ISSN: 1582-5329 pp. 51-61
SOME PROPERTIES OF NEW INTEGRAL OPERATOR
R. M. El-Ashwah and M. K. Aouf
Abstract. Certain integral operator Jm
p (λ, ℓ)(λ≥0;ℓ≥ 0;p ∈ N;m ∈ N0 = N ∪ {0}),where N ={1,2, ...} is introduced for functions of the form f(z) = zp+
∞
X
k=p+1
akzk which are analytic and p-valent in the open unit discU ={z:|z|<1}.
The opject of the present paper is to give an applications of the above operator to the differential inequalities.
2000Mathematics Subject Classification: 30C45. 1.Introduction LetA(p) denote the class of functions of the form:
f(z) =zp+ ∞
X
k=p+1
akzk (p∈N ={1,2, ....}) (1.1)
which are analytic and p-valent in the open unit disc U ={z:|z|<1}.In [3] Catas extended the multiplier transformation and defined the operator Ipm(λ, ℓ)f(z) on
A(p) by the following infinite series
Ipm(λ, ℓ)f(z) =zp+ ∞
X
k=p+1
p+ℓ+λ(k−p)
p+ℓ
m anzn
(λ≥0;ℓ≥0;p∈N, m∈N0;z∈U). (1.2) We note that:
Ip0(1,0)f(z) =f(z) and Ip1(1,0)f(z) = zf ′
(z)
p .
(i) Ipm(1, ℓ) =Ip(m, ℓ)f(z) (see Kumar et al. [9] and Srivastava et al. [16]); (ii)Ipm(1,0)f(z) =Dpmf(z) (see [2], [8] and [11]);
(iii) Im
1 (1, ℓ)f(z) =Iℓmf(z) (see Cho and Srivastava [4] and Cho and Kim [5]);
(iv)I1m(1,0) =Dmf(z) (m∈N0) (see Salagean [14]); (v) I1m(λ,0) =Dλm (see Al-Aboudi [1]);
(vi)Im
1 (1,1) =Imf(z) (see Uralegaddi and Somanatha [17]); (vii) Ipm(λ,0) =Dλ,pm f(z), where Dλ,pm f(z) is defined by
Dmλ,pf(z) =zp+ ∞
X
k=p+1
p+λ(k−p)
p
m akzk.
Furthermore we define the integral operatorJpm(λ, ℓ)f(z) as follows:
Jp0(λ, ℓ)f(z) =f(z),
Jp1(λ, ℓ)f(z) =
p+ℓ
λ
zp−(p+ℓλ )
z Z
0
t(p+ℓλ )−p−1f(t)dt (f(z)∈A(p);z∈U),
Jp2(λ, ℓ)f(z) =
p+ℓ
λ
zp−(p+ℓλ )
z Z
0
t(p+ℓλ )−p−1J1
p(λ, ℓ)f(t)dt (f(z)∈A(p);z∈U),
and, in general,
Jpm(λ, ℓ)f(z) =
p+ℓ
λ
zp−( p+ℓ
λ ) z Z
0
t( p+ℓ
λ )−p−1Jpm−1(λ, ℓ)f(t)dt
= Jp1(λ, ℓ)
zp
1−z
∗Jp1(λ, ℓ)
zp
1−z
∗....Jp1(λ, ℓ)
zp
1−z
∗f(z)
x m−times y
(f(z)∈A(p);m∈N;z∈U). (1.3) We note that iff(z)∈A(p), then from (1.1) and (1.3), we have
Jpm(λ, ℓ)f(z) =zp+ ∞
X
k=p+1
p+ℓ p+ℓ+λ(k−p)
m
akzk (m∈N0=N∪ {0}). (1.4)
From (1.4), it is easy to verify that
λz(Jpm+1(λ, ℓ)f(z))′ = (ℓ+p)Jpm(λ, ℓ)f(z)−[ℓ+p(1−λ)]Jpm+1(λ, ℓ)f(z) (λ >0).
We note that:
(i)J1m(λ,0)f(z) =Iλ−mf(z) (see Patel [12])
=
(
f(z)∈A(1) :Iλ−mf(z) =z+ ∞
X
k=2
[1 +λ(k−1)]−makzk, m∈N0
)
;
(ii)J1α(1,1)f(z) =Iαf(z) (see Jung et al. [7]);
=
(
f(z)∈A(1) :Iαf(z) =z+ ∞
X
k=2
2
k+ 1
α
akzk;α >0;z∈U )
;
(iii) Jα
p(1,1)f(z) =Ipαf(z) (see Shams et al. [15]);
=
f(z)∈A(p) :Ipαf(z) =zp+ ∞
X
k=p+1
p+ 1
k+ 1
α
akzk;α >0;z∈U
;
(iv)Jpm(1,1)f(z) =Dmf(z) (see Patel and Sahoo [13]);
=
f(z)∈A(p) :Dmf(z) =zp+ ∞
X
k=p+1
p+ 1
k+ 1
m
akzk;m is any integer;z∈U
.
