UNIFORMLY STARLIKE AND UNIFORMLY CONVEX FUNCTIONS PERTAINING TO SPECIAL FUNCTIONS
V.B.L. CHAURASIA AND AMBER SRIVASTAVA DEPARTMENT OFMATHEMATICS
UNIVERSITY OFRAJASTHAN, JAIPUR-302004, INDIA
DEPARTMENT OFMATHEMATICS
SWAMIKESHVANANDINSTITUTE OFTECHNOLOGY, MANAGEMENT ANDGRAMOTHAN
JAGATPURA, JAIPUR-302025, INDIA
amber@skit.ac.in
Received 04 September, 2006; accepted 14 July, 2007 Communicated by H.M. Srivastava
ABSTRACT. The main object of this paper is to derive the sufficient conditions for the function z{pψq(z)} to be in the classes of uniformly starlike and uniformly convex functions. Similar results using integral operator are also obtained.
Key words and phrases: Analytic functions, Univalent functions, Starlike functions, Convex functions, Integral operator, Fox- Wright function.
2000 Mathematics Subject Classification. 30C45.
1. I
NTRODUCTIONLet A denote the class of functions of the form
(1.1) f (z) = z +
∞
X
n=2
a
nz
n,
that are analytic in the open unit disk ∆ = {z : |z| < 1}.
Also let S denote the subclass of A consisting of all functions f (z) of the form
(1.2) f(z) = z −
∞
X
n=2
a
nz
n, a
n≥ 0.
A function f ∈ A is said to be starlike of order α, 0 ≤ α < 1, if and only if Re
zf0(z) f(z)
> α, z ∈ ∆. Also f of the form (1.1) is uniformly starlike, whenever
f(z)−f(ξ)(z−ξ)f0(z)
≥ 0, (z,ξ) ∈
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valuable suggestions in the preparation of this paper.
230-06
∆ × ∆. This class of all uniformly starlike functions is denoted by U ST [4] (see also [5], [10]
and [14]).
The function f of the form (1.1) is uniformly convex in ∆ whenever Re
1 + (z − ξ)
ff000(z)(z)≥ 0, (z,ξ) ∈ ∆ × ∆. This class of all uniformly convex functions is denoted by U CV [3]
(also refer [2], [6], [9] and [13]). Further it is said to be in the class U CV (α), α ≥ 0 if Re
1 +
zff(z)0(z)≥ α
zf00(z) f0(z)
.
A function f of the form (1.2) is said to be in the class U ST N (α), 0 ≤ α ≤ 1, if Re
f(z)−f(ξ)(z−ξ)f0(z)
≥ α, (z,ξ) ∈ ∆ × ∆.
In the present paper, we shall use analogues of the lemmas in [8] and [7] respectively in the following form.
Lemma 1.1. A function f of the form (1.1) is in the class U ST (α), if
∞
X
n=2
[(3 − α)n − 2] |a
n| ≤ (1 − α)M
1, where M
1> 0 is a suitable constant. In particular, f ∈ U ST whenever
∞
X
n=2
(3n − 2) |a
n| ≤ M
1.
Lemma 1.2. A sufficient condition for a function f of the form (1.1) to be in the class U CV (α) is that P
∞n=2
n[(α + 1)n − α] a
n≤ M
2, where M
2> 0 is a suitable constant. In particular, f ∈ U CV whenever P
∞n=2
n
2a
n≤ M
2.
The Fox-Wright function [12, p. 50, equation 1.5] appearing in the present paper is defined by
(1.3)
pψ
q(z) =
pψ
q(a
j, α
j)
1,p; (b
j, β
j)
1,q; z
=
∞
X
n=0
Q
pj=1
Γ(a
j+ α
jn)z
nQ
qj=1
Γ(b
j+ β
jn)n! , where α
j(j = 1, . . . , p) and β
j(j = 1, . . . , q) are real and positive and 1+ P
qj=1
β
j> P
p j=1α
j. 2. M
AINR
ESULTSTheorem 2.1. If
q
X
j=1
|b
j| >
p
X
j=1
|a
j| + 1, a
j> 0 and 1 +
q
X
j=1
β
j>
p
X
j=1
α
j,
then a sufficient condition for the function z{
pψ
q(z)} to be in the class U ST (α), 0 ≤ α < 1, is (2.1)
3 − α 1 − α
p
ψ
q(|a
j+ α
j|, α
j)
1,p; (|b
j+ β
j|, β
j)
1,q; 1
+
pψ
q(|a
j|, α
j)
1,p; (|b
j|, β
j)
1,q; 1
≤ M
1+ Q
pj=1
Γa
jQ
qj=1
Γb
j.
