DEMONSTRATIO MATHEMATICA
Vol. XLVI
No 3
2013
10.1515/dema-2013-0477
Lucyna Trojnar-Spelina
APPLICATIONS OF CONVOLUTION PROPERTIES
Abstract. K. I. Noor (2007 Appl. Math. Comput. 188, 814–823) has defined the
classes Qk (a, b, λ, γ) and Tk (a, b, λ, γ) of analytic functions by means of linear operator
connected with incomplete beta function. In this paper, we have extended some of the
results and have given other properties concerning these classes.
1. Introduction
Let A denote the class of functions f analytic in the open unit disc
U = {z : |z| < 1} and normalized by the conditions f (0) = f 0 (0) − 1 = 0.
Denote by S ∗ (α), K(α)(0 ≤ α < 1) the subfamilies consisting of functions
in A that are starlike of order α and convex of order α respectively. For
0 ≤ γ < 1 and k ≥ 2 let Pk (γ) denote the class of functions p analytic in U
satisfying the conditions p(0) = 1 and
2π
p(z) − γ (1)
1 − γ dθ ≤ kπ
0
where z = reiθ . The class Pk (γ) has been introduced by Padmanabhan
and Parvatham (see [16]). For special choices of parameters, we obtain the
known classes of functions. For example, for k = 2 we have the class P(γ)
of functions with real part greater than γ and consequently, for k = 2 and
γ = 0 we obtain the class of functions with positive real part. For γ = 0 we
have the class Pk defined by Pinchuk [19]. From (1), we conclude that
p(z) =
2π
1 1 + (1 − 2γ)ze−it
dµ(t)
2 0
1 − ze−it
2010 Mathematics Subject Classification: 30C45.
Key words and phrases: integral operator, convex function, gamma function, hypergeometric function, incomplete beta function, starlike function, subordination, convolution,
Carlson–Shaffer operator, Dziok–Srivastava operator, Ruscheweyh derivative.
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where µ(t) is a function with bounded variation on [0, 2π] such that
2π
dµ(t) = 2 and
2π
0
|dµ(t)| ≤ k.
0
It follows from (1) that p ∈ Pk (γ) can be expressed in the form
k 1
k 1
(2) p(z) =
+
−
p1 (z) −
p2 (z), pi ∈ P(γ), i = 1, 2, z ∈ U.
4 2
4 2
∞
P
For the functions f and g with the series expansions f (z) =
ak z k and
g(z) =
∞
P
k=0
bk
zk ,
the Hadamard product (or convolution) f ∗ g is defined by
k=0
(f ∗ g)(z) =
∞
X
ak bk z k .
k=0
This product is associative, commutative and distributive over addition and
1
is an identity for it.
the function 1−z
For a > 0, b > 0, a linear operator Ia,b : A −→ A is defined in [2] by
Ia,b f (z) = fa,b (z) ∗ f (z)
where
(3)
z
z
∗ fa,b (z) =
.
(1 − z)a
(1 − z)b
A simple computation leads to the relation
∞
X
(b)k k+1
(4)
fa,b (z) =
z
= φ(a, b; z)
(a)k
k=0
where (x)k denotes the Pochhammer symbol defined by
(
1,
for
k = 0, x ∈ C \ {0},
(x)k =
x(x + 1) . . . (x + k − 1), for k ∈ N = {1, 2, 3, . . . }, x ∈ C,
and φ(a, b; z) is the incomplete beta function connected with the hypergeometric function by the identity
φ(a, b; z) = z2 F1 (1, b; a, z).
Therefore, we have immediately that Ia,b f = L(b, a)f where L(b, a) is
the well known Carlson–Shaffer operator (see [1]). As a special case, we note
that for a = 1 and b = n + 1, we obtain
I1,n+1 f (z) = Dn f (z) =
z(z n−1 f (z))(n)
,
n!