(v) J1m(1,1)f(z) =Imf(z) (see Flett [6]);
=
(
f(z)∈A(1) :Imf(z) =z+ ∞
X
k=2
2
k+ 1
m
akzk;m∈N0;z∈U )
;
(iv)Jm
1 (1,0)f(z) =Imf(z) (see Salagean [14]) =
(
f(z)∈A(1) :Imf(z) =z+ ∞
X
k=2
k−makzk;m∈N0;z∈U
)
.
Also we note that: (i)Jpm(1,0)f(z) =Jpmf(z)
=
f(z)∈A(p) :Jpmf(z) =zp+ ∞
X
k=p+1
p
k m
akzk;m∈N0;z∈U
(ii)Jpm(1, ℓ)f(z) =Jpm(ℓ)f(z)
=
f(z)∈A(p) :Jpm(ℓ)f(z) =zp+ ∞
X
k=p+1
p+ℓ k+ℓ
m
akzk;m∈N0;ℓ≥0;z∈U
;
(iii) Jpm(λ,0)f(z) =Jλ,pm f(z)
=
f(z)∈A(p) :Jλ,pmf(z) =zp+ ∞
X
k=p+1
p p+λ(k−p)
m
akzk;m∈N0;λ≥0;z∈U
.
By using the operatorJm
p (λ, ℓ),we define the following classes of functions.
Definition 1. Let Φ be the set of complex-valued functions ϕ(r, s, t);
ϕ(r, s, t) :C3 →C (C is complex plane)
such that
(i) ϕ(r, s, t) is continuous in a domain D⊂C3; (ii) (0,0,0)∈D and |ϕ(0,0,0)|<1;
(iii)
ϕeiθ, 1
p+ℓ
λ
ζ+hp(1−λλ)+ℓieiθ, 1
p+ℓ
λ 2
hn
1 + 2hp(1−λλ)+ℓi ζ+
hp
(1−λ)+ℓ λ
i2
eiθ+M !
>1,
whenever eiθ, 1 p+ℓ
λ
ζ+hp(1−λλ)+ℓieiθ, 1 p+ℓ
λ 2
1+ 2hp(1−λλ)+ℓiζ+
hp
(1−λ)+ℓ λ
i2
eiθ+M !
∈ D, λ > 0, ℓ ≥ 0, with Re
e−iθM ≥ ζ(ζ −1), for all θ∈R, and for all ζ ≥p≥1.
Definition 2. Let H be a set of complex-valued functions h(r, s, t);
h(r, s, t) :C3 →C
(i) h(r, s, t) is continuous in a domain D⊂C3; (ii) (1,1,1)∈D and |h(1,1,1)|< M (M >1); (iii)
h
M eiθ,
ζ+p+λℓM eiθ
p+ℓ λ
,
1
p+ℓ
λ
"
ζ−ζ2+p+λℓζM eiθ+L ζ+p+λℓM eiθ
≥M,
whenever
M eiθ,
ζ+p+λℓM eiθ
p+ℓ λ
,
1
p+ℓ
λ
"
ζ+p+λℓM eiθ+
+
ζ−ζ2+p+λℓζM eiθ+L ζ+p+λℓM eiθ
∈D,
where Re{L} ≥ζ(ζ−1)for all θ∈R and for all ζ ≥ M−1
M+ 1. 2.Main Results
We recall the following lemmas due to Miller and Mocenu [10].
Lemma 1 [10]. Let w(z) =bpzp+bp+1zp+1+....(p∈N) be regular in the unit disc
U with w(0)6= 0 (z∈U).
If z0 =r0eiθ0(0< r0 <1) and |w(z0)|= max |z|≤z0
|w(z)|,then
z0w ′
(z0) =ζw(z0) (2.1)
and
Re
(
1 +z0w ′′
(z0)
w′ (z0)
)
≥ζ (2.2)
and
where ζ is real and ζ ≥p≥1.
