General Mathematics Vol. 18, No. 2 (2010), 31–46
On a class of p-valent non-Bazilevic functions 1
Khalida Inayat Noor, Ali Muhammad, Muhammad Arif
Abstract
λ,µ
In this paper, we introduce a class Np,α
(a, c, A, B). We investigate
a number of inclusion relationships, distortion theorems for the class
³
´µ
zp
λ,µ
Np,α
(a, c, A, B), the lower and upper bounds of Re I λ (a,c)f
for
(z)
p
f (z) ∈
λ,µ
N p,α
(a, c, A, B) and some other interesting properties of p-valent
functions which are defined here by means of a certain linear integral
operator Ipλ (a, c) f (z).
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and phrases: Multivalent functions, Non-Bazilevic functions,
Choi-Saigo-Srivastava operator, Cho-Kown-Srivastava operator, Differential
Subordination.
1
Introduction
Let A(p) denote the class of functions f (z) normalized by
p
(1)
f (z) = z +
∞
X
ak+p z k+p , (p ∈ N = {1, 2, . . .}),
k=1
1
Received 18 September, 2008
Accepted for publication (in revised form) 10 October, 2008
31
32
K. I. Noor, A. Muhammad, M. Arif
which are analytic and p-valent in the open unit disc E = {z : |z| < 1}. If f (z)
and g(z) are analytic in E, we say that f (z) is subordinate to g(z), written
symbolically as follows:
f ≺ g in E or f (z) ≺ g(z), z ∈ E,
if there exists a Schwarz function w(z), which is analytic in E with
|w(0)| = 0 and |w(z)| < 1, z ∈ E,
such that
f (z) = g(w(z)), z ∈ E.
Indeed it is known that
f (z) ≺ g(z) (z ∈ E) ⇒ f (0) = g(0) and f (E) ⊂ g(E).
Furthermore, if the function g(z) is univalent in E, then we have the following
equivalence, see [6, 7],
f (z) ≺ g(z) (z ∈ E) ⇔ f (0) = g(0) and f (E) ⊂ g(E).
For functions fj (z) ∈ A(p), given by
(2)
p
fj (z) = z +
∞
X
ak+p,j z k+p (j = 1, 2) ,
k=1
we define the Hadamard product (or convolution) of f1 (z) and f2 (z) by
(3)
p
(f1 ? f2 ) (z) = z +
∞
X
ak+p,1 ak+p,2 z k+p = (f2 ? f1 ) (z) (z ∈ E) .
k=1
In our present investigation we shall make use of the Gauss hypergeometric
functions defined by
(4)
2 F1 (a, b; c; z) =
∞
X
(a)k (b)k k
z (z ∈ E) ,
(c)k (1)k
k=0
On a class of p-valent non-Bazilevic functions
33
where a, b, c ∈ C, c ∈
/ Z−
0 = {0, −1, −2, . . .} and (k)n denote the Pochhammer
symbol (or the shifted factorial) given, in terms of the Gamma function Γ, by

