Volume 9 (2008), Issue 1, Article 30, 6 pp.
UNIFORMLY STARLIKE AND UNIFORMLY CONVEX FUNCTIONS
PERTAINING TO SPECIAL FUNCTIONS
V.B.L. CHAURASIA AND AMBER SRIVASTAVA
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF R AJASTHAN ,
JAIPUR -302004, I NDIA
D EPARTMENT OF M ATHEMATICS
S WAMI K ESHVANAND I NSTITUTE OF T ECHNOLOGY, M ANAGEMENT AND G RAMOTHAN
JAGATPURA , JAIPUR -302025, I NDIA
amber@skit.ac.in
Received 04 September, 2006; accepted 14 July, 2007
Communicated by H.M. Srivastava
A BSTRACT. The main object of this paper is to derive the sufficient conditions for the function
z{p ψq (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, FoxWright function.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Let A denote the class of functions of the form
∞
X
(1.1)
f (z) = z +
an z n ,
n=2
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
an z n ,
an ≥ 0.
n=2
0 (z)
A function f ∈ A is said to be starlike of order α, 0 ≤ α < 1, if and only if Re zff (z)
> α,
f (z)−f (ξ)
z ∈ ∆. Also f of the form (1.1) is uniformly starlike, whenever (z−ξ)f
≥ 0, (z,ξ) ∈
0 (z)
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
2
V.B.L. C HAURASIA AND A MBER S RIVASTAVA
∆ × ∆. This class of all uniformly starlike functions is denoted by U ST [4] (see also [5], [10]
and [14]).
00 (z)
The function f of the form (1.1) is uniformly convex in ∆ whenever Re 1 + (z − ξ) ff 0 (z)
≥
0, (z,ξ) ∈ ∆ × ∆. This class of all uniformly convex functions is denoted by U CV [3]
(also
00and [13]). Further it is said to be in the class U CV (α), α ≥ 0 if
refer 0[2], [6], [9]
(z) zf (z)
Re 1 + f (z) ≥ α zff 0 (z)
.
A functionf of the form (1.2) is said to be in the class U ST N (α), 0 ≤ α ≤ 1, if
f (z)−f (ξ)
Re (z−ξ)f
≥ α, (z,ξ) ∈ ∆ × ∆.
0 (z)
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
[(3 − α)n − 2] |an | ≤ (1 − α)M1 ,
n=2
where M1 > 0 is a suitable constant. In particular, f ∈ U ST whenever
∞
X
(3n − 2) |an | ≤ M1 .
n=2
LemmaP1.2. A sufficient condition for a function f of the form (1.1) to be in the class U CV (α)
is that ∞
1)n − α] an ≤ M2 , where M2 > 0 is a suitable constant. In particular,
n=2 n[(α + P
2
f ∈ U CV whenever ∞
n=2 n an ≤ M2 .
The Fox-Wright function [12, p. 50, equation 1.5] appearing in the present paper is defined
by
X
∞ Qp
n
(aj , αj )1,p ;
j=1 Γ(aj + αj n)z
Q
(1.3)
ψ
(z)
=
ψ
z
=
,
p q
p q
q
(bj , βj )1,q ;
Γ(b
+
β
n)n!
j
j
j=1
n=0
P
P
where αj (j = 1, . . . , p) and βj (j = 1, . . . , q) are real and positive and 1+ qj=1 βj > pj=1 αj .
2. M AIN R ESULTS
Theorem 2.1. If
q
X
j=1
|bj | >
p
X
|aj | + 1, aj > 0
j=1
and
1+
q
X
j=1
βj >
p
X
αj ,
j=1
then a sufficient condition for the function z{ p ψq (z)} to be in the class U ST (α), 0 ≤ α < 1, is
Qp
3−α
(|aj + αj |, αj )1,p ;
(|aj |, αj )1,p ;
j=1 Γaj
(2.1)
1 + p ψq
1 ≤ M1 + Q q
.
p ψq
(|bj + βj |, βj )1,q ;
(|bj |, βj )1,q ;
1−α
j=1 Γbj
Proof. Since
Qp
Qp
∞
X
Γ[aj + αj (n − 1)]z n
j=1 Γaj
Qq j=1
z{p ψq (z)} = Qq
z+
j=1 Γbj
j=1 Γ[bj + βj (n − 1)](n − 1)!
n=2
so by virtue of Lemma 1.1, we need only to show that
Qp
∞
X
Q
j=1 Γ[aj + αj (n − 1)]
(2.2)
[(3 − α)n − 2] q
≤ (1 − α)M1 .
j=1 Γ[bj + βj (n − 1)](n − 1)! n=2
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 30, 6 pp.
