General Mathematics Vol. 18, No. 2 (2010), 15–29
Properties of a class of multivalent functions starlike
with respect to symmetric and conjugate points 1
S. M. Khairnar, S. M. Rajas
Abstract
Let S be the class of functions analytic and multivalent in the open
unit disc given by
p
f (z) = z +
∞
X
ap+n z p+n and an ∈ C.
n=1
In this paper we have studied three subclasses
∗
Ss∗ M (α, β, δ), Sc∗ M (α, β, δ) and Ssc
M (α, β, δ)
consisting of analytic functions with negative coefficients and starlike
with respect to symmetric points, starlike with respect to conjugate
points and starlike with respect to symmetric conjugate points, respectively. Here, we discuss coefficient inequality, growth and distortion theorems, extreme points, closure theorems and convolution properties of
these classes.
2000 Mathematics Subject Classification: 30C45, 30C50.
1
Received 12 September, 2008
Accepted for publication (in revised form) 10 October, 2008
15
16
S. M. Khairnar, S. M. Rajas
Key words and phrases: Analytic functions, Multivalent functions,
Starlike with respect to symmetric points, distortion bounds, Convolution
theorem.
1
Introduction
Let S be the class of functions analytic and multivalent in the open unit disc
given by
(1)
f (z) = z p +
∞
X
ap+n z p+n and an ∈ C.
n=1
Let M be the subclass of S consisting of functions f of the form
(2)
f (z) = z p −
∞
X
ap+n z p+n
n=1
where an ≥ 0, an ∈ IR.
In [5], Sakaguchi introduced the class of analytic functions which are univalent
and starlike with respect to symmetric points. This class is denoted by Ss∗ and
n
o
zf 0
satisfies Re f (z)−f
(−z) > 0 for z ∈ ID.
This definition has given rise to many generalized and extended classes of
∗ M (α, β, δ) confunctions. The subclasses Ss∗ M (α, β, δ), Sc∗ M (α, β, δ) and Ssc
sisting of analytic functions with negative coefficients were introduced by
Halim and A. Janteng in [1] and are respectively starlike with respect to
symmetric points, starlike with respect to conjugate points and starlike with
respect to symmetric conjugate points. Here α, β and δ satisfy the conditions
0 ≤ α < 1, 0 < β < 1, 0 ≤ δ < p and 0 <
2(1−β)
1+αβ
< 1. This paper extends the
result in [2] to other properties namely distortion, convex combination and
convolution.
Properties of a class of multivalent functions starlike...
17
Let S ∗ be the subclass of S consisting of functions starlike in D. Notice that
³ 0´
f ∈ S ∗ iff Re zff > 0 for z ∈ D.
Consider Ss∗ , the subclass of S consisting of function given by (1) consisting
of function given by (1) satisfying
½
¾
zf 0 (z)
Re
> 0, z ∈ D.
f (z) − f (−z)
These functions are called starlike with respect to symmetric points and were
introduced by Sakaguchi in [5]. The same class is also considered by Robertson
[8], Stankiewicz [4], Z. Wu [14], Owa and Z. Wu [13] and Aini Janteng, M.
Darus [2]. K. M. El-Ashwah and Thomas in [10], have introduced two other
∗ .
subclasses namely Sc∗ and Ssc
In [14], Sudarshan et. al and Aini Janteng, M. Darus [2] have discussed the
subclasses Ss∗ M (α, β, δ) of functions f analytic and univalent in ID and satisfying the condition
¯
¯
¯
¯
¯ zf 0 (z)
¯
¯
¯ αzf 0 (z)
¯
¯
¯
¯
¯ f (z) − (−z) − (p + δ)¯ < β ¯ f (z) − f (−z) + (p − δ)¯
for some 0 ≤ α ≤ 1, 0 < β ≤ 1, 0 ≤ δ < p and z ∈ ID.
However, in this paper we consider the subclass M defined by (2).
Definition 1 A function f ∈ Ss∗ M (α, β, δ) is said to be starlike with respect
to symmetric points if it satisfies
¯
¯
¯
¯
¯
¯
¯
¯ αzf 0 (z)
zf 0 (z)
¯
¯
¯
¯
¯ f (z) − f (−z) − (p + δ)¯ < β ¯ f (z) − f (−z) + (p − δ)¯ for p ∈ N and z ∈ D.
