UNIFORMLY STARLIKE AND UNIFORMLY
CONVEX FUNCTIONS PERTAINING TO SPECIAL
FUNCTIONS
V.B.L. CHAURASIA
AMBER SRIVASTAVA
Department of Mathematics
University of Rajasthan
Jaipur-302004, India
Department of Mathematics
Swami Keshvanand Institute of Technology,
Management and Gramothan
Jagatpura, Jaipur-302025, India
EMail: amber@skit.ac.in
Received:
04 September, 2006
Accepted:
14 July, 2007
Communicated by:
H.M. Srivastava
2000 AMS Sub. Class.:
30C45.
Key words:
Analytic functions, Univalent functions, Starlike functions, Convex functions,
Integral operator, Fox-Wright function.
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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Abstract:
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.
Acknowledgements:
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.
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Contents
1
Introduction
3
2
Main Results
5
3
An Integral Operator
9
4
Particular Cases
12
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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1.
Introduction
Let A denote the class of functions of the form
(1.1)
f (z) = z +
∞
X
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
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
an z n ,
an ≥ 0.
n=2
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A
f ∈ A is said to be starlike of order α, 0 ≤ α < 1, if and only if
function
zf 0 (z)
Re f (z) > α, z ∈ ∆. Also f of the form (1.1) is uniformly starlike, whenever
f (z)−f (ξ)
≥ 0, (z,ξ) ∈ ∆ × ∆. This class of all uniformly starlike functions is
0
(z−ξ)f (z)
denoted by U ST [4] (see also [5], [10] and [14]).
The function f of the form (1.1) is uniformly convex in ∆ whenever
f 00 (z)
≥ 0,
(z,ξ) ∈ ∆ × ∆.
Re 1 + (z − ξ) 0
f (z)
This class of all uniformly convex functions is denoted by U CV [3] (also refer
[2],[6], [9] and
it is said to be in the class U CV (α), α ≥ 0 if
00 Further
[13]).
zf (z) zf 0 (z)
Re 1 + f (z) ≥ α f 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)
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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
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
(3n − 2) |an | ≤ M1 .
n=2
Lemma 1.2. A sufficient
P∞condition for a function f of the form (1.1) to be in the
class U CV (α) is that n=2 n[(α + 1)n − α]P
an ≤ M2 , where M2 > 0 is a suitable
2
constant. In particular, f ∈ U CV whenever ∞
n=2 n an ≤ M2 .
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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)
z =
p ψq (z) = p ψq
q
(bj , βj )1,q ;
j=1 Γ(bj + βj n)n!
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where
1, . . . , p) and βj (j = 1, . . . , q) are real and positive and 1 +
Pq αj (jP=
p
j=1 βj >
j=1 αj .
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n=0
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2.
Main Results
Theorem 2.1. If
q
X
|bj | >
j=1
p
X
|aj | + 1, aj > 0
and
j=1
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
3−α
(|aj |, αj )1,p ;
(|aj + αj |, αj )1,p ;
1
(2.1)
1 + p ψq
p ψq
(|bj |, βj )1,q ;
(|bj + βj |, βj )1,q ;
1−α
Qp
j=1 Γaj
≤ M1 + Q q
.
j=1 Γbj
Proof. Since
Qp
Qp
∞
n
X
j=1 Γ[aj + αj (n − 1)]z
j=1 Γaj
Qq
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
Γ[a
+
α
(n
−
1)]
j
j
Q
j=1
(2.2)
[(3 − α)n − 2] q
≤ (1 − α)M1 .
j=1 Γ[bj + βj (n − 1)](n − 1)! n=2
Now, we have
∞
X
n=2
Qp
Q
j=1 Γ[aj + αj (n − 1)]
[(3 − α)n − 2] q
j=1 Γ[bj + βj (n − 1)](n − 1)! Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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Qp
Γ[a
+
α
(n
+
1)]
j
j
j=1
=
[(3 − α)(n + 2) − 2] Qq
j=1 Γ[bj + βj (n + 1)](n + 1)! n=0
∞ Qp
X
Γ[(a
+
α
)
+
nα
]
j
j
j
Q j=1
= (3 − α)
q
j=1 Γ[(bj + βj ) + nβj ]n! n=0
#
" ∞ Qp
Qp
X j=1 Γ(aj + αj n) 1
Γa
j
j=1
+ (1 − α)
− Qq
Qq
j=1 Γ(bj + βj n) n!
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
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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≤ (1 − α)M1
which in view of Lemma 1.1 gives the desired result.
Theorem 2.2. If
q
X
j=1
bj >
p
X
j=1
aj + 1, aj > 0
and
1+
q
X
j=1
βj >
p
X
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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.
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Theorem 2.3. If
q
X
j=1
bj >
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
(aj + 2αj , αj )1,p ;
(2.3) (1 + α) p ψq
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
(2.5)
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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Now, we have
∞
X
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Qp
j=1 Γ[aj + αj (n − 1)]
n[(α + 1)n − α] Qq
j=1 Γ[bj + βj (n − 1)](n − 1)!
n=2
Qp
∞
X
j=1 Γ(aj + αj n)
2
= (1 + α)
(n + 1) Qq
j=1 Γ[(bj + βj n)n!
n=1
Qp
∞
X
j=1 Γ(aj + αj n)
−α
(n + 1) Qq
.