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that is the Ruscheweyh derivative of order n. We recall here the fact that
Dziok and Srivastava [5] have introduced and considered more general the
Dziok–Srivastava operator
H : Ap,k → Ap,k
such that
Hf (z) = H(a1 , . . . , aq ; c1 , . . . , cs )f (z)
= [z p · q Fs (a1 , . . . , aq ; c1 , . . . , cs ; z)] ∗ f (z),
where q Fs is given by
q Fs (a1 , . . . , aq ; c1 , . . . , cs ; z) =
∞
X
(a1 )n · . . . · (aq )n z n
n=0
(c1 )n · . . . · (cs )n n!
(z ∈ U),
and Ap,k denotes the class of functions with the series expansion
f (z) = z p +
∞
X
an z n
(p < k; p, k ∈ N = {1, 2, . . . }).
n=k
It is easy to observe that for p = s = 1, q = 2 and a2 = 1, the Dziok–
Srivastava operator becomes the Carlson–Shaffer operator and consequently
Ic1 ,a1 f (z) = H(a1 , 1; c1 ). Many interesting subclasses of analytic functions,
associated with the Dziok–Srivastava operator, were studied recently by (for
example) Srivastava et al. [8], [9], [17], [20], (see also [3], [4], [18]).
The following subclasses have been defined in [13], for k ≥ 2, 0 ≤ λ ≤ 1
and 0 ≤ γ < 1, by using the operator Ia,b :
z(Ia,b f )0 + λz 2 (Ia,b f )00
Qk (a, b, λ, γ) = f ∈ A :
∈ Pk (γ), z ∈ U ,
(1 − λ)(Ia,b f ) + λz(Ia,b f )0
Tk (a, b, λ, γ) = f ∈ A : (Ia,b f )0 + λz(Ia,b f )00 ∈ Pk (γ), z ∈ U .
Note that
1) Qk (a, a, 1, 0) = Vk where Vk is the class of functions of bounded boundary rotation introduced by Loewner [10] and deeply examined by Paatero
[14, 15].
2)Q2 (a, a, 0, γ) = S ∗ (γ).
3)Q2 (a, a, 1, γ) = K(γ).
In [13], many of interesting results concerning the classes Qk (a, b, λ, γ) and
Tk (a, b, λ, γ) have been obtained. In particular, inclusion results, covering
theorem and radius problems have been studied. In this paper, we continue
and extend the investigation of the paper [13].
2. Main results
In proofs of our main results, we will use the following lemmas
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Lemma 2.1. ([21], p. 54) If f ∈ K, g ∈ S ∗ , then for each analytic
function h,
(f ∗ hg)(U )
⊂ coh(U ),
(f ∗ g)(U )
where coh(U ) denotes the closed convex hull of h(U ).
Lemma 2.2. [7] Let a > 0. If b ≥ max{2, a} then the function φ(a, b; z) ∈ K
for z ∈ U .
Lemma 2.3. [13] Let f ∈ Pk (α), g ∈ Pk (β), for α ≤ 1, β ≤ 1. Then
(f ∗ g) ∈ Pk (δ), where δ = 1 − 2(1 − α)(1 − β), for z ∈ U .
Lemma 2.4. [13] We have
Tk (a, b, λ, γ) ⊂ Tk (a, b, 0, δ)
where
(5)
δ = γ + (1 − γ)(2η − 1)
and
1
η = (1 + tλ )−1 dt.
0
Theorem 2.1. We have
(i) for b1 > 0 and b2 ≥ max{2, b1 },
Q2 (a, b2 , λ, γ) ⊂ Q2 (a, b1 , λ, γ),
(ii) for a1 > 0 and a2 ≥ max{2, a1 },
Q2 (a1 , b, λ, γ) ⊂ Q2 (a2 , b, λ, γ).
Proof. (i) Let f ∈ Q2 (a, b2 , λ, γ) and set
Fi (z) =
z(Ia,bi f )0 + λz 2 (Ia,bi f )00
,
(1 − λ)(Ia,bi f ) + λz(Ia,bi f )0
i = 1, 2.
