TAIWANESE JOURNAL OF MATHEMATICS
Vol. 18, No. 1, pp. 39-51, February 2014
DOI: 10.11650/tjm.18.2014.2986
This paper is available online at http://journal.taiwanmathsoc.org.tw
GENERALIZATIONS OF STRONGLY STARLIKE FUNCTIONS
Jacek Dziok
Abstract. By using functions of bounded variation we generalize the class of
strongly starlike functions and related classes. The main object is to obtain characterizations and inclusion properties of these classes of functions.
1. INTRODUCTION
Let A denote the class of functions which are analytic in U := {z ∈ C : |z| < 1}
and let Ap (p ∈ N0 := {0, 1, 2, . . .}) denote the class of functions f ∈ A of the form
p
f (z) = z +
(1)
∞
an z n
(z ∈ U) .
n=p+1
Let a, δ ∈ C, |a| < 1, α < p, 0 < β ≤ 1, k ≥ 2, p ∈ N, ϕ ∈ Ap .
A function f ∈ Ap is said to be in the class Sβ∗ of multivalent strongly starlike
function of order β if
Arg zf (z) < β π (z ∈ U).
pf (z) 2
We denote by Mk the class of real-valued functions m of bounded variation on [0, 2π]
which satisfy the conditions
2π
2π
|dm (t)| ≤ k.
dm (t) = 2,
(2)
0
0
It is clear that M2 is the class of nondecreasing functions on [0, 2π] satisfying (2) or
2π
equivalently dm (t) = 2.
0
Received February 22, 2013, accepted May 16, 2013.
Communicated by Hari M. Srivastava.
2010 Mathematics Subject Classification: 30C45, 30C50, 30C55.
Key words and phrases: Analytic functions, Bounded variation, Bounded boundary rotation, Strongly
starlike functions.
39
40
Jacek Dziok
Let Pk (a, β) denote the class of functions q ∈ A0 for which there exists m ∈ Mk
such that
1
q (z) =
2
(3)
2π 0
1 + (1 − 2a) ze−it
1 − ze−it
β
dm (t)
(z ∈ U) .
Here and throughout we assume that all powers denote principal determinations.
Moreover, let us denote
Pk (a) := Pk (a, 1) , Pk (a, β) := q ∈ A0 : q 1/β ∈ Pk (a) .
In particular, P := P2 (0) is the well-known class of Carathéodory functions. The
classes Pk := Pk (0) , Pk (ρ) (0 ≤ ρ < 1) were investigated by Paatero [19] (see also
Pinchuk [23]) and Padmanabhan and Parvatham [21], respectively. We note that
f ∈ Sβ∗ ⇐⇒
zf (z)
∈ P (β) ,
pf (z)
π
.
P (β) := P2 (β) = q ∈ A0 : |Arg q (z)| < β
2
Now, we generalize the class of strongly starlike functions. We denote by Mk (a, β; δ, ϕ)
the class of functions f ∈ Ap such that
z (ϕ ∗ f ) (z)
z (ϕ ∗ f ) (z)
δ
+
(1
−
δ)
1+
∈ Pk (a, β) ,
Jδ,ϕ (f ) (z) :=
p
p (ϕ ∗ f ) (z)
(ϕ ∗ f ) (z)
where
where ∗ denote the Hadamard product (or convolution). Moreover, let us denote
M (a, β; δ, ϕ) := M2 (a, β; δ, ϕ) , Wk (a, β; ϕ) := Mk (a, β; 0, ϕ),
W (a, β; ϕ) := W2 (a, β; ϕ) , Wk (a, β) := Wk (a, β; z p/ (1 − z)) ,
Sp∗ (ϕ, a) := W2 (a, 1; ϕ) .
We see that Sβ∗ = W2 (0, β) and
f ∈ Wk (a, β; ϕ) ⇔ ϕ ∗ f ∈ Wk (a, β) .