Definition 2 A function f ∈ Sc∗ M (α, β, δ) is said to be starlike with respect
to conjugate points if it satisfies
¯
¯
¯
¯
¯ αzf 0 (z)
¯ zf 0 (z)
¯
¯
¯
¯
¯
¯
− (p + δ)¯ < β ¯
+ (p − δ)¯ for p ∈ N and z ∈ D.
¯
¯ f (z) + f (z)
¯ f (z) + f (z)
¯
¯
18
S. M. Khairnar, S. M. Rajas
∗ M (α, β, δ) is said to be starlike with respect
Definition 3 A function f ∈ Ssc
to symmetric conjugate points if it satisfies
¯
¯
¯
¯
¯
¯
¯ αzf 0 (z)
¯
0 (z)
zf
¯
¯
¯
¯
− (p + δ)¯ < β ¯
+ (p − δ)¯ for p ∈ N and z ∈ D.
¯
¯ f (z) − f (−z)
¯
¯ f (z) − f (−z)
¯
Notice that the above conditions imposed on α, β and δ in the introduction
are necessary to ensure that these classes form a subclass of S.
First we state the preliminary results similar to those obtained by Halim et.
al. in [1], required for proving our main results.
2
Preliminaries
Theorem 1 f ∈ Ss∗ M (α, β, δ) if and only if
(3)
∞
X
(1+βα)(p+n)+[1−βp+δ(1+β)][(−1)p+n −1]
β{αp+(p−δ)[1−(−1)p ]}+(−1)p (p+δ)−δ
n=1
ap+n ≤ 1 for p ∈ N
Corollary 1 If f ∈ Ss∗ M (α, β, δ) then
ap+n ≤
β{αp + (p − δ)[1 − (−1)]} + (−1)p (p + δ) − δ
,
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
for p ∈ N and n ≥ 1.
Theorem 2 f ∈ Sc∗ M (α, β, δ) if and only if
(4)
∞
X
(1 + βα)(p + n) + 2[p(β − 1) − δ(1 + β)]
n=1
β{αp + 2(p − δ)} − (p + 2δ)
ap+n ≤ 1 for p ∈ N.
Corollary 2 If f ∈ Sc∗ M (α, β, δ) then
ap+n ≤
β{αp + 2(p − δ)} − (p + 2δ)
for p ∈ N and n ≥ 1.
(1 + βα)(p + n) + 2[p(β − 1) − δ(1 + β)]
Properties of a class of multivalent functions starlike...
19
∗ M (α, β, δ) if and only if
Theorem 3 f ∈ Ssc
(5)
∞
X
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
β{αp + (p − δ)[1 − (−1)p } + (−1)p (p + δ) − δ
n=1
ap+n ≤ 1,
for p ∈ N.
∗ M (α, β, δ) then
Corollary 3 If f ∈ Ssc
ap+n ≤
for p ∈ N
3
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
,
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
and n ≥ 1.
Growth and Distortion Theorems
Theorem 4 Let the function f be defined by (2) and belong to the class
Ss∗ M (α, β, δ). Then for {z : 0 < |z| = r < 1},
r−
β{α + 2(1 − δ)} − (1 + 2δ) 2
β{α + 2(1 − δ)} − (1 + 2δ) 2
r ≤ |f (z)| ≤ r+
r ,
2(1 + βα)
2(1 + βα)
for p ∈ N.
Proof. Let f (z) = z p −
∞
P
n=1
(6) |f (z)| ≤|z|p +
∞
X
ap+n z p+n , p ∈ N.
ap+n |z|p+n
n=1
≤ |z|p + |z|p+1
∞
X
ap+n
n=1
β{αp+(p − δ)(1 − (−1)p )}+(−1)p (p + δ) − δ
(1 + βα)(p + 1) + [1−βp+δ(1 + β)(−1)p+1 − 1]
β{αp + (p − δ)(1 − (−1)p )} + (−1)p (p + δ) − δ p+1
= rp +
r .
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
≤ |z|p +|z|p+1
20
S. M. Khairnar, S. M. Rajas
Similarly,
(7)|F (z)| ≥ |z|p −
∞
X
ap+n |z|p+n
n=1
≥ |z|p − |z|p+1
∞
X
ap+n
n=1
β{αp + (p − δ)(1 − (−1)p )} + (−1)p (p + δ) − δ
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
β{αp + (p − δ)(1 − (−1)p )} + (−1)p (p + δ) − δ p+1
= rp −
r .