From the definition of the class Q2 (a, b2 , λ, γ), we have F2 (z) ≺ p(z) =
1+(1−2γ)z
. Thus, F2 (z) = p(ω(z)) where |ω(z)| < 1 and ω(0) = 0. Note
1−z
that
z(φ(b1 , a; z) ∗ f (z))0 + λz 2 (φ(b1 , a; z) ∗ f (z))00
F1 (z) =
(1 − λ)(φ(b1 , a; z) ∗ f (z)) + λz(φ(b1 , a; z) ∗ f (z))0
z(φ(b1 , b2 ; z) ∗ Ia,b2 f (z))0 + λz 2 (φ(b1 , b2 ; z) ∗ Ia,b2 f (z))00
=
(1 − λ)(φ(b1 , b2 ; z) ∗ Ia,b2 f (z)) + λz(φ(b1 , b2 ; z) ∗ Ia,b2 f (z))0
φ(b1 , b2 ; z) ∗ z(Ia,b2 f (z))0 + λz 2 (Ia,b2 f (z))00
=
φ(b1 , b2 ; z) ∗ [(1 − λ)(Ia,b2 f (z)) + λz(Ia,b2 f (z))0 ]
φ(b1 , b2 ; z) ∗ [F2 (z) · q(z)]
φ(b1 , b2 ; z) ∗ [p(ω(z)) · q(z)]
=
,
=
φ(b1 , b2 ; z) ∗ q(z)
φ(b1 , b2 ; z) ∗ q(z)
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517
where q(z) = (1−λ)(Ia,b2 f (z))+λz(Ia,b2 f (z))0 . It follows from the definition
0 (z)
of the class Q2 (a, b, λ, γ) that Re zqq(z)
= ReF2 (z) > γ ≥ 0 that is, q is
the starlike function. It is easily seen that, by Lemma 2.2, the function
φ(b1 , b2 ; z) is convex function, hence, from Lemma 2.1, we have
φ(b1 , b2 ; z) ∗ [p(ω(z)) · q(z)]
⊂ cop(U ) ⊂ p(U )
φ(b1 , b2 ; z) ∗ q(z)
because p is convex univalent. Therefore F1 ≺ p and the result follows.
(ii) Since the desired inclusion relation follows, by applying the method
used in the proof of the part (i), we omit the details.
The proof of Theorem 2.1 is completed.
Theorem 2.2. We have
(i) for b1 > 0 and b2 ≥ max{2, b1 }
Tk (a, b2 , λ, γ) ⊂ Tk (a, b1 , λ, γ),
(ii) for a1 > 0 and a2 ≥ max{2, a1 }
Tk (a1 , b, λ, γ) ⊂ Tk (a2 , b, λ, γ).
Proof. (i) Let us define Gi (z) = (Ia,bi f )0 + λz(Ia,bi f )00 , i = 1, 2, and let
f ∈ Tk (a, b2 , λ, γ). Thus G2 ∈ Pk (γ) or equivalently
k 1
k 1
G2 (z) =
+
p1 (z) −
−
p2 (z),
4 2
4 2
where pi ∈ P(γ), i = 1, 2. Note that
G1 (z) = (φ(b1 , a; z) ∗ f (z))0 + λz (φ(b1 , a; z) ∗ f (z))00
= [φ(b1 , b2 ; z) ∗ (φ(b2 , a; z) ∗ f (z))]0 + λz [φ(b1 , b2 ; z) ∗ (φ(b2 , a; z) ∗ f (z))]00
φ(b1 , b2 ; z)
k 1
k 1
φ(b1 , b2 ; z)
∗ G2 (z) =
∗
+
p1 (z) −
−
p2 (z)
=
z
z
4 2
4 2
k 1
φ(b1 , b2 ; z)
k 1
φ(b1 , b2 ; z)
=
+
∗ p1 (z) −
−
∗ p2 (z) .