→
−
→
, a2 ) , β = (β1 , β2 ) . We say that a function f ∈ Ap belongs to the
Let −
a =
(a1→
−
→
a , β ; δ, ϕ , if there exists a function g ∈ Wk (a2 , β2 ; ϕ) such that
class CW k −
(4)
δ
p
z (ϕ ∗ f ) (z)
1+
(ϕ ∗ g) (z)
+ (1 − δ)
z (ϕ ∗ f ) (z)
∈ Pk (a1 , β1 ) .
p (ϕ ∗ g) (z)
These classes generalize well-known classes of functions, which were defined in
ealier works, see for example [1-10] and [14-26]. We note that
41
Generalizations of Strongly Starlike Functions
• Mk (a, β; α, ϕ) is related to the class of functions with the bounded Mocanu
variation defined by Coonce and Ziegler [4] and intensively investigated by Noor
et al. [15-18]
• Vk := Wk 0, 1; z 2 is the well-known class of functions of bounded bound(1−z)
ary rotation (for details, see, [2, 7, 14, 21]).
•
p
p
z
z (p + (1 − p) z)
, α/p , Spc (α) := Sp∗
,
α/p
Sp∗ (α) := Sp∗
1−z
p (1 − z)2
are the classes of multivalent starlike functions of order α and multivalent convex
functions of order α, respectively.
•
(α < p)
Rp (α) := Sp∗ z p / (1 − z)2(p−α) , α/p
will be called the class of multivalent prestarlike functions of order α. In particular, R (α) := R1 (α) is the well-know class of prestarlike functions of order
α introduced by Ruscheweyh [24].
z
is the well-known class of close-to-convex func• CC := CW k 0, 1; 0 (1−z)
2
tions.
The main object of the paper is to obtain some characterizations and inclusion
properties for the defined classes of functions. Some applications of the main results
are also considered.
2. CHARACTERIZATION THEOREMS
Let us define
Bk (a, β) :=
where
(5)
ha,β (z) :=
k 1
k 1
+
q1 −
−
q2 : q1 , q2 ≺ ha,β ,
4 2
4 2
1 + (1 − 2a) z
1−z
β
, ha := ha,1
(z ∈ U) .
From the result of Hallenbeck and MacGregor ([13], pp. 50) we have the following
lemma.
Lemma 1. q ≺ ha,β if and only if there exists m ∈ M2 such that
1
q (z) =
2
2π 0
1 + (1 − 2a) ze−it
1 − ze−it
β
dm (t)
(z ∈ U) .
42
Jacek Dziok
Theorem 1.
Bλ (a, β) ⊂ Bk (a, β)
λLet 1q ∈ Bλ (a, β) .
λ Proof.
1
+
−
q
1
4
2
4 − 2 q2 or
k
k 1
+
q1 −
−
q=
4 2
4
(2 ≤ λ < k) .
Then there exist q1 , q2 ≺ ha,β such that q =
1
2
q2
k−λ
λ−2
q1 +
q2 .
q2 =
k−2
k−2
Since q2 ≺ ha,β , we have q ∈ Bk (a, β) .
Theorem 2. The class Bk (a, β) is convex.
Proof.
Let q, r ∈ Bk (a, β) , α ∈ [0, 1], μ :=
ha,β (j = 1, 2) such that
k
4
+ 12 . Then there exist qj , rj ≺
q = μq1 − (1 − μ) q2 , r = μr1 − (1 − μ) r2 .
It follows that
αq + (1 − α) r = μ (αq1 + (1 − α) r1 ) − (1 − μ) (αq2 + (1 − α) r2 ) .
Since αqj +(1 − α) rj ≺ ha,β (j = 1, 2) , we conclude that αq+(1 − α) r ∈ Bk (a, β) .
Hence, the class Bk (a, β) is convex.
Theorem 3.
Pk (a, β) = Bk (a, β) .
Proof. Let q ∈ Pk (a, β). Then q satisfy (3) for some m ∈ Mk . If m ∈ M2 , then
by Lemma 1 and Theorem 1 we have q ∈ P2 (h) ⊂ Pk (h) . Let now m ∈ Mk M2 .
Since m is the function with bounded variation, by the Jordan theorem there exist
real-valued functions μ1 , μ2 which are nondecreasing and nonconstant on [0, 2π] such
that
2π
2π
2π
|dm (t)| = dμ1 (t) + dμ2 (t) .
(6)
m = μ1 − μ2 ,
0
0
0
Thus, putting
αj =
μj (2π) − μj (0)
1
, mj :=
μj
2
αj
we get m1 , m2 ∈ M2 and
(7)
m = α1 m1 − α2 m2 .