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
≥ |z|p − |z|p+1
Hence the result.
The result is sharp for
(8)
f (z) = z p −
β{αp + (−δ)(1 − (−1)p )} + (−1)p (p + δ) − δ
z p+1
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
at z = ±r.
Next we state similar results for functions belonging to Sc∗ M (α, β, δ) and
∗ M (α, β, δ). Method of proof is same as in Theorem 4.
Ssc
Theorem 5 Let the function f be defined by (2) and belong to the class
Sc∗ M (α, β, δ), then for {z : 0 < |z| = r < 1},
β{αp + 2(p − δ)} − (p + 2δ)
rp+1 ≤ |f (z)|
(1 + βα)(p + 1) + 2[p(β − 1) − δ(1 + β)]
β{αp + 2(p − δ)} − (p + 2δ)
≤ rp +
rp+1 .
(1 + βα)(p + 1) + 2[p(β − 1) − δ(1 + β)]
rp −
The result is sharp for
(9)
f (z) = z p −
β{αp + 2(p − δ)} − (p + 2δ)
z p+1 .
(1 + βα)(p + 1) + 2[p(β − 1) − δ(1 + β)]
Theorem 6 Let the function f be defined by (2) and belong to the class
∗ M (α, β, δ). Then for {z : 0 < |z| = r < 1},
Ssc
β{αp + 2(p − δ)}[1 − (−1)p ]} + (−1)p (p + δ) − δ p+1
r
≤ |f (z)|
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
≤ rp +
rp+1 .
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
rp −
Properties of a class of multivalent functions starlike...
21
The result sharp for
(10)
f (z) = z p −
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
z p+1
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
at z = ±r.
Theorem 7 Let f ∈ Ss∗ M (α, β, δ), then for {z : 0 < |z| = r < 1}
prp−1 −
(p + 1)β{αp + (p − δ)(1 − (−1)p )} + (−1)p (p + δ) − δ p
r ≤ |f 0 (z)| ≤
(1 + βα)(p + 1) + (1 − βp + δ(1 + β)][(−1)p+1 − 1]
prp−1 +
(p + 1)β{αp + (p − δ)(1 − (−1)p )} + (−1)p (p + δ) − δ p
r .
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
The result is sharp for function given by (8).
Theorem 8 Let f be the function defined by (2) and belonging to the class
Sc∗ M (α, β, δ), then for {z : 0 < |z| = r < 1}
(p + 1)β{αp + 2(p − δ)} − (p + 2δ)
rp ≤ |f 0 (z)|
(1 + βα)(p + 1) + 2[p(β − 1) − δ(1 + β)]
β(p + 1){αp + 2(p − δ)} − (p + 2δ)
rp .
≤ prp−1 +
(1 + βα)(p + 1) + 2[p(β − 1) − δ(1 + β)]
prp−1 −
The result is sharp for the function given by (9).
Theorem 9 Let f be the function defined by (2) and belonging to the class
∗ M (α, β, δ), then for {z : 0 < |z| = r < 1}
Ssc
(p + 1)β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ p
r ≤ |f 0 (z)|
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)](−1)p+1 − 1]
(p + 1)β{αp + (p − δ)[1 − (−1)p ]} + (−1p )(p + δ) − δ p
≤ prp−1 +
r .
(1 + βα)(p + 1) + [1 − βp + δ(1 + β)][(−1)p+1 − 1]
prp−1 −
The result is sharp for the function given by (10).
1−
β{α + 2(1 − δ)} − (1 + 2δ)
β{α + 2(1 − δ)} − (1 + 2δ)
r ≤ |f 0 (z) ≤ 1 +
r.
(1 + βα)
(1 + βα)
22
S. M. Khairnar, S. M. Rajas
The result is sharp for
β{α + 2(1 − δ)} − (1 + 2δ) 2
z
2(1 + βα)
f (z) = z −
at z = ±r and for p = 1 ∈ N .
4
Closure Theorems
All three subclasses discussed here are closed under convex linear combinations. We prove for the class Ss∗ M (α, β, δ). It can be proved similarly for
∗ M (α, β, δ).
