U = {z â C : \z\ < 1} and let Ap be the subclass of A consisting of the functions f of ...... [8] Shams, S., Kulkarni, S. R. and Jahangiri, Jay M.: On a class of univalent.
Proyecciones Journal of Mathematics Vol. 33, No 2, pp. 189-204, June 2014. Universidad Cat´olica del Norte Antofagasta - Chile
A class of multivalent functions defined by generalized Ruscheweyh derivatives involving a general fractional derivative operator Hari Singh Parihar Central University of Rajasthan, India and Ritu Agarwal Malaviya National Institute of Technology, India Received : March 2013. Accepted : September 2013
Abstract The main aim of the present paper is to obtain a new class of multivalent functions which is defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator.We study the region of starlikeness and convexity of the class Ωp (α, β, γ). Also we apply the Fractional calculus techniques to obtain the applications of the class Ωp (α, β, γ). Finally, the familiar concept of δ-neighborhoods of p-valent functions for above mentioned class are employed. Subject class (2010) : Primary 26A33, Secondary 30C45. Keywords : Starlike function, p-valent function, Convolution,Generalized fractional derivative operator, Generalized Ruscheweyh derivatives
190
Hari Singh Parihar and Ritu Agarwal
1. Introduction Let A denote the class of functions that are analytic in the open unit disk U = {z ∈ C : |z| < 1} and let Ap be the subclass of A consisting of the functions f of the form p
(1.1)
f (z) = z −
∞ X
k=n+p
ak z k , (n ∈ N )
where p is some positive integer and f is analytic and p-valent in U. The generalized fractional derivative operator of order λ, introduced by Srivastava and Saxena [9], [10], is defined as
(1.2)
λ,µ,ν f (z) J0,z
=
⎧ ⎪ ⎪ n ⎪ R ⎪ ⎪ 1 d ⎪ z λ−µ 0z (z − ζ)−λ ⎪ dz Γ(1−λ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ³ ⎪ ⎨
. 2 F1 µ − λ, 1 − ν; 1 − λ; 1 −
ζ z
´
o
f (ζ)dζ ,
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (0 ≤ λ < 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dn J λ−n,µ,ν f (z), (n ≤ λ < n + 1, n ∈ N) dz n 0,z
where f is an analytic function in a simply connected region of the zplane containing the origin, and the multiplicity of (z − ζ)−λ is removed by requiring log(z − ζ) to be real when z − ζ > 0, provided further that, f (z) = O(|z|k ),
(1.3)
(z → 0)
In terms of gamma function, we have
(1.4)
λ,µ,ν ρ J0,z z =
Γ(ρ+1)Γ(ρ−µ+ν+2) ρ−µ , Γ(ρ−µ+1)Γ(ρ−λ+ν+2) z
(0 ≤ λ < 1, ρ > max{0, µ − ν − 1} − 1) It follows at once from the above definition that (1.5)
λ,λ,ν f (z) = Dzλ f (z) = J0,z
1 Γλ
Z z 0
f (t) dt, (z − t)1−δ
(0 ≤ λ < 1).
A class of multivalent functions defined by generalized ...
191
where Dzλ f (z) is the fractional derivative operator of order λ. Furthermore, in terms of gamma function, we have (1.6)
Dzλ z ρ =
Γ(ρ + 1) ρ−λ z , (0 ≤ λ) Γ(ρ − δ + 1)
Similary, the fractional integral opearator of order λ is (1.7)
Dz−λ f (z) =
d 1 Γ(1 − λ) dz
Z z 0
f (t) dt, (z − t)δ
where f is an analytic function in a simply connected region of the zplane containing the origin, and the multiplicity of (z − t)−λ is removed by requiring log(z − t) to be real when z − t > 0. In terms of gamma function, Γ(ρ + 1) ρ+λ z . Γ(ρ + δ + 1)