Γ (k + n)  k (k + 1) (k + 2) . . . (k + n − 1) , n ∈ N
(k)n =
=
 1, n = 0.
Γ (k)
We note that the series defined by (4) converges absolutely for z ∈ E and
hence 2 F1 (a, b; c; z) represents an analytic function in E, see [13].
We define a function Φp (a, c; z) by
Φp (a, c; z) = z p +
∞
X
(a)
k
k=0
(c)k
z k+p
¡
¢
a ∈ R; c ∈ R\Z−
0 = {0, −1, . . .} .
With the aid of the function Φp (a, c; z), we consider a function Φ†p (a, c; z)
defined by
Φp (a, c; z) ? Φ†p (a, c; z) =
zp
(1 − z)λ+p
, z ∈ E,
where λ > −p. This function yields the following family of linear operators
(5)
Ipλ (a, c) f (z) = Φ†p (a, c; z) ? f (z), z ∈ E,
where a, c ∈ R\Z−
0 . For a function f (z) ∈ A(p), given by (1), it follows from
(5) that for λ > −p and a, c ∈ R\Z−
0
(6)
Ipλ (a, c) f (z)
∞
X
(c)k (λ + p)k
= z +
ap+k z p+k
(a)k (1)k
p
k=0
= z p 2 F1 (c, λ + p; a; z) ? f (z), z ∈ E.
From equation (6) we deduce that
(7)
³
´0
z Ipλ (a, c) f (z) = (λ + p) Ipλ+1 (a, c) f (z) − λIpλ (a, c) f (z),
and
(8)
³
´0
z Ipλ (a + 1, c) f (z) = aIpλ (a, c) f (z) − (a − p) Ipλ (a + 1, c) f (z).
34
K. I. Noor, A. Muhammad, M. Arif
We also note that
Zz
Ip0 (a
+ 1, 1) f (z) = p
Ip0 (p, 1) f (z)
=
Ip1 (p, 1) f (z) =
f (t)
dt,
t
0
1
Ip (p +
zf 0 (z)
1, 1) f (z) = f (z),
,
p
2zf 0 (z) + z 2 f 00 (z)
Ip2 (p, 1) f (z) =
,
p (p + 1)
f (z) + zf 0 (z)
Ip2 (p + 1, 1) f (z) =
,
p (p + 1)
Ipn (a, a) f (z) = Dn+p−1 f (z), n ∈ N, n > −p,
where Dn+p−1 f (z) is the Ruscheweyh derivative of (n + p − 1)th order, see [4].
¡
¢
The operator Ipλ (a, c) λ > −p, a; c ∈ R\Z−
was recently introduced by
0
Cho et al [1], who investigated (among other things) some inclusion relationships and argument properties of various subclasses of multivalent functions
in A(p), which were defined by means of the operator Ipλ (a, c).
For λ = c = 1 and a = n + p, the Cho-Kown-Srivastava operator yields
Ip1 (n + p, 1) f (z) = In,p (n > −p) ,
where In,p denotes an integral operator of the (n + p − 1)th order, which was
studied by Liu and Noor [5], see also [9, 10]. The linear operator I1λ (µ + 2, 1)
(λ > −1, µ > −2) was also recently introduced and studied by Choi et al [2].
For relevant details about further special cases of the Choi-Saigo-Srivastava
operator I1 (λ + 2, 1), the interested reader may refer to the works by Cho et
al [2] and Choi et al [1], see also [3].
Using the Cho-Kown-Srivastava operator Ipλ (a, c), we now define a subclass
of A(p) as follows:
On a class of p-valent non-Bazilevic functions
35
Definition 1 Assume that 0 < µ < 1, α ∈ C, −1 ≤ B ≤ 1, A 6= B, A ∈ R,
λ,µ
we say that a function f (z) ∈ A(p) is in the class Np,α
(a, c, A, B) if it satisfies:
(
µ
zp
(1−α) λ
Ip (a, c) f (z)
Ã
¶µ
−α
Ipλ+1 (a, c)
Ipλ (a, c)
!µ
zp
Ipλ (a, c) f (z)
¶µ )
≺
1+Az
, z ∈ E,
1+Bz
where the powers are understood as a principal values.
λ,µ
λ,µ
In particular, we let Np,α
(a, c, 1 − 2ρ, −1) = Np,α
(a, c, ρ) denote the subλ,µ