Γ(b
+
β
n)n!
j
j
j=1
n=1
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Using (n + 1)2 = n(n + 1) + (n + 1), (2.5) may be expressed as
Qp
∞
X
j=1 Γ(aj + αj n)
(2.6) (1+α)
(n + 1) Qq
j=1 Γ(bj + βj n)(n − 1)!
n=1
Qp
∞
X
j=1 Γ(aj + αj n)
+
(n + 1) Qq
j=1 Γ(bj + βj n)n!
n=1
Q
p
∞
X
Γ(aj + αj n)
Qq j=1
= (1 + α)
j=1 Γ(bj + βj n)(n − 2)!
n=2
∞ Qp
X
Γ[(aj + αj ) + αj n]
Qqj=1
+ (2α + 3)
j=1 Γ[(bj + βj ) + βj n]n!
n=0
Q
p
∞
X
j=1 Γ(aj + αj n)
Q
+
q
j=1 Γ(bj + βj n)n!
n=1
(aj + 2αj , αj )1,p ;
= (1 + α) p ψq
1
(bj + 2βj , βj )1,q ;
Qp
(aj + αj , αj )1,p ;
j=1 Γaj
+ (2α + 3) p ψq
1 + p ψq (1) − Qq
,
(bj + βj , βj )1,q ;
j=1 Γbj
which is bounded above by M2 if and only if (2.3) holds. Hence the theorem is
proved.
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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3.
An Integral Operator
In this section we obtain sufficient conditions for the function
Z z
(aj , αj )1,p ;
z =
p ψq (x)dx
p φq
(bj , βj )1,q ;
0
to be in the classes U ST and U CV .
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Theorem 3.1. If
q
X
Amber Srivastava
bj >
j=1
p
X
aj , aj > 0
j=1
and
1+
q
X
j=1
βj >
p
X
vol. 9, iss. 1, art. 30, 2008
αj ,
j=1
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Rz
then a sufficient condition for the function p φq (z) = 0 p ψq (x)dx to be in the class
U ST is
(aj − αj , αj )1,p ;
(3.1)
3 p ψq (1) − 2 p ψq
1
(bj − βj , βj )1,q ;
Qp
Qp
Γ(a
−
α
)
j
j
j=1 Γaj
j=1
+ 2 Qq
≤ M1 + Q q
.
j=1 Γ(bj − βj )
j=1 Γbj
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Proof. Since
Z
(3.2)
Contents
p φq (z)
=
Close
z
p ψq (x)dx
0
Qp
∞ Qp
X
Γ[(aj − αj ) + αj n] z n
j=1 Γaj
Qj=1
= Qq
z+
,
q
Γ[(b
−
β
)
+
β
n]
n!
j
j
j
j=1 Γbj
j=1
n=2
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
−
α
)
Γ(a
Γa
j
j
j
j=1
j=1
− Qq
− Qq
Γ(b
−
β
)
j
j
j=1
j=1 Γbj
(aj − αj , αj )1,p ;
= 3 p ψq (1) − 2 p ψq
1
(bj − βj , βj )1,q ;
Qp
Qp
j=1 Γ(aj − αj )
j=1 Γaj
+ 2 Qq
− Qq
.
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
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
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vol. 9, iss. 1, art. 30, 2008
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q
X
j=1
bj >
p
X
j=1
aj , aj > 0
and
1+
q
X
j=1
Rz
βj >
p
X
αj ,
j=1
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
1 + p ψq (1) ≤ M2 + Qq
(3.4)
.
p ψq
(bj + βj , βj )1,q ;
j=1 Γbj
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Proof. Since p φq (z) has the form (3.2), then
Qp
∞
X
j=1 Γ[(aj − αj ) + αj n]
(3.5)
n2 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
Q
p
(aj + αj , αj )1,p ;
j=1 Γaj
= p ψq
1 + p ψq (1) − Qq
,
(bj + βj , βj )1,q ;
j=1 Γbj
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
Title Page
which in view of Lemma 1.2 gives the desired result (3.4).
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4.
Particular Cases
4.1. By setting α1 = α2 = · · · = αp = 1; β1 = β2 = · · · = βq = 1 and
Qp
j=1 Γaj
M1 = M 2 = 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].
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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References
[1] K.K. DIXIT AND V. VERMA, Uniformly starlike and uniformly convexity
properties for hypergeometric functions, Bull. Cal. Math. Soc., 93(6) (2001),
477–482.
[2] A. GANGADHARAN, T.N. SHANMUGAM AND H.M. SRIVASTAVA, Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. Math. Appl., 44 (2002), 1515–1526.
Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
[3] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
vol. 9, iss. 1, art. 30, 2008
[4] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. and Appl.,
155 (1991), 364–370.
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[5] S. KANAS AND F. RONNING, Uniformly starlike and convex functions
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[6] S. KANAS AND H.M. SRIVASTAVA, Linear operators associated with kuniformly convex functions, Integral Transform Spec. Funct., 9 (2000), 121–
132.
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[7] G. MURUGUSUNDARAMOORTHY, Study on classes of analytic function
with negative coefficients, Thesis, Madras University (1994).
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(1997), 92–109.
Close
[9] C. RAMACHANDRAN, T.N. SHANMUGAM, H.M. SRIVASTAVA AND A.
SWAMINATHAN, A unified class of k-uniformly convex functions defined by
the Dziok-Srivastava linear operator, Appl. Math. Comput., 190 (2007), 1627–
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[10] S. SHAMS, S.R. KULKARNI AND J.M.JAHANGIRI, Classes of uniformly
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AND A. GANGADHARAN, Generalized hypergeometric functions associated
with uniformly starlike and uniformly convex functions, Acta Ciencia Indica,
XXXIM(2) (2005), 469–476.
[12] H.M. SRIVASTAVA AND H.L. MANOCHA, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons,
New York, Chichester, Brisbane, and Toronto, 1984.
[13] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional calculus
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[14] H.M. SRIVASTAVA, A.K. MISHRA AND M.K. DAS, A class of parabolic
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Uniformly Starlike and
Uniformly Convex Functions
V.B.L. Chaurasia and
Amber Srivastava
vol. 9, iss. 1, art. 30, 2008
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