Sc∗ M (α, β, δ) and Ssc
Theorem 10 Consider
fj (z) = z p −
∞
X
ap+n z p+n ∈ Ss∗ M (α, β, δ)
n=1
for j = 1, 2, 3, · · · , ` and p ∈ N then
g(z) =
X̀
cj fj (z) ∈ Ss∗ M (α, β, δ) where
j=1
X̀
j=1
Proof. Let
g(z) =
X̀
Ã
cj
z −
= zp −
∞
X
n=1
∞
X
n=1
where ep+n =
P̀
j=1
cj , ap+n,j .
∞
X
!
ap+n,j z
p+n
n=1
j=1
= zp −
p
z p+n
X̀
cj ap+n,j
j=1
ep+n z p+n
cj = 1.
Properties of a class of multivalent functions starlike...
23
Now g(z) ∈ Ss∗ M (α, β, δ) since
∞
X
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
β{αp + (p − δ)[1 − (−1)p ] + (−1)p (p + δ) − δ
n=1
∞ X̀
X
≤
n=1 j=1
≤
X̀
ep+n
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
cj ap+n,j
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
cj = 1 since,
j=1
∞
X
n=1
5
(1 + βα)(p + n)[1 − βp + δ(1 + β)[(−1)p+n − 1]
ap+n,j ≤ 1.
β{αp + (p − δ)[1 − (−1)p ] + (−1)p (p + δ) − δ
Extreme Points
Theorem 11 Let fp (z) = z p ,
fp+n (z) = z p −
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
z p+n
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
for n ≥ 1 and p ∈ N , then f ∈ Ss∗ M (α, β, δ) if and only if it can be expressed
as in the form
f (z) =
∞
X
λp+n fp+n (z)
n=0
where λp+n ≥ 0 and
∞
P
n=0
λp+n = 1, p ∈ N .
Proof. Let
(11)f (z) =
∞
X
λp+n fp+n (z), p ∈ N
n=0
= zp −
∞
X
β{αp+(p−δ)[1−(−1)p ]}+(−1)p (p+δ)−δ p+n
z
λp+n
(1+βα)(p+n)+[1−βp+δ(1+p)][(−1)p+n −1)
n=1
24
S. M. Khairnar, S. M. Rajas
Now f (z) ∈ Ss∗ M (α, β, δ) since
¸
∞ ·
X
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
·
¸
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
λp+n
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
∞
X
λp+n = 1 − λp ≤ 1, p ∈ N.
=
n=1
n=1
Conversely, suppose that f ∈ Ss∗ M (α, β, δ). Then by Corollary 1
ap+n ≤
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
, n ≥ 1.
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
Set
(12)
λp+n =
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
ap+n , n ≥ 1, p ∈ N
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
and λp = 1 −
∞
P
n=1
λp+n then f (z) =
∞
P
n=0
λp+n fp+n (z).
Similarly extreme points for functions belonging to Sc∗ M (α, β, δ) and
∗ M (α, β, δ) are found.
Ssc
Method of proving Theorem 12 and Theorem 13 are similar to that of Theorem
11.
Theorem 12 Let fp (z) = z p ,
fp+n (z) = z p −
β(αp + 2(p − δ)) − (p + 2δ)
z p+n
(1 + βα)(p + n) + 2[p(β − 1) − δ(1 + β)]
for n ≥ 1 and p ∈ N . Then f ∈ Sc∗ M (α, β, δ) if and only if it can be expressed
in the form
f (z) =
∞
X
n=0
λp+n fp+n (z) where λp+n ≥ 0 and
∞
X
n=0
λp+n = 1.
Properties of a class of multivalent functions starlike...
25
Theorem 13 Let fp (z) = z p
fp+n (z) = z p −
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
z p+n
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
∗ M (α, β, δ) if and only if it can be expressed
for n ≥ 1 and p ∈ N . Then f ∈ Ssc
in the form
f (z) =
∞
X
λp+n fp+n (z) where λp+n ≥ 0 and
n=0
6
∞
X
λp+n = 1.
n=0
Convolution Theorems
∗ M (α, β, δ) are closed
The three subclasses Ss∗ M (α, β, δ), Sc∗ M (α, β, δ) and Ssc
under convolution. We prove first for the class Ss∗ M (α, β, δ).
Theorem 14 Let f, g ∈ Ss∗ M (α, β, δ) where
p
f (z) = z −
∞
X
ap+n z
p+n
and g(z) = z −
n=1
then f ∗ g ∈
Ss∗ M (α, β, ν).
p
∞
X
bp+n z p+n
n=1
For
[(p + n) + (1 + δ)[(−1)p+n − 1]][N ]2 + [δ − (−1)p (p + δ)][D]2
<ν
[αp + (p − δ)[1 − (−1)p ]][D]2 − {α(p + n) + (δ − p)[(−1)p+n − 1][N ]2
where
N
= β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
D = (1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1].