Proof. Since
z{
pψ
q(z)} = Q
pj=1
Γa
jQ
qj=1
Γb
jz +
∞
X
n=2
Q
pj=1
Γ[a
j+ α
j(n − 1)]z
nQ
qj=1
Γ[b
j+ β
j(n − 1)](n − 1)!
so by virtue of Lemma 1.1, we need only to show that (2.2)
∞
X
n=2
[(3 − α)n − 2]
Q
pj=1
Γ[a
j+ α
j(n − 1)]
Q
qj=1
Γ[b
j+ β
j(n − 1)](n − 1)!
≤ (1 − α)M
1.
Now, we have
∞
X
n=2
[(3 − α)n − 2]
Q
pj=1
Γ[a
j+ α
j(n − 1)]
Q
qj=1
Γ[b
j+ β
j(n − 1)](n − 1)!
=
∞
X
n=0
[(3 − α)(n + 2) − 2]
Q
pj=1
Γ[a
j+ α
j(n + 1)]
Q
qj=1
Γ[b
j+ β
j(n + 1)](n + 1)!
= (3 − α)
∞
X
n=0
Q
pj=1
Γ[(a
j+ α
j) + nα
j] Q
qj=1
Γ[(b
j+ β
j) + nβ
j]n!
+ (1 − α)
"
∞X
n=0
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)
1 n! −
Q
p j=1Γa
jQ
qj=1
Γb
j#
= (3 − α)
pψ
q(|a
j+ α
j|, α
j)
1,p; (|b
j+ β
j|, β
j)
1,q; 1
+ (1 − α)
pψ
q(|a
j|, α
j)
1,p; (|b
j|, β
j)
1,q; 1
− (1 − α) Q
pj=1
Γa
jQ
q j=1Γb
j≤ (1 − α)M
1which in view of Lemma 1.1 gives the desired result.
Theorem 2.2. If
q
X
j=1
b
j>
p
X
j=1
a
j+ 1, a
j> 0 and 1 +
q
X
j=1
β
j>
p
X
j=1
α
j,
then a sufficient condition for the function z{
pψ
q(z)} to be in the class U ST N (α), 0 ≤ α < 1, is:
3 − α 1 − α
p
ψ
q(a
j+ α
j, α
j)
1,p; (b
j+ β
j, β
j)
1,q; 1
+
pψ
q(a
j, α
j)
1,p; (b
j, β
j)
1,q; 1
≤ M
1+ Q
pj=1
Γa
jQ
qj=1
Γb
j. Proof. The proof of Theorem 2.2 is a direct consequence of Theorem 2.1.
Theorem 2.3. If
q
X
j=1
b
j>
p
X
j=1
a
j+ 2, a
j> 0 and 1 +
q
X
j=1
β
j>
p
X
j=1
α
j,
then a sufficient condition for the function z{
pψ
q(z)} to be in the class U CV (α), 0 ≤ α < 1, is (2.3) (1 + α)
pψ
q(a
j+ 2α
j, α
j)
1,p; (b
j+ 2β
j, β
j)
1,q; 1
+ (2α + 3)
pψ
q(a
j+ α
j, α
j)
1,p; (b
j+ β
j, β
j)
1,q; 1
+
pψ
q(1) ≤ M
2+ Q
pj=1
Γa
jQ
qj=1
Γb
j. Proof. By virtue of Lemma 1.2, it suffices to prove that
(2.4)
∞
X
n=2
n[(α + 1)n − α]
Q
pj=1
Γ[a
j+ α
j(n − 1)]
Q
qj=1
Γ[b
j+ β
j(n − 1)](n − 1)! ≤ M
2.
Now, we have
(2.5)
∞
X
n=2
n[(α + 1)n − α]
Q
pj=1
Γ[a
j+ α
j(n − 1)]
Q
qj=1
Γ[b
j+ β
j(n − 1)](n − 1)!
= (1 + α)
∞
X
n=1
(n + 1)
2Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ[(b
j+ β
jn)n! − α
∞
X
n=1
(n + 1) Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n! . Using (n + 1)
2= n(n + 1) + (n + 1), (2.5) may be expressed as
(1 + α)
∞
X
n=1
(n + 1)
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)(n − 1)! +
∞
X
n=1
(n + 1) Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n!
(2.6)
= (1 + α)
∞
X
n=2
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)(n − 2)! + (2α + 3)
∞
X
n=0
Q
pj=1
Γ[(a
j+ α
j) + α
jn]
Q
qj=1
Γ[(b
j+ β
j) + β
jn]n!
+
∞
X
n=1
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n!