class Np,α
(a, c, A, B) for A = 1 − 2ρ, B = −1 and 0 ≤ ρ < p. It is obvious
λ,µ
that f (z) ∈ Np,α
(a, c, ρ) if and only if f (z) ∈ A(p) and it satisfies
(
µ
Re (1−α)
zp
Ipλ (a, c) f (z)
Ã
¶µ
−α
Ipλ+1 (a, c)
Ipλ (a, c)
!µ
zp
Ipλ (a, c) f (z)
¶µ )
> ρ, z ∈ E.
Special Cases
0,µ
(i) When a = c = p = 1, λ = 0, then N1,α
(1, 1, A, B) is the class studied
by Z. Wang et al [14].
0,µ
(ii) The subclass N1,−1
(1, 1, 1, −1) = N (µ) has been studied by Obradovic
[11].
(iii) If a = c = p = 1, λ = 0, α = B = −1 and A = 1 − 2ρ, then the
0,µ
class N1,−1
(1, 1, 1 − 2ρ, −1) reduces to the class of non-Bazilevic func-
tions of order ρ(0 ≤ ρ < 1). The Fekete-Szegö problem of the class
0,µ
N1,−1
(1, 1, 1 − 2ρ, −1) were considered by N. Tuneski and M .Darus [12].
2
Preliminary Results
In this section we recall some known results.
36
K. I. Noor, A. Muhammad, M. Arif
Lemma 1 Let the function h(z) be analytic and convex (univalent) in E with
h(0) = 1. Suppose also that the function Φ(z) given by
Φ(z) = 1 + c1 z + c2 z 2 + . . .
is analytic in E. If
(9)
Φ(z) +
z Φ0 (z)
≺ h(z) (z ∈ E; Reγ ≥ 0; γ 6= 0) ,
γ
then
γ
Φ(z) ≺ Ψ(z) = γ
z
Zz
tγ−1 h(t)dt ≺ h(z) (z ∈ E) ,
0
and Ψ(z) is the best dominant of (9).
3
Main Result
λ,µ
(a, c, A, B). Then
Theorem 1 Let Reα > 0 and f (z) ∈ Np,α
µ
(10)
zp
Ipλ (a, c) f (z)
¶µ
(λ + p) µ
≺
α
Z1
0
1 + Az
1 + Azu (λ+p)µ −1
u α
du ≺
.
1 + Bzu
1 + Bz
Proof. Let
µ
(11)
Φ(z) =
zp
Ipλ (a, c) f (z)
¶µ
.
Then Φ(z) is analytic in E with Φ(0) = 1. Taking logarithmic differentiation
of (11) both sides and using the identity (7) in the resulting equation, we
deduce that
(
µ
(1 − α)
zp
Ipλ (a, c) f (z)
Ã
¶µ
−α
Ipλ+1 (a, c)
Ipλ (a, c)
!µ
zp
Ipλ (a, c) f (z)
¶µ )
On a class of p-valent non-Bazilevic functions
= Φ(z) +
Now, by Lemma 1 for γ =
µ
zp
Ipλ (a, c) f (z)
¶µ
αz Φ0 (z)
1 + Az
≺
.
(λ + p) µ
1 + Bz
(λ+p)µ
,
α
we deduce that
(λ + p) µ − (λ+p)µ
α
≺ q(z) =
z
α
=
37
(λ + p) µ
α
Z1
0
Zz
t
(λ+p)µ
−1
α
µ
0
1 + At
1 + Bt
¶
dt
1 + Azu (λ+p)µ −1
1 + Az
u α
du ≺
,
1 + Bzu
1 + Bz
and the proof is complete.
Theorem 2 Let 0 ≤ α2 ≤ α1 . Then
λ,µ
λ,µ
Np,α
(a, c, A, B) ⊂ Np,α
(a, c, A, B) .
1
2
λ,µ
Proof. Let f (z) ∈ Np,α
1 (a, c, A, B). Then by Theorem 3.1 we have
λ,µ
f (z) ∈ Np,0
(a, c, A, B) .
Since
(
µ
(1 + α2 )
zp
Ipλ (a, c) f (z)
Ã
¶µ
− α2
Ipλ+1 (a, c)
Ipλ (a, c)
!µ
zp
Ipλ (a, c) f (z)
¶µ )
¶µ
½
µ
µ
¶µ
¶µ
α2
zp
zp
α2
= 1+
(1 + α1 )
−
α1
α1
Ipλ (a, c) f (z)
Ipλ (a, c) f (z)
Ã
−α1
Ipλ+1 (a, c)
Ipλ (a, c)
λ,µ
Wee see that f (z) ∈ Np,α
2 (a, c, A, B).
!µ
zp
Ipλ (a, c) f (z)
¶µ )
≺
1 + Az
.
1 + Bz
38
K. I. Noor, A. Muhammad, M. Arif
Theorem 3 Let Reα > 0, 0 < µ < 1, −1 ≤ B < A ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then
(λ + p) µ
α
Z1
0
1 + Au (λ+p)µ −1
u α
du < Re
1 + Bu
(12)
µ
zp
Ipλ (a, c) f (z)
(λ + p) µ
α
<
Z1
0
¶µ
1 − Au (λ+p)µ −1
u α
du,
1 − Bu
and the inequality (12) is sharp, with the extremal function defined by
 −1