Combining (6) and (7) we obtain
(j = 1, 2)
43
Generalizations of Strongly Starlike Functions
2π
2α1 − 2α2 =
2π
|dm (t)| ≤ k,
dm (t) = 2, 2α1 + 2α2 =
0
0
and so
α1 =
λ 1
+
4 2
, α2 =
λ 1
−
4 2
⎛
⎝λ =
2π
⎞
|dm (t)| ≤ k⎠ .
0
Therefore, by (3) and (7) we obtain
λ 1
λ 1
+
q1 −
−
q2 ,
q=
4 2
4 2
where
1
qj (z) =
2
2π 0
1 + (1 − 2a) ze−it
1 − ze−it
β
(z ∈ U, j = 1, 2) .
dmj (t)
Thus, by Lemma 1 and Theorem 1 we have q ∈ Bλ (a, β) ⊂ Bk (a, β) . Conversely, let
q ∈ Bk (a, β) . Then there exist q1 , q2 ≺ ha,β such that q is of the form
k 1
k 1
+
q1 −
−
q2 .
q=
4 2
4 2
Thus, by Lemma 1 there exist m1 , m2 ∈ M2 such that q is of the form (3) with
k 1
k 1
+
m1 −
−
m2 .
m=
4 2
4 2
Since
2π
dm (t) =
0
k 1
+
4 2
2π
|dm (t)| ≤
0
2π
k 1
+
4 2
dm1 −
0
2π
k 1
−
4 2
dm1 +
0
2π
k 1
−
4 2
dm2 = 2,
0
2π
dm2 = k,
0
we have m ∈ Mk and consequently q ∈ Pk (a, β) .
Lemma 2. [7]. Let q ∈ A0 . Then q ∈ Pk (a) if and only if
2π it −
a
q
re
dt ≤ kπ
1−a 0
(0 < r < 1) .
44
Jacek Dziok
From Lemma 2 we have the following corollary.
Corollary 1. Let q ∈ A0 . Then q ∈ Pk (a, β) if and only if
2π 1/β it
re
−
a
q
dt ≤ kπ
1−a
(8)
(0 < r < 1) .
0
3. THE MAIN INCLUSION RELATIONSHIPS
From now on we make the assumptions: 0 ≤ δ ≤ 1 and
ha,β (z) > α
(9)
(z ∈ U) .
Then we have
(10)
Wk (a, β) ⊂ Sp∗ (α) .
Let Φp(b, c) denote the multivalent incomplete hipergeometric function defined by
(11)
Φp(b, c) (z) := z p 2 F1 (b, 1; c; z) =
∞
(b)n−p
(c)n−p
n=p
zn
(z ∈ U) .
Lemma 3. [11]. Let h ∈ K, q ∈ A0 and λ > 0. If
q(z) + λ
zq (z)
≺ h (z) ,
q(z)
then q ≺ h.
Lemma 4. [5]. Let f ∈ Rp (α), g ∈ Sp∗ (α). Then
f ∗ (hg)
(U ) ⊆ co {h(U)} ,
f ∗g
where co {h(U )} denotes the closed convex hull of h(U).
Lemma 5. [5]. Let p ∈ N. If either
(12)
or
(13)
[b] ≤ [c], [b] = [c] and
1
(2p + 1 − b − c) ≤ α < p
2
c
≤ α < p,
0 < b ≤ c and p −
2
then
Φp (b, c) ∈ Rp (α) .
Generalizations of Strongly Starlike Functions
45
Theorem 4. If ψ ∈ Rp (α), then
Wk (a, β; ϕ) ⊂ Wk (a, β; ψ ∗ ϕ) .
(14)
Proof.
that
Let f ∈ Wk (a, β; ϕ) . Thus, by Theorem 3 there exist q1 , q2 ≺ ha,β such
k 1
z (ϕ ∗ f ) (z)
k 1
=
+
q1 (z) −
−
q2 (z) (z ∈ U) .
p (ϕ ∗ f ) (z)
4 2
4 2
Moreover, H = ϕ ∗ f ∈ Wk (a, β) ⊂ Sp∗ (α). Thus, applying the properties of
convolution, we get
z [(ψ ∗ ϕ) ∗ f ] (z)
k 1 ψ (z) ∗ [q1 (z) H(z)]
=
+
p [(ψ ∗ ϕ) ∗ f ] (z)
4 2 ψ (z) ∗ H(z)
(15)
k 1 ψ (z) ∗ [q2 (z) H(z)]
−
(z ∈ U).