Proof. We have f ∈ Ss∗ M (α, β, δ) if and only if
(13)
∞
X
(1 + βα)(p + n) + (1 − βp + δ(1 + β)][(−1)p+n − 1]
n=1
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
ap+n ≤ 1.
Similarly
(14)
∞
X
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
n=1
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
bp+n ≤ 1.
26
S. M. Khairnar, S. M. Rajas
To find a smallest number ν such that
(15)
∞
X
(1 + να)(p + n) + [1 − νp + δ(1 + ν)][(−1)p+n − 1]
ν{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
n=1
ap+n bp+n ≤ 1.
By Cauchy Schwarz inequality (13) and (14) imply
(16)
∞
X
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1] p
ap+n bp+n ≤ 1
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
n=1
(15) will hold for
(1 + να)(p + n) + [1 − νp + δ(1 + ν)][(−1)p+n − 1]
ap+n bp+n
ν{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
≤
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
q
p
ap+n bp+n ]β{αp + (p − δ)[1 − (−1)p ] + (−1)p (p + δ) − δ
ap+n bp+n ,
that is if
p
(17) ap+n bp+n ≤ {[ν{αp+(p−δ)[1−(−1)p ]}+(−1)p (p+δ)−δ][(1+βα)(p+n)
+[1 − βp + δ(1 + δ)][(−1)p+n − 1]}/{[β{αp + (p − δ)[1 − (−1)p ]}
+(−1)p (p + δ) − δ][(1 + να)(p + n) + [1 − νp + δ(1 + ν)][(−1)p+n − 1]}
(16) implies
(18)
p
ap+n bp+n ≤
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
.
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
Thus it is enough to show that
β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
(1 + βα)(p + n) + [1 − βp + δ(1 + β)][(−1)p+n − 1]
≤ {[ν{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ][(1 + βα)(p + n)
+[1 − βp + δ(1 + β)][(−1)p+n − 1]}/{[β{αp + (p − δ)[1 − (−1)p ]}
+(−1)p (p + δ) − δ][(1 + να)(p + n) + [1 − νp + δ(1 + ν)][(−1)p+n − 1]}
Properties of a class of multivalent functions starlike...
27
which implies to
(p + n) + (1 + δ)[(−1)p+n − 1][N ]2 + [δ − (−1)p (p + δ)][D]2
≤ν
[αp + (p − δ)[1 − (−1)p ]][D]2 − {α(p + n) + (δ − p)[(−1)p+n − 1][N ]2
where
N
= β{αp + (p − δ)[1 − (−1)p ]} + (−1)p (p + δ) − δ
D = (1 + βα)(p + n) + [1 − βp + δ(1 + β)[(−1)p+n − 1]
Theorem 15 Let f, g ∈ Sc∗ M (α, β, δ) then f ∗ g ∈ Sc∗ M (α, β, ν) for
[(p + n) − 2(p + δ)][N ]2 − (p + 2δ)[D]2
≤ν
[αp + 2(p − δ)][D]2 − [α(p + n) + 2(p − δ)][N ]2
where
N
= β{αp + 2(p − δ)} + (p + 2δ)
D = (1 = βα)(p + n) + 2[p(β − 1) − δ(1 + β)].
∗ M (α, β, δ) is similar to Theorem 14.
Convolution theorem for subclass Ssc
Acknowledgements:
The first author is thankful and acknowledges the
support from the research projects funded from the Department of Science
and Technology, (Ref. No. SR/S4/MS:544/08), Government of India, National Board of Higher Mathematics, Department of Atomic Energy, (Ref.
No.NBHM/DAE/R.P.2/09), Government of India and BCUD, University of
Pune (UOP), Pune, (Ref No.BCUD, Engg.2009), and UGC New Delhi.
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S. M. Khairnar
Department of Mathematics
Maharashtra Academy of Engineering
Alandi, Pune - 412105, M. S., India
e-mail: smkhairnar2007@gmail.com
S. M. Rajas
Department of Mathematics
G. H. Raisoni Institute of Engineering and Technology
Pune - 412207, M. S. India
e-mail: sachin− rajas@rediffmail.com