µ
 (λ + p) µ Z1 1 + Azu (λ+p)µ
−1
λ
p
Ip (a, c) Fα,µ,A,B (z) = z
u α
du
.


α
1 + Bzu
(13)
0
λ,µ
Proof. Since f (z) ∈ Np,α
(a, c, A, B), according to Theorem 1, we have
µ
zp
λ
Ip (a, c) f (z)
¶µ
(λ + p) µ
≺
α
Z1
0
1 + Azu (λ+p)µ −1
u α
du,
1 + Bzu
Therefore it follows from the definition of subordination and A > B that
µ
Re
zp
Ipλ (a, c) f (z)
¶µ



 (λ + p) µ Z1 1 + Azu (λ+p)µ
< sup Re
u α −1 du


α
1 + Bzu
z∈E
0
≤
<
(λ + p) µ
α
(λ + p) µ
α
½
Z1
sup Re
0
Z1
0
z∈E
1 + Azu
1 + Bzu
¾
u
1 + Au (λ+p)µ −1
u α
du.
1 + Bu
(λ+p)µ
−1
α
du
On a class of p-valent non-Bazilevic functions
Also
µ
Re


 (λ + p) µ Z1 1 + Azu (λ+p)µ

> inf Re
u α −1 du


z∈E
α
1 + Bzu
¶µ
zp
39
Ipλ (a, c) f (z)
0
≥
>
(λ + p) µ
α
(λ + p) µ
α
½
Z1
inf Re
z∈E
0
Z1
0
1 + Azu
1 + Bzu
¾
u
(λ+p)µ
−1
α
du
1 − Au (λ+p)µ −1
du.
u α
1 − Bu
Note that the function Ipλ (a, c) Fα,µ,A,B (z) defined by (13) belongs to the
λ,µ
class Np,α
(a, c, A, B) and hence we obtain that the inequality (12) is sharp.
By applying the similar techniques that we used in proving Theorem 12, we
have the following result.
Theorem 4 Let Reα > 0, 0 < µ < 1, −1 ≤ A < B ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then
(λ + p) µ
α
Z1
0
1 + Au (λ+p)µ −1
u α
du < Re
1 + Bu
<
(14)
µ
zp
Ipλ (a, c) f (z)
(λ + p) µ
α
Z1
0
¶µ
1 − Au (λ+p)µ −1
u α
du,
1 − Bu
and the inequality (14) is sharp, with the extremal function defined by (13).
Theorem 5 Let 0 < µ < 1, Reα ≥ 0, −1 ≤ B < A ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then

(15)
 (λ + p) µ
α

<
Z1
0
(λ + p) µ
α
 12
1 − Au (λ+p)µ −1 
u α
du
< Re
1 − Bu
Z1
0
1 + Au
u
1 + Bu
 21
(λ+p)µ
−1
α
du ,
µ
zp
Ipλ (a, c) f (z)
¶µ
2
40
K. I. Noor, A. Muhammad, M. Arif
and inequality (15) is sharp, with the extremal function defined by (13).
Proof. According to Theorem 1, we have
µ
zp
Ipλ (a, c) f (z)
¶µ
≺
1 + Az
.
1 + Bz
Since −1 ≤ B < A ≤ 1, we have
1−A
0<
< Re
1−B
µ
zp
Ipλ (a, c) f (z)
¶µ
<
1+A
.
1+B
Hence the result follows by Theorem 3.
λ,µ
Note that the function defined by (13) belongs to Np,α
(a, c, A, B), we
obtain that the inequality (15) is sharp. By applying the similar arguments
as in Theorem 5, we have the following Theorem.
Theorem 6 Let 0 < µ < 1, Reα ≥ 0, −1 ≤ A < B ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then