−
4 2
ψ (z) ∗ H(z)
By Lemma 4 we conclude that
Fj (z) :=
ψ (z) ∗ [qj (z) H(z)]
∈ co {qj (U)} ⊂ ha,β (U ) (z ∈ U, j = 1, 2).
ψ (z) ∗ H(z)
Therefore, Fj ≺ ha,β and by (15) we have f ∈ Wk (a, β; ψ ∗ ϕ), which proves the
theorem.
Theorem 5. Let ψ ∈ Rp (α) , 0 ≤ δ ≤ 1. Then
Mk (a, β; δ, ϕ) ∩ Wk (a, β; ϕ) ⊂ Mk (a, β; δ, ψ ∗ ϕ) .
(16)
Proof. Let f ∈ Mk (a, β; δ, ϕ) ∩ Wk (a, β; ϕ). Then, applying Theorem 4, we
obtain f ∈ Wk (a, β; ψ ∗ ϕ). Thus, we have
F1 (z) :=
z [(ψ ∗ ϕ) ∗ f ] (z)
z (ϕ ∗ f ) (z)
, F2 (z) :=
∈ Pk (a, β) .
p [(ψ ∗ ϕ) ∗ f ] (z)
p (ϕ ∗ f ) (z)
Since the class Pk (a, β) is convex by Theorem 2, we conclude that (1 − δ) F1 + δF2 ∈
Pk (a, β) . Hence, f ∈ Mk (a, β; δ, ψ ∗ ϕ) and, in consequence, we get (16).
Lemma 6. If 0 ≤ γ ≤ δ, then
M (a, β; δ, ϕ) ⊂ M (a, β; γ, ϕ).
Proof.
Let f ∈ M (a, β; δ, ϕ) and let
q (z) :=
z (ϕ ∗ f ) (z)
p (ϕ ∗ f ) (z)
(z ∈ U) .
46
Jacek Dziok
Then, we obtain
zq (z)
= Jδ,ϕ (f ) (z) (z ∈ U) .
q (z)
Since Jδ,ϕ (f ) ≺ ha,β , we have q ≺ ha,β by Lemma 3. Moreover,
q (z) + δ
γ
δ−γ
Jδ,ϕ (f ) +
q.
δ
δ
Because ha,β is convex and univalent in U , then we obtain Jγϕ (f ) ≺ ha,β or equivalently f ∈ M (a, β; γ, ϕ).
From Theorem 5 and Lemma 6 we have the following corollary.
Jγ,ϕ (f ) =
Corollary 2. Let ψ ∈ Rp (α) , 0 ≤ δ ≤ 1. Then
M (a, β; δ, ϕ) ⊂ M (a, β; δ, ψ ∗ ϕ) .
Theorem 6. If ψ ∈ Rp (α) , then
→
→
−
−
→
→
a , β ; δ, ϕ ⊂ CW k −
a , β ; δ, ψ ∗ ϕ .
(17)
CW k −
→
−
→
a , β ; δ, ϕ . Then there exist g ∈ Wk (a2 , β2 ; ϕ) and
Proof.
Let f ∈ CW k −
q1 , q2 ≺ ha1 ,β1 such that
k 1
k 1
z (ϕ ∗ f ) (z)
=
+
q1 (z) −
−
q2 (z) (z ∈ U)
p (ϕ ∗ g) (z)
4 2
4 2
and F = ϕ ∗ g ∈ Wk (a2 , β2) ⊂ Sp∗ (α). Thus, applying the properties of convolution,
we get
k 1 ψ ∗ (q1 F )
z [(ψ ∗ ϕ) ∗ f ] (z)
=
+
(z)
p [(ψ ∗ ϕ) ∗ g] (z)
4 2 ψ ∗ F
(18)
k 1 ψ ∗ (q2 F )
−
(z) (z ∈ U) .
−
4 2
ψ ∗F
By Lemma 4 we conclude that
ψ ∗ (qj F )
(z) ∈ co {qj (U )} ⊂ ha1,β1 (U ) (z ∈ U, j = 1, 2).
ψ∗F
→
−
→
a , β ; δ, ψ ∗ ϕ .
