if we set
λ = −1, µ = 0 and η = β − 1,
in (1.10), we obtain the generalized Bernardi–Libera integral operator Fβ,p : Ap → Ap(β > −p) defined by
S
−1,0,β−1
0,z f (z) = Fβ,pf (z) =
p+ β
zβ
∫ z
0
tβ−1f (t)dt
= zp +
∞−
n=1
p+ β
p+ β + nap+nz
p+n (β > −p). (1.13)
For the choice p = 1, where β ∈ N, the operator defined by (1.13) reduces to the well-known Bernardi integral
operator [10].
Using the generalized fractional differintegral operator Sλ,µ,η0,z , we now introduce the following subclass ofAp:
Definition 3. A function f (z) ∈ Ap is said to be in the classMp,jλ,µ,η(A, B; γ ), if and only if
z(Sλ,µ,η0,z f (z))
(j+1)
(S
λ,µ,η
0,z f (z))(j)
− p+ j ≺ γ (A− B)z
1+ Bz
(z ∈ U;−1 ≤ B < A ≤ 1; j ∈ N0 = N ∪ {0}; γ ∈ C− {0};µ, η ∈ R;µ < p+ 1;−∞ < λ < η + p+ 1). (1.14)
For simplicity, we put
N
p,j
λ,µ,η(γ ) =Mp,jλ,µ,η(1,−1; γ ),
whereN p,jλ,µ,η(γ ) denote the class of functions f ∈ Ap satisfying the following inequality:
ℜ

1
γ

z(Sλ,µ,η0,z f (z))
(j+1)
(S
λ,µ,η
0,z f (z))(j)
− p+ j

> −1. (1.15)
Clearly, we have the following relationships:
(i) N p,jλ,µ,0(γ ) ≡ Sλ;jp (γ );
(ii) N p,j0,0,0(γ ) ≡ Sp,j(γ );
(iii) N 1,00,0,0(γ ) ≡ S(γ );
(iv) N 1,01,0,0(γ ) ≡ K(γ );
(v) N 1,00,0,0(1− α) ≡ S∗(α), 0 ≤ α < 1.
The class Sλ;jp (γ ) was studied recently by Goyal and Goswami [11], and the class Sp,j(γ ); was studied by Altintas and
Srivastava [12]. Further, the class S(γ ), is the class of starlike function of complex order γ in U and the classK(γ ), is the
class of convex functions of complex order γ in U, which were considered by Nasr and Aouf [13] and Waitrowski [14],
respectively. Moreover S∗(α) denotes the class of starlike functions of order α in U.
A majorization problem for the class S(γ ), has been investigated by Altintas et al. [15] and a majorization problem
for the class Sp,j(γ ), has recently been investigated by Altintas et al. [12]. Very recently, Goswami and Wang [16], Goyal
and Goswami [11] and Goyal et al. [17] generalized these results for different function classes. In this present paper we
investigate a majorization problem for the classMp,jλ,µ,η(A, B; γ ), and give some special cases of our main result.
J.K. Prajapat, M.K. Aouf / Computers and Mathematics with Applications 63 (2012) 42–47 45
2. Majorization problem for the classMp,jλ,µ,η(A, B;γ)
We begin by proving
Theorem 1. Let the function f (z) ∈ Ap and suppose that g(z) ∈Mp,jλ,µ,η(A, B; γ ). If
(S
λ,µ,η
0,z f (z))
(j) ≪ (Sλ,µ,η0,z g(z))(j) (z ∈ U),
then (Sλ+1,µ,η0,z f (z))(j) ≤ (Sλ+1,µ,η0,z g(z))(j) for |z| ≤ r0, (2.1)
where r0 = r0(A, B, j, γ , λ, µ, η) is the smallest positive root of the equation
r3|γ (A− B)+ ξB| − (ξ + 2|B|)r2 − (|γ (A− B)+ ξB| + 2)r + ξ = 0
(ξ = p+ η − λ)(−1 ≤ B < A ≤ 1; j ∈ N0; µ, η ∈ R;
µ < p+ 1; −∞ < λ < η + p+ 1; p ∈ N; γ ∈ C− {0}). (2.2)
Proof. Since g ∈Mp,jλ,µ,η(A, B; γ ), we find from (1.13) that
z(Sλ,µ,η0,z g(z))
(j+1)
(S
λ,µ,η
0,z g(z))(j)
− p+ j = γ (A− B)w(z)
1+ Bw(z) , (2.3)
where w(z) = c1z + c2z2 + · · · , w ∈ P ,P denotes the well known class of bounded analytic functions in U (see [18]),
which satisfies the conditions
w(0) = 0 and |w(z)| ≤ |z| (z ∈ U).
On using (1.11) in (2.3), we get
|(Sλ,µ,η0,z g(z))(j)| ≤
ξ(1+ |B| |z|)
ξ − |γ (A− B)+ ξB| |z| |(S
λ+1,µ,η
0,z g(z))
(j)|, (2.4)
where ξ = p+ η − λ. Next, since (Sλ,µ,η0,z f (z))(j) is majorized by (Sλ,µ,η0,z g(z))(j) in the unit disk U, from (1.3), we have
(S
λ,µ,η
0,z f (z))
(j) = ϕ(z)(Sλ,µ,η0,z g(z))(j), (2.5)
where |φ(z)| ≤ 1. Differentiating (2.5) w. r. to z and multiplying by z, we get
z(Sλ,µ,η0,z f (z))
(j+1) = zϕ′(z)(Sλ,µ,η0,z g(z))(j) + ϕ(z)z(Sλ,µ,η0,z g(z))(j+1),
which on using (1.11) once again, gives
(S
λ+1,µ,η
0,z f (z))
(j) = zϕ
′(z)
ξ
(S
λ,µ,η
0,z g(z))
(j) + ϕ(z)(Sλ+1,µ,η0,z g(z))(j). (2.6)
Noting that ϕ ∈ P satisfying the inequality (see, e. g., [19])
|ϕ′(z)| ≤ 1− |ϕ(z)|
2
1− |z|2 , (z ∈ U), (2.7)
and making use of (2.5) and (2.7) in (2.6), we have
|(Sλ+1,µ,η0,z f (z))(j)| ≤