(16)
 (λ + p) µ
α

<
Z1
0
(λ + p) µ
α
1 + Au
u
1 + Bu
Z1
0
 12
(λ+p)µ
−1
α
du < Re
µ
zp
Ipλ (a, c) f (z)
¶µ
2
 12
1 − Au (λ+p)µ −1 
u α
du
,
1 − Bu
and inequality (16) is sharp, with the extremal function defined by (13).
Theorem 7 Let 0 < µ < 1, Reα ≥ 0, −1 ≤ B < A ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then
(i)If α = 0, the for |z| = r < 1, we have
µ
(17)
r
p
1 + Br
1 + Ar
¶1
µ
µ
¶1
¯
¯
µ
¯ λ
¯
p 1 − Br
≤ ¯Ip (a, c) f (z)¯ ≤ r
1 − Ar
On a class of p-valent non-Bazilevic functions
41
and inequality (17) is sharp , with the extremal function defined by
µ
Ipλ (a, c) f (z)
(18)
=z
p
1 + Bz
1 + Az
¶1
µ
.
(ii) If α 6= 0, the for |z| = r < 1, we have

(19)
rp 
(λ + p) µ
α

≤ rp 
Z1
0
(λ + p) µ
α
− µ1
1 + Aru (λ+p)µ −1 
du
u α
1 + Bru
Z1
0
¯
¯
¯
¯
≤ ¯Ipλ (a, c) f (z)¯
− µ1
1 − Aru (λ+p)µ −1 
u α
du
1 − Bru
,
and inequality (19) is sharp with the extremal function defined by (13).
λ,µ
Proof. (i) If α = 0. Since f (z) ∈ Np,α
(a, c, A, B), −1 ≤ B < A ≤ 1, we
λ,µ
obtain from the definition of Np,α
(a, c, A, B) that
µ
zp
Ipλ (a, c) f (z)
¶µ
≺
1 + Az
.
1 + Bz
Therefore it follows from the definition of the subordination that
µ
zp
Ipλ (a, c) f (z)
¶µ
=
1 + Aw(z)
,
1 + Bw(z)
where w(z) = c1 z + c2 z 2 + . . . is analytic E and |w(z)| ≤ |z|, so when |z| =
r < 1, we have
¯µ
¯
¶¯µ ¯
¯
¯
¯
¯
zp
¯
¯ = ¯ 1 + Aw(z) ¯ ≤ 1 + A |w(z)| ≤ 1 + Ar ,
¯ I λ (a, c) f (z) ¯
¯ 1 + Bw(z) ¯ 1 + B |w(z)|
1 + Br
p
and
¯µ
µ
¶¯µ
¶µ
p
p
¯
¯
z
z
1 − Ar
¯
¯
≥
.
¯ I λ (a, c) f (z) ¯ ≥ Re I λ (a, c) f (z)
1 − Br
p
p
It is obvious that (17) is sharp, with the extremal function defined by (18).
42
K. I. Noor, A. Muhammad, M. Arif
(ii) If α 6= 0. according to Theorem 1 we have
µ
zp
λ
Ip (a, c) f (z)
¶µ
(λ + p) µ
≺
α
Z1
0
1 + Azu (λ+p)µ −1
u α
du.
1 + Bzu
Therefore it follows from the definition of the subordination
µ
zp
Ipλ (a, c) f (z)
¶µ
(λ + p) µ
=
α
Z1
0
1 + Aw(z)u (λ+p)µ −1
u α
du,
1 + Bw(z)u
where w(z) = c1 z + c2 z 2 + . . . is analytic E and |w(z)| ≤ |z|, so when |z| =
r < 1, we have
¯µ
¶¯µ
¯
¯
zp
¯
¯
¯ I λ (a, c) f (z) ¯ ≤
(λ + p) µ
α
≤
(λ + p) µ
α
p
≤
(λ + p) µ
α
¯
Z1 ¯
¯ 1 + Aw(z)u ¯ (λ+p)µ −1
¯
¯
du
¯ 1 + Bw(z)u ¯ u α
0
Z1
0
Z1
0
1 + Au |w(z)| (λ+p)µ −1
u α
du
1 + Bu |w(z)|
1 + Aur (λ+p)µ −1
u α
du,
1 + Bur
and
¯µ
¶¯µ
¶µ
µ
Z1
¯
¯
zp
1 − Aur (λ+p)µ −1
zp
(λ + p) µ
¯
¯ ≥ Re
u α
du.
≥
¯ I λ (a, c) f (z) ¯
λ
Ip (a, c) f (z)
α
1 − Bur
p
0
λ,µ
Note that the function defined by (13) belongs to the class Np,α
(a, c, A, B),
we obtain that the inequality (19) is sharp. By applying the similar method
as in Theorem 5 we have
Theorem 8 Let 0 < µ < 1, Reα ≥ 0, −1 ≤ A < B ≤ 1 and f (z) ∈
λ,µ
Np,α
(a, c, A, B). Then
On a class of p-valent non-Bazilevic functions
43
(i) If α = 0, the for |z| = r < 1, we have
µ
(20)
r
p
1 − Br
1 − Ar
¶1
µ
¶1
¯
¯
µ
¯ λ
¯
p 1 + Br
≤ ¯Ip (a, c) f (z)¯ ≤ r
1 + Ar
µ
and inequality (20) is sharp, with the extremal function defined by (18).
(ii) If α 6= 0, the for |z| = r < 1, we have