4 2
z
4 2
z
By Lemma 2.2, we have that φ(b
2 ; z) ∈ K so using the well known relation
1 , b
f (z)
1
∗
f ∈ K =⇒ f ∈ S 2 =⇒ Re z
> 12 , we immediately obtain φ(b1 z,b2 ;z) ∈
P 21 . Therefore, from Lemma 2.3 with k = 2, we conclude that φ(b1 z,b2 ;z) ∗
pi (z) ∈ P(δ) where δ = 1 − 2(1 − 12 )(1 − γ) = γ, i = 1, 2, what means that
G1 ∈ Pk (γ). The proof is thus completed.
(ii) The proof is similar to that of (i) so we omit the details.
Remark 2.1. In [13], there are no results concerning inclusion relationships between the classes Qk (a, b, λ, γ) and Tk (a, b, λ, γ) with respect to the
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parameter a. Thus, the results (ii) of Theorem 2.1 and (ii) of Theorem 2.2
become the essential supplement of the results of [13].
Theorem 2.3. Let f ∈ Tk (a, b, λ, γ), g ∈ A and Re g(z)
> 21 for z ∈ U .
z
Then f ∗ g ∈ Tk (a, b, λ, γ).
> 12
Proof. We first suppose that f ∈ Tk (a, b, λ, γ), g ∈ A and Re g(z)
z
in U . Let us define
H1 (z) = [Ia,b f (z)]0 + λz [Ia,b f (z)]00
and
H2 (z) = [Ia,b (f ∗ g)(z)]0 + λz [Ia,b (f ∗ g)(z)]00 .
From the definition of the class Tk (a, b, λ, γ), we immediately get
k 1
k 1
H1 (z) =
+
p1 (z) −
−
p2 (z),
4 2
4 2
where pi ∈ P(γ), i = 1, 2. Note that
H2 (z) = [φ(b, a; z) ∗ (f ∗ g)(z)]0 + λz [φ(b, a; z) ∗ (f ∗ g)(z)]00
g(z)
g(z) =
∗ (φ(b, a; z) ∗ f (z))0 + λz(φ(b, a; z) ∗ f (z))00 =
∗ H1 (z).
z
z
1
1
Since ( g(z)
z ) ∈ P( 2 ) and H1 ∈ Pk (γ) then putting α = 2 and β = γ in
Lemma 2.3, we conclude that H2 ∈ Pk (γ). The proof of Theorem 2.3 is
completed.
Remark 2.2. We remark that the class of functions in A with Re f (z)
>
z
1
2
is known to be equal to the closed convex hull of the convex functions (coK)
([6], p. 52). Thus, the previous theorem shows that the class Tk (a, b, λ, γ) is
invariant under the convolution with functions of coK.
By applying the relation f ∈ K =⇒ f ∈ S ∗ 21 =⇒ Re f (z)
> 12 , we
z
get the following
Corollary 2.1.
Let f ∈ Tk (a, b, λ, γ). We have
(i) if g ∈ S ∗ 12 then (f ∗ g) ∈ Tk (a, b, λ, γ),
(ii) if g ∈ K then (f ∗ g) ∈ Tk (a, b, λ, γ).
Remark 2.3. Let a and c be the complex numbers with c 6= 0, −1, −2, . . . .
We consider the function defined by
Φ(a, c; z) =1 F1 (a, c; z) =
∞
X
(a)k z k
k=0
(c)k k!
.
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This function is called the confluent (or Kummer) hypergeometric function.
If Re c > Re a > 0 then
Γ(c)
ta−1 (1 − t)c−a−1 etz dt,
Γ(a)Γ(c − a) 0
1
Φ(a, c; z) =
where Γ denotes the Gamma function. Miller and Mocanu showed in [11]
that for a, c ∈ R, c ≥ R(a) where

if a < 14 ,

 2(1 − a),
(6)
R(a) =
(1 − 2a)2 + 54 , if 14 ≤ a < 34 ,


2a,
if 34 ≤ a,
the function zΦ(a, c; z) is starlike of order
obtain
1
2
in U . Thus, we immediately
Corollary 2.2. If a, c ∈ R, and satisfy c ≥ R(a) where R(a) is given by
(6) then
f ∈ Tk (a, b, λ, γ) =⇒ (zΦ(a, c; z) ∗ f (z)) ∈ Tk (a, b, λ, γ).