= (1 + α)
pψ
q(a
j+ 2α
j, α
j)
1,p; (b
j+ 2β
j, β
j)
1,q; 1
+ (2α + 3)
pψ
q(a
j+ α
j, α
j)
1,p; (b
j+ β
j, β
j)
1,q; 1
+
pψ
q(1) − Q
pj=1
Γa
jQ
qj=1
Γb
j,
which is bounded above by M
2if and only if (2.3) holds. Hence the theorem is proved.
3. A
NI
NTEGRALO
PERATORIn this section we obtain sufficient conditions for the function
p
φ
q(a
j, α
j)
1,p; (b
j, β
j)
1,q; z
= Z
z0
p
ψ
q(x)dx to be in the classes U ST and U CV .
Theorem 3.1. If
q
X
j=1
b
j>
p
X
j=1
a
j, a
j> 0 and 1 +
q
X
j=1
β
j>
p
X
j=1
α
j,
then a sufficient condition for the function
pφ
q(z) = R
z0 p
ψ
q(x)dx to be in the class U ST is (3.1) 3
pψ
q(1) − 2
pψ
q(a
j− α
j, α
j)
1,p; (b
j− β
j, β
j)
1,q; 1
+ 2
Q
pj=1
Γ(a
j− α
j) Q
qj=1
Γ(b
j− β
j) ≤ M
1+ Q
pj=1
Γa
jQ
qj=1
Γb
j. Proof. Since
(3.2)
pφ
q(z) = Z
z0
p
ψ
q(x)dx = Q
pj=1
Γa
jQ
qj=1
Γb
jz +
∞
X
n=2
Q
pj=1
Γ[(a
j− α
j) + α
jn]
Q
qj=1
Γ[(b
j− β
j) + β
jn]
z
nn! ,
we have
∞
X
n=2
(3n − 2) Q
pj=1
Γ[(a
j− α
j) + α
jn]
Q
qj=1
Γ[(b
j− β
j) + β
jn]n!
(3.3)
= 3
∞
X
n=1
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n! − 2
"
∞X
n=0
Q
pj=1
Γ[(a
j− α
j) + α
jn]
Q
qj=1
Γ[(b
j− β
j) + β
jn]n!
− Q
pj=1
Γ(a
j− α
j) Q
qj=1
Γ(b
j− β
j) − Q
pj=1
Γa
jQ
qj=1
Γb
j#
= 3
pψ
q(1) − 2
pψ
q(a
j− α
j, α
j)
1,p; (b
j− β
j, β
j)
1,q; 1
+ 2
Q
pj=1
Γ(a
j− α
j) Q
qj=1
Γ(b
j− β
j) − Q
pj=1
Γa
jQ
qj=1
Γb
j.
In view of Lemma 1.1, (3.3) leads to the result (3.1).
Theorem 3.2. If
q
X
j=1
b
j>
p
X
j=1
a
j, a
j> 0 and 1 +
q
X
j=1
β
j>
p
X
j=1
α
j,
then a sufficient condition for the function
pφ
q(z) = R
z0 p
ψ
q(x)dx to be in the class U CV is
(3.4)
pψ
q(a
j+ α
j, α
j)
1,p; (b
j+ β
j, β
j)
1,q; 1
+
pψ
q(1) ≤ M
2+ Q
pj=1
Γa
jQ
qj=1
Γb
j. Proof. Since
pφ
q(z) has the form (3.2), then
∞
X
n=2
n
2Q
pj=1
Γ[(a
j− α
j) + α
jn]
Q
qj=1
Γ[(b
j− β
j) + β
jn]n!
(3.5)
=
∞
X
n=1
(n + 1) Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n!
=
∞
X
n=0
Q
pj=1
Γ[(a
j+ α
j) + α
jn]
Q
qj=1
Γ[(b
j+ β
j) + β
jn]n! +
∞
X
n=0
Q
pj=1
Γ(a
j+ α
jn) Q
qj=1
Γ(b
j+ β
jn)n! − Q
pj=1
Γa
jQ
qj=1
Γb
j=
pψ
q(a
j+ α
j, α
j)
1,p; (b
j+ β
j, β
j)
1,q; 1
+
pψ
q(1) − Q
pj=1
Γa
jQ
qj=1
Γb
j,
which in view of Lemma 1.2 gives the desired result (3.4).
4. P
ARTICULARC
ASES4.1. By setting α
1= α
2= · · · = α
p= 1; β
1= β
2= · · · = β
q= 1 and M
1= M
2= M
3=
Q
p j=1Γa
jQ
qj=1
Γb
j,
Theorems 2.1, 2.3, 3.1 and 3.2 reduce to the results recently obtained by Shanmugam, Ra- machandran, Sivasubramanian and Gangadharan [11].
4.2. By specifying the parameters suitably, the results of this paper readily yield the results due
to Dixit and Verma [1].
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