Therefore, Fj ≺ ha1 ,β1 and by (18) we have f ∈ CW k −
Fj (z) :=
Combining Theorems 4-6 with Lemma 5 we obtain the following theorem.
Theorem 7. If either (12) or (13), then
Wk (a, β; ϕ) ⊂ Wk (a, β; Φp(b, c) ∗ ϕ) ,
M (a, β; δ, ϕ) ⊂ M (a, β; δ, Φp(b, c) ∗ ϕ) ,
→
→
−
−
→
→
a , β ; δ, ϕ ⊂ CW −
a , β ; δ, Φ (b, c) ∗ ϕ .
CW −
k
k
p
Generalizations of Strongly Starlike Functions
47
Since Φp(b, c) ∗ Φp (c, b) ∗ ϕ = ϕ, by Theorem 7 we obtain the next result.
Theorem 8. If either (12) or (13), then
Wk (a, β; Φp(c, b) ∗ ϕ) ⊂ Wk (a, β; ϕ) ,
M (a, β; δ, Φp(c, b) ∗ ϕ) ⊂ M (a, β; δ, ϕ) ,
→
→
−
−
→
−
→
−
CW k a , β ; δ, Φp(c, b) ∗ ϕ ⊂ CW k a , β ; δ, ϕ .
Let us define the linear operators Jλ : Ap −→ Ap ,
(19)
Jλ (f ) (z) := λ
zf (z)
+ (1 − λ) f (z) ,
p
(z ∈ U, (λ) > 0) .
Since Jλ (f ) = Φp ( λp + 1, λp ) ∗ f , putting b = λp , c =
the following theorem.
Theorem 9. If p − λp ≤ α < p, then
p
λ
+ 1 in Theorem 8, we have
Wk (a, β; Jλ (ϕ)) ⊂ Wk (a, β; ϕ) ,
M (a, β; δ, Jλ (ϕ)) ⊂ M (a, β; δ, ϕ) ,
→
→
−
−
→
→
a , β ; δ, Jλ (ϕ) ⊂ CW k −
a , β ; δ, ϕ .
CW k −
In particular, for λ = 1 we get the following theorem.
Theorem 10. If 0 ≤ α < p, then
Wk a, β; zϕ (z) ⊂ Wk (a, β; ϕ) ,
M a, β; δ, zϕ (z) ⊂ M (a, β; δ, ϕ) ,
→
→
−
−
→
→
a , β ; δ, zϕ (z) ⊂ CW −
a , β ; δ, ϕ .
CW −
k
k
4. APPLICATIONS TO CLASSES DEFINED BY LINEAR OPERATORS
For real numbers λ, t (λ > −p), we define the function
(20)
Ψ(a1 , b1, t) (z) := (z p q Fs (a1 , ..., aq; b1 , ..., bs; z)) ∗ fλ,t (z) (z ∈ U),
where q Fs (a1 , ..., aq; b1 , ..., bs; z) is the generalized hypergeometric function and
fλ,t (z) =
∞ n+λ t
n=p
p+λ
z n (z ∈ U).
It is easy to verify that
(21)
bΨ(b + 1, c, t) = zΨ (b, c, t) + (b − p)Ψ(b, c, t),
48
Jacek Dziok
(22)
bΨ(b, c, t) = zΨ (b, c + 1, t) + (b − p)Ψ(b, c + 1, t),
(23)
(p + λ) Ψ(b, c, t + 1) = zΨ (b, c, t) + λΨ(b, c, t),
(24)
Ψ(b, c, t) = Φp(b, d) ∗ Ψ(d, c, t).
where Φp(b, d) is defined by (11).
Corresponding to the function Ψ(b, c, t) we consider the following classes of functions:
Vk (a, β; b, c, t) := Wk (a, β; Ψ(b, c, t)) ,
→
→
−
−
→
→
a , β ; b, c, t) := CW −
a , β ; δ, Ψ(b, c, t) .
CV (−
k
k
By using the linear operator
Θp [b, c, t] f = Ψ(b, c, t) ∗ f ( f ∈ Ap )
(25)
we can define the class Vk (a, β; b, c, t) alternatively in the following way:
f ∈ Vk (a, β; b, c, t) ⇐⇒ b
Θp [b + 1, c, t] f (z)
+ p − b ∈ Pk (a, β) .