Example 2.1. From the last result, we deduce that the class Tk (a, b, λ, γ)
is invariant under convolution with the function
1
gδ (z) = zΦ(1, δ + 1; z) = δz (1 − t)δ−1 etz dt;
δ ≥ 1,
0
that is, if f (z) = z+
∞
P
ak+1 z k+1 ∈ Tk (a, b, λ, γ) then g(z) = z+
∞
P
k=1
k=1
ak+1 k+1
(δ+1)k z
∈ Tk (a, b, λ, γ).
Corollary 2.3. Let α > 0. If β ≥ max{2, α} then
f ∈ Tk (a, b, λ, γ) =⇒ Iβ,α f ∈ Tk (a, b, λ, γ).
Proof. By applying Lemma 2.2 and the definition of Ia,b , we immediately
deduce desired assertion from (ii) of Corollary 2.1.
Corollary 2.4. Let, for µ ≥ 0
µ + 1 µ−1
t
f (t)dt.
zµ 0
z
Fµ (f ) = Fµ (f )(z) =
Then
f ∈ Tk (a, b, λ, γ) =⇒ Fµ (f ) ∈ Tk (a, b, λ, γ).
Proof. It is sufficient to note that Fµ (f ) = L(µ + 1, µ + 2)f . Thus the
desired implication follows directly from Corollary 2.3.
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Corollary 2.5. Let f has the series expansion f (z) =
∞
P
ak z k and let
k=2
sn (z) := z +
n
X
ak z k
(n ∈ N \ {1}).
k=2
Then
f ∈ Tk (a, b, λ, γ) =⇒
1
sn (rn z) ∈ Tk (a, b, λ, γ)
rn
where
n−1
X 1
rn = sup r : Re
zk > ,
2
(
(7)
)
(n ∈ N \ {1}).
(|z| < 1) ,
k=0
n
P
Proof. Let f ∈ Tk (a, b, λ, γ). Putting hn (z) =
z k , we can write sn (z) =
k=1
(f ∗ hn )(z). Thus from (7), we get Re hnr(rn nz z) > 12 , for z ∈ U, n ∈ N \ {1}.
Hence, by applying Theorem 2.3, we have
1
1
1
sn (rn z) = (f ∗ hn )(rn z) = f (z) ∗ hn (rn z) ∈ Tk (a, b, λ, γ).
rn
rn
rn
The proof is thus completed.
Theorem 2.4. Let f1 ∈ T2 (a, b, λ, γ1 ) and f2 ∈ Tk (a, b, λ, γ2 ),
1, i = 1, 2). If f ∈ A is defined by
(0 ≤ γi <
z
Ia,b f (z) = (Ia,b f1 (z))0 ∗ (Ia,b f2 (z))0 dt
0
then f ∈ Tk (a, b, λ, κ) where κ = 1 − 8(1 − η)2 (1 − γ1 )(1 − γ2 ) and η is given
by (5).
Proof. Let f1 ∈ T2 (a, b, λ, γ1 ) and f2 ∈ Tk (a, b, λ, γ2 ). It follows from
Lemma 2.4 that (Ia,b f1 (z))0 ∈ P2 (δ1 ) and (Ia,b f2 (z))0 ∈ Pk (δ2 ) where δi =
γi + (1 − γi )(2η − 1) and the parameter η is given by (5). Using the definition
of the class Pk (γ), we thus obtain that
(Ia,b f1 (z))0 = p(z),
p ∈ P(δ1 )
and
(Ia,b f2 (z))0 =
k 1
+
4 2
p1 (z) −
k 1
−
4 2
p2 (z),
pi ∈ P(δ2 ),
i = 1, 2,
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which leads to
0
(Ia,b f (z))
k
=
+
4
k 1
+
4 2
k 1
−
4 2
p1 (z) −
p2 (z)
= p(z) ∗
k 1
1
(p(z) ∗ p1 (z)) −
(p(z) ∗ p2 (z)).