Dz−λ z ρ =
(1.8)
The generalized Ruscheweyh derivatives Jλ,µ p f , µ > −1 of f ∈ Ap is defined by Goyal and Goyal [2] as follows: Jλ,µ p f (z) = (1.9) = zp − where Bpλ, µ (k) =
Γ(µ−λ+ν+2) p λ,µ,ν µ−p f (z)) Γ(ν+2)Γ(µ+1) z J0,z (z
P∞
λ, µ k k=n+p ak Bp (k)z
Γ(k − p + 1 + µ)Γ(ν + 2 + µ − λ)Γ(k + ν − p + 2) Γ(k − p + 1)Γ(k + ν − p + 2 + µ − λ)Γ(ν + 2)Γ(1 + µ)
(1.10)
For λ = µ, this generalized Ruscheweyh derivatives get reduced to Ruscheweyh derivatives of f(z) of order λ (see, e.g. [12]): Dλ f (z) = = zp + (1.11)
zp dλ λ−p f (z)) Γ(λ+1) dz λ (z
P∞
k=n+p ak Bk (λ)z
k
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Hari Singh Parihar and Ritu Agarwal
where (1.12)
Bk (λ) =
Γ(λ + k) Γ(λ + p)Γ(k − p + 1)
For p=1, (1.11) reduces to ordinary Ruscheweyh derivatives for univalent functions [8]. The operation * is the convolution (Hadamard product) of two power series ∞ ∞ f (z) = z p −
X
k=n+p
ak z k and g(z) = z p −
defined as (f ∗ g)(z) = z p −
(1.13)
∞ X
X
bk z k
k=n+p
ak bk z k
k=n+p
A function f ∈ Ωp is said to be in the class Ωp (α, β, λ) if and only if (1.14)
Re
(
0 z(Jλ,µ p f (z)) 2 λ,µ 00 (1 − α)(Jλ,µ p f (z)) + αz (Jp f (z))
)
>β
for z ∈ U and 0 ≤ α < 1, 0 ≤ β < p and λ > −1. The class Ωp (α, β, λ) contains many well-known classes of analytic functions such as: • For α = λ = 0, Ωp (α, β, λ) reduces to the class S ∗ (β) of starlike functions of order β. • For α = λ = −1, Ωp (α, β, λ) reduces to the class K(β) of convex functions of order β.
2. Main Results The coefficient bounds for the functions f ∈ Ωp (α, β, λ) are found in the following theorem: Theorem 2.1. Let f ∈ Ap , z ∈ U be of the form (1.1). Then f ∈ Ωp (α, β, λ) iff (2.1)
∞ X αβ(1 + k − k2 ) + k − β
k=n+p
p − β + αβ(1 + p − p2 )
where 0 ≤ α < 1, 0 ≤ β < p and λ > −1.
Bpλ,µ (k)ak < 1
A class of multivalent functions defined by generalized ... Proof. Since f ∈ Ωp (α, β, λ) Re
(2.2)
(
0 z(Jλ,µ p f (z)) 2 λ,µ 00 (1 − α)(Jλ,µ p f (z)) + αz (Jp f (z))
)
193
>β
Making use of equation (1.9) in the above inequality, we obtain
Re
(
pz p−1 −
P∞
λ, µ k−1 k=n+p kak Bp (k)z P λ, µ k (1 − α + αp(p − 1))z p − ∞ k=n+p [(1 − α) + α(k(k − 1))]ak Bp (k)z
)
(2.3) Therefore, we obtain
∞ X αβ(1 + k − k2 ) + k − β
(2.4)
k=n+p
p − β + αβ(1 + p − p2 )
Bpλ,µ (k)ak < 1.
In this theorem, we will show that this class is closed under linear combination. Theorem 2.2. Let for j ∈ {1, 2, 3, ..., m} fj (z) = z p − Then for 0 < Pj < 1,
Pm
j=1 Pj
∞ X
k=n+p
= 1, the function F (z) defined by
F (z) =
(2.5)
ak,j z k ∈ Ωp (α, β, λ)
m X
Pj fj (z)
j=1
is also in Ωp (α, β, λ). Proof. For every j ∈ {1, 2, 3, ..., m}, we obtain (2.6)
∞ X αβ(1 + k − k2 ) + k − β
k=n+p
p − β + αβ(1 + p − p2 )
Bpλ,µ (k)ak,j < 1.
Since (2.7) F (z) =
m X
j=1
⎛
Pj ⎝z p −
∞ X
k=n+p
⎞
ak,j z k ⎠ = z p −
∞ X
k=n+p
⎛ ⎝
m X
j=1
⎞
Pj ak,j ⎠ z k
>β
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Hari Singh Parihar and Ritu Agarwal Therefore,
(2.8)
⎛
∞ X αβ(1 + k − k2 ) + k − β
k=n+p
Bpλ,µ (k) ⎝
p − β + αβ(1 + p − p2 )
m X
j=1
which proves the Theorem. Theorem 2.3. Let
f (z) = z p − and p
g(z) = z −
∞ X
⎞
Pj ak,j ⎠