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U NIFORMLY S TARLIKE AND U NIFORMLY C ONVEX F UNCTIONS
3
Now, we have
Qp
Γ[a
+
α
(n
−
1)]
j
j
Q
j=1
[(3 − α)n − 2] q
Γ[b
+
β
(n
−
1)](n
−
1)!
j
j
j=1
n=2
Qp
∞
X
Γ[a
+
α
(n
+
1)]
j
j
j=1
=
[(3 − α)(n + 2) − 2] Qq
Γ[b
+
β
(n
+
1)](n
+
1)!
j
j
j=1
n=0
Qp
∞ X
Q j=1 Γ[(aj + αj ) + nαj ] = (3 − α)
q
Γ[(b
+
β
)
+
nβ
]n!
j
j
j
j=1
n=0
" ∞ Qp
#
Qp
X j=1 Γ(aj + αj n) 1
Γa
j
j=1
+ (1 − α)
Qq
− Qq
Γ(b
+
β
n)
n!
j
j
j=1
j=1 Γbj
n=0
(|aj + αj |, αj )1,p ;
= (3 − α) p ψq
1
(|bj + βj |, βj )1,q ;
Qp
(|aj |, αj )1,p ;
j=1 Γaj
+ (1 − α) p ψq
1 − (1 − α) Qq
(|bj |, βj )1,q ;
j=1 Γbj
∞
X
≤ (1 − α)M1
which in view of Lemma 1.1 gives the desired result.
Theorem 2.2. If
q
X
bj >
j=1
p
X
aj + 1, aj > 0
and
1+
j=1
q
X
βj >
j=1
p
X
αj ,
j=1
then a sufficient condition for the function z{p ψq (z)} to be in the class U ST N (α), 0 ≤ α < 1,
is:
Qp
3−α
(aj + αj , αj )1,p ;
(aj , αj )1,p ;
j=1 Γaj
1 + p ψq
1 ≤ M1 + Q q
.
p ψq
(bj + βj , βj )1,q ;
(bj , βj )1,q ;
1−α
j=1 Γbj
Proof. The proof of Theorem 2.2 is a direct consequence of Theorem 2.1.
Theorem 2.3. If
q
X
bj >
j=1
p
X
aj + 2, aj > 0
j=1
and
1+
q
X
j=1
βj >
p
X
αj ,
j=1
then a sufficient condition for the function z{p ψq (z)} to be in the class U CV (α), 0 ≤ α < 1, is
(2.3) (1 + α) p ψq
(aj + 2αj , αj )1,p ;
1
(bj + 2βj , βj )1,q ;
Qp
(aj + αj , αj )1,p ;
j=1 Γaj
+ (2α + 3) p ψq
1 + p ψq (1) ≤ M2 + Qq
.
(bj + βj , βj )1,q ;
j=1 Γbj
Proof. By virtue of Lemma 1.2, it suffices to prove that
Qp
∞
X
j=1 Γ[aj + αj (n − 1)]
(2.4)
n[(α + 1)n − α] Qq
≤ M2 .
j=1 Γ[bj + βj (n − 1)](n − 1)!
n=2
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 30, 6 pp.
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4
V.B.L. C HAURASIA AND A MBER S RIVASTAVA
Now, we have
(2.5)
∞
X
Qp
Γ[aj + αj (n − 1)]
j=1 Γ[bj + βj (n − 1)](n − 1)!
n=2
Qp
Qp
∞
∞
X
X
j=1 Γ(aj + αj n)
j=1 Γ(aj + αj n)
2
= (1 + α)
(n + 1) Qq
−α
(n + 1) Qq
.
Γ[(b
+
β
n)n!
Γ(b
+
β
n)n!
j
j
j
j
j=1
j=1
n=1
n=1
n[(α + 1)n − α] Qq
j=1
Using (n + 1)2 = n(n + 1) + (n + 1), (2.5) may be expressed as
Qp
Qp
∞
∞
X
X
j=1 Γ(aj + αj n)
j=1 Γ(aj + αj n)
(n + 1) Qq
(2.6) (1 + α)
+
(n + 1) Qq
j=1 Γ(bj + βj n)(n − 1)!
j=1 Γ(bj + βj n)n!
n=1
n=1
Q
Q
p
p
∞
∞
X
X
Γ[(aj + αj ) + αj n]
j=1 Γ(aj + αj n)
Qq
Qqj=1
= (1 + α)
+ (2α + 3)
j=1 Γ(bj + βj n)(n − 2)!
j=1 Γ[(bj + βj ) + βj n]n!
n=0
n=2
Q
p
∞
X
Γ(aj + αj n)
Qqj=1
+
j=1 Γ(bj + βj n)n!
n=1
(aj + αj , αj )1,p ;
(aj + 2αj , αj )1,p ;
1
= (1 + α) p ψq
1 + (2α + 3) p ψq
(bj + 2βj , βj )1,q ;
(bj + βj , βj )1,q ;
Qp
j=1 Γaj
+ p ψq (1) − Qq
,
j=1 Γbj
which is bounded above by M2 if and only if (2.3) holds. Hence the theorem is proved.