−
2
4 2
Applying Lemma 2.3, we deduce that (Ia,b f (z))0 ∈ Pk (κ) where
κ = 1 − 2(1 − δ1 )(1 − δ2 )
= 1 − 2[1 − γ1 − (1 − γ1 )(2η − 1)] · [1 − γ2 − (1 − γ2 )(2η − 1)]
= 1 − 8(1 − η)2 (1 − γ1 )(1 − γ2 )
where the parameter η is given in (5). The proof of theorem is thus completed.
Theorem 2.5. For a fixed number n, n ∈ N , let
∞
X
akn+1 z kn+1 .
(8)
f (z) = z +
k=1
If f ∈ Tk (a, b, 0, γ), then f ∈ Tk (a, b, λ, γ), for |z| < rn , where rn =
p
1
n
1 + (λn)2 − λn . The result is best possible.
Proof. Under the hypothesis that f of the form (8) is in Tk (a, b, 0, γ), we
have
(Ia,b f (z))0 ∈ Pk (γ)
and consequently
(Ia,b f (z))0 =
(9)
k 1
+
4 2
p1 (z) −
k 1
−
4 2
p2 (z),
where pi ∈ P(γ), i = 1, 2. Let us put
F (z) = [Ia,b f (z)]0 + λz [Ia,b f (z)]00 .
We can easily prove that
F (z) = [Ia,b f (z)]0 ∗
(10)
gn (z)
z
where
∞
gn (z) = (1 − nλ)
X
z
z
+ nλ
=z+
(1 + knλ)z kn+1 .
n
n
2
1−z
(1 − z )
k=1
Let us set
zn
=
1 − ρeiθ ,
Re
(ρ > 0) and |z| = r < 1. It was shown in [22] that
gn (z)
1
1
≥ + 2 (1 − 2nλrn − r2n ).
z
2 2ρ
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p
1
1+
− λn , we have gnz(z) ∈ P( 21 ). It follows
Thus, for |z| < rn =
from (9) and (10) that
gn (z)
k 1
gn (z)
k 1
p1 (z) ∗
−
p2 (z) ∗
,
F (z) =
+
−
4 2
z
4 2
z
where pi ∈ P(γ), i = 1, 2. Hence, an application of Lemma 2.3 leads to the
result F ∈ Pk (γ) and consequently, f ∈ Tk (a, b, λ, γ) for |z| < rn . The proof
is completed.
(λn)2
n
If n = 1 then from Theorem 2.5, we deduce
∞
P
ak z k . If f ∈ Tk (a, b, 0, γ), then
Corollary 2.6. Let f (z) = z +
k=2
√
f ∈ Tk (a, b, λ, γ), for |z| < 1 + λ2 − λ.
Remark 2.4. In [13], the following result was proved (Theorem 4.5): Let
(Ia,b f (z))0 ∈ Pk (γ). Then f ∈ Tk (a, b, λ, γ) for |z| < rλ where
1
1
√
rλ =
; λ 6= .
2
2
2λ + 4λ − 2λ + 1
The result is best possible.
In the light of the Corollary 2.6, the previous result of [13] seems to be not
the best possible because, for example, for λ = 1, we have
p
1
√
1 + λ2 − λ >
.
2λ + 4λ2 − 2λ + 1
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DEPARTMENT OF MATHEMATICS
RZESZÓW UNIVERSITY OF TECHNOLOGY
Wincentego Pola 2
35-959 RZESZÓW, POLAND
E-mail: [email protected]
Received January 24, 2011.
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