(21)
rp 
(λ + p) µ
α

≤ rp 
Z1
0
(λ + p) µ
α
− µ1
1 − Au (λ+p)µ −1 
du
u α
1 − Bu
Z1
0
¯
¯
¯
¯
≤ ¯Ipλ (a, c) f (z)¯
− µ1
1 + Au (λ+p)µ −1 
u α
du
1 + Bu
,
and inequality (21) is sharp with the extremal function defined by (13).
λ,µ
λ,µ
(a,c,A, B) .Then f (z) ∈ Np,α
(a, c, A, B)
Theorem 9 Let Reα ≥ 0 and f (z) ∈ Np,0
for |z| < R (λ, α, µ, p), where
(22)
(λ + p) µ
q
R (λ, α, µ, p) =
α+
α2 + (λ + p)2 µ2
.
Proof. Set
µ
(23)
zp
Ipλ (a, c) f (z)
¶µ
= ρ + (p − ρ) h(z).
Then clearly, h(z) is analytic in E and h(0) = 1. Taking logarithmic differentiation of (23) both sides and using identity (7) in the resulting equation, we
observe that
(
Ã
!µ
)
¶µ
¶µ
Ipλ+1 (a, c)
zp
zp
(24) Re (1 − α)
−α
−ρ
Ipλ (a, c) f (z)
Ipλ (a, c)
Ipλ (a, c) f (z)
½
¾
½
¾
αzh0 (z)
α |zh0 (z)|
= (p − ρ) Re h(z) +
≥ (p − ρ) Re h(z) −
.
(λ + p) µ
(λ + p) µ
µ
44
K. I. Noor, A. Muhammad, M. Arif
Now by using the following well known estimate, see [8],
¯ 0 ¯ 2rReh(z)
¯zh (z)¯ ≤
(|z| = r < 1) ,
1 − r2
in (24), we have
(
µ
Re (1 − α)
zp
Ipλ (a, c) f (z)
Ã
¶µ
−α
Ipλ+1 (a, c)
Ipλ (a, c)
½
= (p − ρ) Reh(z) 1 −
(25)
!µ
zp
Ipλ (a, c) f (z)
2αr
(λ + p) µ (1 − r2 )
¶µ
)
−ρ
¾
.
The right hand side of (25) is positive if r < R (λ, α, µ, p) where R (λ, α, µ, p)
is given by (22).
Sharpness of this result follows by taking
¶µ
µ
1+z
zp
= ρ + (p − ρ)
.
1−z
Ipλ (a, c) f (z)
where 0 ≤ ρ < p, λ > −p and z ∈ E.
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[3] N. E. Cho, O. H. Kown, H. M. Srivastava, Inclusion and argument properties for certain subclasses of meromorphic functions associated with a
On a class of p-valent non-Bazilevic functions
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family of multiplier transformations, J. Math. Anal. Appl. 300, 2004, 505520.
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K. I. Noor, A. Muhammad, M. Arif
[13] E. T. Whittaker, G. N. Watson, A course on Modern Analysis: An introduction to the general Theory of Infinite processes and of Analytic Functions; With an account of the principle of Transcendental Functions, 4th
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Khalida Inayat Noor
Department of Mathematics
COMSATS Institute of Information Technology, Islamabad Pakistan
email: khalidanoor@hotmail.com
Ali Muhammad
Department of Basic Sciences
University of Engineering and Technology, Peshawar Pakistan
email: ali7887@gmail.com
Muhammad Arif
Department of Mathematics
Abdul Wali Khan University Mardan Pakistan
email: marifmaths@yahoo.com