Θp [b, c, t] f (z)
Corollary 3. If p − [b] ≤ α < p, m ∈ N, then
(26)
Vk (a, β; b + m, c, t) ⊂ Vk (a, β; b, c, t) ,
(27)
CV k (a, β; b + m, c, t) ⊂ CV k (a, β; b, c, t) .
Proof.
It is clear that it is sufficient to prove the corollary for m = 1. Let
Jλ and Ψ(b,c, t) be defined by (19) and (20), respectively. Then, by (21) we have
Ψ(b+1,c, t) = J p (Ψ(b, c, t)). Hence, by using Theorem 9 we conclude that
b
Wk (a, β; Ψ(b + 1, c, t)) ⊂ Wk (a, β; Ψ(b, c, t)) ,
→
→
−
−
→
→
a , β ; δ, Ψ(b + 1, c, t) ⊂ CW −
a , β ; δ, Ψ(b, c, t) .
CW −
k
k
This clearly forces the inclusion relations (26) and (27) for m = 1.
Analogously to Corollary 3, we prove the following corollary.
Corollary 4. Let m ∈ N. If p − [c] ≤ α < p, then
Vk (a, β; b, c, t)⊂ Vk (a,β; b,c + m, t) ,
→
→
−
−
→
→
a , β ; b, c, t ⊂ CV −
a , β ; b, c + m, t .
CV −
k
k
If − [λ] ≤ α < p, then
Vk (a, β; b, c, t + m) ⊂ Vk (a, β; b, c, t) ,
→
→
−
−
→
→
a , β ; b, c, t + m ⊂ CV −
a , β ; b, c, t .
CV −
k
k
Generalizations of Strongly Starlike Functions
49
It is natural to ask about the inclusion relations in Corollaries 3 and 4 when m is
positive real. Using Theorems 4 and 6, we shall give a partial answer to this question.
Corollary 5. If the multivalent incomplete hipergeometric function Φp (b, d) defined
by (11) belongs to the class Rp (α), then
−
−
→
→
→
→
(28)
a , β ;d, c, t ⊂ CV k −
a , β ; b, c,t ,
Vk (a, β; d, c, t) ⊂ Vk (a, β;b,c, t) , CV k −
(29)
−
→
→
−
−
→
Vk (a, β; c, b, t)⊂ Vk (a,β;c, d,t) , CV k →
a , β ;c, b, t ⊂ CV k −
a , β ; c, d, t .
Proof. Let us put ψ = Φp (b, d), ϕ = Ψ(d, c, t). Then, by (22) and Theorems 4
and 6 we obtain
Wk (a, β; Ψ(d, c, t)) ⊂ Wk (a, β; Ψ(b, c, t)) ,
→
→
−
−
→
→
a , β ; δ, Ψ(d, c, t) ⊂ CW −
a , β ; δ, Ψ(b, c, t) .
CW −
k
k
Thus, we get the inclusion relations (28). Analogously, we prove the inclusions (29).
Combining Corollary 5 with Lemma 5, we obtain the following result.
Corollary 6. If either (12) or (13), then the inclusion relations (28) and (29) hold
true.
The linear operator Θp [b, c, t] defined by (25) includes (as its special cases) other
linear operators of geometric function theory which were considered in earlier works.
In particular, we can mention here the Dziok-Srivastava operator, the Hohlov operator, the Carlson-Shaffer operator, the Ruscheweyh derivative operator, the generalized
Bernardi-Libera-Livingston operator, the fractional derivative operator, and so on for
the precise relationships, see, Dziok and Srivastava ([10], pp. 3-4). Moreover, the
linear operator Θp [b, c, t] includes also the Sălăgean operator, the Noor operator, the
Choi-Saigo-Srivastava operator, the Kim-Srivastava operator, and others (for the precise
relationships, see, Cho et al. [3]). By using these linear operators we can consider
→
−
→
a , β ; b, c, t), see for examseveral subclasses of the classes Vk (a, β; b, c, t) and CV k (−
ple [1-10, 12, 20, 22, 26]. Also, the obtained results generalize several results obtained
in these classes of functions.
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Jacek Dziok
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Jacek Dziok
Institute of Mathematics
University of Rzeszów
35-310 Rzeszów
Poland
E-mail: [email protected]