|ϕ(z)| + 1− |ϕ(z)|
2
1− |z|2
(1+ |B| |z|)|z|
ξ − |γ (A− B)+ ξB| |z|

|(Sλ+1,µ,η0,z g(z))(j)|,
which upon setting
|z| = r and |ϕ(z)| = ρ (0 ≤ ρ ≤ 1),
leads us to the inequality
|(Sλ+1,µ,η0,z f (z))(j)| ≤
ϑ(ρ)
(1− r2)(ξ − |γ (A− B)+ ξB|r) |(S
λ+1,µ,η
0,z g(z))
(j)|,
46 J.K. Prajapat, M.K. Aouf / Computers and Mathematics with Applications 63 (2012) 42–47
where
ϑ(ρ) = −r(1+ |B|r)ρ2 + (1− r2)(ξ − |γ (A− B)+ ξB|r)ρ + r(1+ |B|r) (2.8)
takes its maximum value at ρ = 1, with r0 = r0(A, B, j, γ , λ, µ, η), where r0 is the smallest positive root of Eq. (2.2).
Furthermore, if 0 ≤ σ ≤ r0, then the function Ψ (ρ) defined by
Ψ (ρ) = −σ(1+ |B|σ)ρ2 + (1− σ 2)(ξ − |γ (A− B)+ ξB|σ)ρ + σ(1+ |B|σ) (2.9)
is an increasing function on the interval 0 ≤ ρ ≤ 1, so that
Ψ (ρ) ≤ Ψ (1) = (1− σ 2)(ξ − |γ (A− B)+ ξB|σ) (0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r0). (2.10)
Hence upon setting ρ = 1, in (2.9), we conclude that (2.1) of Theorem 1 holds true for
|z| ≤ r0 = r0(A, B, j, γ , λ, µ, η),
where r0(A, B, j, γ , λ, µ, η) is the smallest positive root of Eq. (2.2). This complete the proof of Theorem 1. 
Remark 1. Setting η = 0 in Theorem 1, we get a result due to Goswami and Wang [16, Theorem 1; pp. 100].
Putting λ = −α,µ = 0 and η = β − 1 in Theorem 1, we get
Corollary 1. Let the function f (z) ∈ Ap and suppose that g(z) satisfy
z(Qαβ,pg(z))
(j+1)
(Qαβ,pg(z))(j)
≺ p− j+ [γ A+ (p− j− γ )B]z
1+ Bz .
If
(Qαβ,pf (z))
(j) ≪ (Qαβ,pg(z))(j) (z ∈ U),
then (Qα+1β,p f (z))(j) ≤ (Qα+1β,p g(z))(j) for |z| ≤ r1,
where r1 = r1(j, γ , α, β, p) is the smallest positive root of the equation
r3|γ (A− B)+ χB| − (χ + 2|B|)r2 − (|γ (A− B)+ χB| + 2)r + χ = 0.
(χ = p+ β + α − 1; β > −p; α + β > −p; p, j ∈ N; γ ∈ C− {0}).
Setting A = 1 and B = −1 in Theorem 1, we have
Corollary 2. Let the function f (z) ∈ Ap and suppose that g(z) ∈ N p,jλ,µ,η(γ ). If
(S
λ,µ,η
0,z f (z))
(j) ≪ (Sλ,µ,η0,z g(z))(j) (z ∈ U),
then (Sλ+1,µ,η0,z f (z))(j) ≤ (Sλ+1,µ,η0,z g(z))(j) for |z| ≤ r1,
where
r2 = r2(j, γ , λ, µ, η) = τ −
√
τ − 4ξ |2γ − ξ |
2|2γ − ξ |
(τ = 2+ ξ + |2γ − ξ |; ξ = p+ η − λ; λ < η + p+ 1; µ < p+ 1;µ, η ∈ R; p, j ∈ N; γ ∈ C− {0}).
Remark 2. (i) Setting η = 0 in Corollary 2, we get a result due to Goyal and Goswami [11, Theorem 2.1; pp. 1856].
(ii) Putting j = λ = µ = η = 0 and p = 1 in Corollary 2, we get the result obtained by [15].
(iii) For j = λ = µ = η = 0 and γ = p = 1 Corollary 2, gives the result obtained by MacGregor [18].
Acknowledgments
The author is grateful to the referees for the valuable comments.
J.K. Prajapat, M.K. Aouf / Computers and Mathematics with Applications 63 (2012) 42–47 47
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Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator

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