Lemma 2 [10]. Let w(z) = a+wνzν +.... be regular in U with w(z) 6= a and
ν ≥1 If z0=r0eiθ0(0< r0 <1)and |w(z0)|= max |z|≤z0
|w(z)|, then
z0w ′
(z0) =ζw(z0)
and
Re
(
1 +z0w ′′
(z0)
w′ (z0)
)
where ζ is a real number and
ζ≥ν |w(z0)−a|
2
|w(z0)|2− |a|2 ≥ν
|w(z0)| − |a| |w(z0)|+|a|
. (2.3)
Theorem 1. Let ϕ(r, s, t)∈Φ and let f(z) belonging to the class A(p) satisfy
(i) (Jpm(λ, ℓ)f(z), Jpm−1(λ, ℓ)f(z) , Jpm−2(λ, ℓ)f(z))∈D⊂C3
and
(ii) ϕ(Jpm(λ, ℓ)f(z), Ipm−1(λ, ℓ)f(z) , Ipm−2(λ, ℓ)f(z))<1
for m >2, p∈N, λ >0, ℓ≥0 and z∈U.Then we have
Jpm(λ, ℓ)f(z)<1 (z∈U). (2.4)
Proof. We define the analytic function w(z) by
Jpm(λ, ℓ)f(z) =w(z) (m >2;p∈N;λ >0;ℓ≥0) (2.5) for f(z) belonging to the class A(p). Then, it folllows that w(z) ∈ A(p) and
w(z)6= 0 (z∈U).With the aid of the idenlitty (1.5), we have
Jpm−1(λ, ℓ)f(z) = 1 p+ℓ
λ
p(1−λ) +ℓ λ
w(z) +zw′(z)
(2.6)
and
Jpm−2(λ, ℓ)f(z) = 1
p+ℓ λ
2 (
p(1−λ) +ℓ λ
2
w(z) +
1 + 2
p(1−λ) +ℓ λ
zw′(z) +z2w′′(z)
. (2.7) Suppose that z0=r0eiθ0 (0< r0 <1;θ0 ∈R) and
w(z0) = max |z|≤|z0|
|w(z)|= 1. (2.8)
Then, letting w(z0) =eiθ0 and using (2.1) of Lemma 1, we obtain
Jpm−1(λ, ℓ)f(z0) = 1 (p+λℓ)
p(1−λ) +ℓ λ
w(z0) +z0w ′
(z0)
= 1
(p+λℓ)
ζ+
p(1−λ) +ℓ λ
eiθ0
and
Jpm−2(λ, ℓ)f(z0) = 1 (p+λℓ)2
1 + 2hp(1−λλ)+ℓiζ+hp(1−λλ)+ℓi2
eiθ0 +
z20w′′(z0)
i
= 1
(p+λℓ)2
"(
1 + 2
p(1−λ) +ℓ λ
ζ+
p(1−λ) +ℓ λ
2)
eiθ0 +M
#
, (2.11) where M =z02w′′(z0) and ζ ≥p≥1.
Further, an application of (2.2) in Lemma 1, gives
Re
(
z0w′′(z0)
w′ (z0)
)
= Re
(
z02w′′(z0)
ζeiθ0
)
≥ζ−1, (2.12)
or
Rene−iθ0M
o
≥ζ(ζ−1) (θ0 ∈R;ζ ≥1). (2.13) Since ϕ(r, s, t)∈Φ,we also have
ϕ(Jpm(λ, ℓ)f(z0), Jm−1(λ, ℓ)f(z0), Jm−2(λ, ℓ)f(z0))
=
ϕ eiθ0, 1
(p+λℓ)
ζ+
p(1−λ) +ℓ λ
eiθ0, 1
(p+λℓ)2
1 + 2
p(1−λ) +ℓ λ
ζ
+
p(1−λ) +ℓ λ
2)
eiθ0 +M
#!
>1 (2.14)
which contradicts the condition (ii) of Theorem 1 Therefore, we conclude that |w(z)|=Jpm(λ, ℓ)f(z)<1 (z∈U; m >2).
Corollary 1. Let ϕ1(r, s, t) = s and let f(z) ∈ A(p) satisfy the conditions in Theorem 1 for m >2;p∈N;λ >0;ℓ≥0 and z∈U. Then
(Jpm+i(λ, ℓ)f(z)
Proof. Note that ϕ1(r, s, t) =s in Φ,with the aid of Theorem 1, we have
Jpm−1(λ, ℓ)f(z)<1⇒
Jpm(λ, ℓ)f(z)<1 (m >2;p∈N;λ >0;ℓ≥0)
⇒Jpm+i(λ, ℓ)f(z)<1 (i= 0,1,2, ...;m >2;p∈N;λ >0;ℓ≥0;z∈U).