3. A N I NTEGRAL O PERATOR
In this section we obtain sufficient conditions for the function
Z z
(aj , αj )1,p ;
z =
p φq
p ψq (x)dx
(bj , βj )1,q ;
0
to be in the classes U ST and U CV .
Theorem 3.1. If
q
X
bj >
j=1
p
X
aj , aj > 0
j=1
q
X
βj >
j=1
p
X
αj ,
j=1
Rz
to be in the class U ST is
Qp
Qp
Γ(a
−
α
)
j
j
(aj − αj , αj )1,p ;
j=1
j=1 Γaj
3 p ψq (1) − 2 p ψq
1 + 2 Qq
≤ M1 + Q q
.
(bj − βj , βj )1,q ;
j=1 Γ(bj − βj )
j=1 Γbj
then a sufficient condition for the function p φq (z) =
(3.1)
1+
and
ψ (x)dx
0 p q
Proof. Since
Z
(3.2)
p φq (z)
=
0
z
Qp
∞ Qp
X
Γ[(aj − αj ) + αj n] z n
j=1 Γaj
Qj=1
z+
,
p ψq (x)dx = Qq
q
j=1 Γbj
j=1 Γ[(bj − βj ) + βj n] n!
n=2
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 30, 6 pp.
http://jipam.vu.edu.au/
U NIFORMLY S TARLIKE AND U NIFORMLY C ONVEX F UNCTIONS
5
we have
(3.3)
Qp
j=1 Γ[(aj − αj ) + αj n]
(3n − 2) Qq
j=1 Γ[(bj − βj ) + βj n]n!
n=2
" ∞ Qp
∞ Qp
X
X
Γ(a
+
α
n)
Γ[(aj − αj ) + αj n]
j
j
Qqj=1
Qqj=1
=3
−2
j=1 Γ(bj + βj n)n!
j=1 Γ[(bj − βj ) + βj n]n!
n=1
n=0
#
Qp
Qp
Γ(a
−
α
)
Γa
j
j
j
j=1
j=1
− Qq
− Qq
j=1 Γ(bj − βj )
j=1 Γbj
Qp
Qp
Γ(a
−
α
)
j
j
(aj − αj , αj )1,p ;
j=1
j=1 Γaj
= 3 p ψq (1) − 2 p ψq
1 + 2 Qq
− Qq
.
(bj − βj , βj )1,q ;
j=1 Γ(bj − βj )
j=1 Γbj
∞
X
In view of Lemma 1.1, (3.3) leads to the result (3.1).
Theorem 3.2. If
q
X
j=1
bj >
p
X
aj , aj > 0
1+
and
j=1
q
X
j=1
βj >
p
X
αj ,
j=1
Rz
then a sufficient condition for the function p φq (z) = 0 p ψq (x)dx to be in the class U CV is
Qp
(aj + αj , αj )1,p ;
j=1 Γaj
(3.4)
1 + p ψq (1) ≤ M2 + Qq
.
p ψq
(bj + βj , βj )1,q ;
j=1 Γbj
Proof. Since p φq (z) has the form (3.2), then
Qp
∞
X
j=1 Γ[(aj − αj ) + αj n]
2
(3.5)
n Qq
j=1 Γ[(bj − βj ) + βj n]n!
n=2
Qp
∞
X
j=1 Γ(aj + αj n)
=
(n + 1) Qq
j=1 Γ(bj + βj n)n!
n=1
Q
Qp
p
∞
∞ Qp
X
X
j=1 Γ[(aj + αj ) + αj n]
j=1 Γ(aj + αj n)
j=1 Γaj
Qq
Qq
=
+
− Qq
j=1 Γ[(bj + βj ) + βj n]n!
j=1 Γ(bj + βj n)n!
j=1 Γbj
n=0
n=0
Qp
(aj + αj , αj )1,p ;
j=1 Γaj
= p ψq
1 + p ψq (1) − Qq
,
(bj + βj , βj )1,q ;
j=1 Γbj
which in view of Lemma 1.2 gives the desired result (3.4).
4. PARTICULAR C ASES
4.1. By setting α1 = α2 = · · · = αp = 1; β1 = β2 = · · · = βq = 1 and
Qp
j=1 Γaj
M1 = M2 = M3 = Q q
,
j=1 Γbj
Theorems 2.1, 2.3, 3.1 and 3.2 reduce to the results recently obtained by Shanmugam, Ramachandran, 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].
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 30, 6 pp.
http://jipam.vu.edu.au/
6
V.B.L. C HAURASIA AND A MBER S RIVASTAVA
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J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 30, 6 pp.
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