Theorem 2. Let h(r, s, t)∈H, and let f(z) belonging to A(p) satisfying
(i) J
m−1
p (λ, ℓ)f(z) Jm
p (λ, ℓ)f(z) ,J
m−2
p (λ, ℓ)f(z) Jpm−1(λ, ℓ)f(z)
,J m−3
p (λ, ℓ)f(z) Jpm−2(λ, ℓ)f(z)
!
∈D⊂C3
and
(ii)
h J m−1
p (λ, ℓ)f(z) Jm
p (λ, ℓ)f(z) ,J
m−2
p (λ, ℓ)f(z) Jpm−1(λ, ℓ)f(z)
,J m−3
p (λ, ℓ)f(z) Jpm−2(λ, ℓ)f(z)
! < M
for some λ, ℓ, m, p, M (λ >0;ℓ≥0;m >3;p∈N;M >1)and for all z∈U. Then
we have
Jm−1
p (λ, ℓ)f(z) Jm
p (λ, ℓ)f(z)
< M (z∈U) (2.16)
Proof. We define the function w(z) by
Jpm−1(λ, ℓ)f(z)
Jm
p (λ, ℓ)f(z)
=w(z) (λ >0, ℓ≥0;m >3;p∈N) (2.17)
for f(z) belonging to the class A(p). Then, it follows that w(z) is either analytic or meromorphic in U,w(0) = 1, and w(z) 6= 1. With the aid of the identity (1.5), we have
Jpm−2(λ, ℓ)f(z)
Jpm−1(λ, ℓ)f(z)
= 1 p+ℓ
λ
"
p+ℓ λ
w(z) +zw ′
(z)
w(z)
# (2.18) and
Jm−3
p (λ, ℓ)f(z) Jpm−2(λ, ℓ)f(z)
= 1 p+ℓ
λ
"
p+ℓ λ
w(z) +zw ′
(z)
w(z)
#
+
p+ℓ
λ
zw′(z) +zw ′
(z)
w(z) +
z2w′′(z)
w(z) −(
zw′(z)
w(z) )2
p+ℓ
λ
w(z) +zww(′(zz))
Suppose that z0 = r0eiθ0(0 < r0 < 1;θ0 ∈ R) and |w(z0)| = max |z|≤|z0|
|w(z)| = M.
Letting w(z0) =M eiθ0 and applying Lemma 2 witha=ν = 1,we see that
Jpm−2(λ, ℓ)f(z)
Jpm−1(λ, ℓ)f(z)
= 1 p+ℓ
λ
h
ζ+p+λℓM eiθ0
i
(2.20)
and
Jpm−3(λ, ℓ)f(z0)
Jpm−2(λ, ℓ)f(z0)
= 1 p+ℓ
λ
ζ+p+λℓM eiθ0+
ζ−ζ2+p+λℓζM eiθ0+L
ζ+p+λℓM eiθ0
,
where L= z02w
′′ (z0)
w(z0) and ζ ≥
M−1
M+1.
Further, an application of (ii) in Lemma 2 gives Re{L} ≥ζ(ζ−1).
Since h(r, s, t)∈H,we have
h J
m−1
p (λ, ℓ)f(z0)
Jm
p (λ, ℓ)f(z0)
,J m−2
p (λ, ℓ)f(z0)
Jpm−1(λ, ℓ)f(z0)
,J m−3
p (λ, ℓ)f(z0)
Jpm−2(λ, ℓ)f(z0)
!
=
h
M eiθ0,
ζ+p+λℓM eiθ0
p+ℓ
λ
,
1
p+ℓ
λ
ζ+p+λℓM eiθ0+
ζ−ζ2+p+λℓζM eiθ0+L
ζ+p+λℓM eiθ0
≥M,
which contradicts condition (ii) of Theorem 2. Therefore, we conclude that
|w(z)|=
Jm−1
p (λ, ℓ)f(z) Jm
p (λ, ℓ)f(z)
< M (2.23)
for all λ >0, ℓ≥0, m >3, p∈N and z∈U. This completes the proof of Theorem 2.
Remarks. (i)Puttingλ= 1 in the above results, we obtain the correspording results for the operatr Jm
p (ℓ)f(z);
(iii)Puttingλ=ℓ= 1 in the above results, we obtain the correspording results for the operatrDmf(z).
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R. M. El-Ashwah
Department of Mathematics
Faculty of Science (Damietta Branch) University of Mansoura University New Damietta 34517, Egypt email:r [email protected]
M. K. Aouf
Department of Mathematics Faculty of Science
University of Mansoura University Mansoura 35516, Egypt
