Dunkl generalization of Phillips operators and
approximation in weighted spaces
M. Mursaleen1,2,3 , Md. Nasiruzzaman4 , A. Kilicman5 and S. H. Sapar6
1 Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan),
Taichung, Taiwan
2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
3 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan;
4 Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia;
5,6 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang,
Selangor, Malaysia
* Correspondence: [email protected]; Tel.: +60-89466813
Abstract:Purpose of this article is to introduce a modification of Phillips operators on the intervalh12,∞ via Dunkl generalization. This type of modification enables a better error estimation on the interval h
1 2,∞
rather than the classical Dunkl Phillips operators on[0,∞). We discuss the convergence results and obtain the degrees of approximations. Furthermore, we calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre’s K-functional and second order modulus of continuity.
Keywords: Szász operator; Dunkl analogue; generalization of exponential function; Korovkin type theorem; modulus of continuity; order of convergence.
1. Preliminaries and Introduction
Szász operators [18] provide an extension to Bernstein operators [4] on the interval [0,∞). In the present years, several authors have studied the Dunkl type generalization of Szász operators (see [1,2,9–12,14–16,19]).
Sucu [17] introduced Dunkl analogue of Szász operators. That is, for f ∈C[0,∞),x≥0, υ≥0 and
n∈N,
Sn∗(f;x):= 1
eυ(nx) ∞
∑
`=0(nx)` γυ(`) f
`+2υθ` n
, (1)
whereNis the set of all natural numbers and
eυ(x) = ∞
∑
`=0x`
γυ(`)
. (2)
The coefficientsγυare given as
γυ(2`) =
22``!Γ`+
υ+12
Γυ+12
, γυ(2`+1) =
22`+1`!Γ `+υ+32
Γυ+12
(3)
with recursion
γυ(`+1)
(`+1+2υθ`+1) =γυ(`) (4)
where
θ` = (
0 if`=0, 2, 4,· · ·,
1 if`=1, 3, 5,· · ·, .. (5)
Studies on Dunkl type generalizations [13] and previous studies of Szász type operators [6,7] demonstrate an error estimation to the operators which allow us much faster approximation to the function which is being approximated. In this paper, we modify the Phillips operators given by [13] via Dunkl generalization. Our main idea is to approximate these operators by well known Korovkin’s and weighted Korovkin’s theorems. We estimate the degrees of approximations and calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre’sK-functional and second order modulus of continuity.
2. New operators and their moments
Let{χn(x)}be a sequence of nonnegative continuous functions on[0,∞)as
χn(x) =
x− 1
2n
+, n∈N, (6)
where
κ+= (
κ ifκ≥0,
0 ifκ<0.
(7)
Moreover, suppose
Jn,υ(x) =
eυ(−nχn(x))
eυ(nχn(x))
(8)
For f ∈Cζ(R+) ={f ∈C[0,∞): f(t) =O(tζ),ζ>n, n∈
N}, we define
Pn,υ(f;x) = n2 eυ(nχn(x))
∞
∑
`=0(nχn(x))`
γυ(`) ∞ Z
0
e−ntn`+2υθ`−1t`+2υθ` γυ(`)
f(t)dt, υ≥0
Lemma 1. Let e`=t`−1,`=1, 2, 3, 4, 5andJn,υ(x)defined by(8). Then for x ≥0,Pn,υ(e1;x) =1and for any x≥ 1
2n we have
(1) Pn,υ(e2;x) = x+ 1 2n,
(2) Pn,υ(e3;x) = x2+ 1
n(3+2υJn,υ(x))x− 1
4n2Jn,υ(x),
(3) Pn,υ(e4;x) = x3+ 1
2n(15−4υJn,υ(x))x 2
+ 1
4n2(39+16υ+72υJn,υ(x))x
− 1
8n3(7+16υ+68υJn,υ(x)),
(4) Pn,υ(e5;x) = x4+1
n(14+4υJn,υ(x))x 3
+ 1
2n2
99−68υJn,υ(x) +8υ2
x2
+ 1
2n3
131+294υJn,υ(x) +96υ2+16υ3Jn,υ(x)
x
+ 1
16n4
−367+808υJn,υ(x) +432υ2+64υ3Jn,υ(x)
.
Remark 1. For any 0 ≤ x ≤ 1
2n, we have Pn,υ(e2;x) = 1n; Pn,υ(e3;x) = n22; Pn,υ(e4;x) = 6
n3; Pn,υ(e5;x) = 24n4.
Here we also introduce the Stancu type generalization to the operators defined by (9). Thus, for each f ∈Cζ(R+)the modified version of the operators (9) is defined as
Sn∗,υ(f;x) = n 2
eυ(nχn(x))
∞
∑
`=0(nχn(x))`
γυ(`) ∞ Z
0
e−ntn`+2υθ`−1t`+2υθ` γυ(`)
f nt+
α
n+β
dt
where 0≤α≤β. Note that if we takeα=β=0 in (9), then the operatorsSn∗,υreduce to operators defined
by (9) and if takeχn(x) =xinPn,υ, then we get the operators defined studied in [13].
Lemma 2. For x≥0,S∗
n,υ(e1;x) =1and for x≥ 1
2n, we have
1◦ Sn∗,υ(e2;x) = n
n+βPn,υ(e2;x) + α
n+β,
2◦ Sn∗,υ(e3;x) = n 2
(n+β)2Pn,υ(e3;x) + 2 αn
(n+β)2Pn,υ(e2;x) + α2 (n+β)2, 3◦ Sn∗,υ(e4;x) = n
3
(n+β)3Pn,υ(e4;x) + 3 αn2
(n+β)3Pn,υ(e3;x) + 3 α2n
(n+β)3Pn,υ(e2;x) + α3 (n+β)3, 4◦ Sn∗,υ(e5;x) = n
4
(n+β)4Pn,υ(e5;x) + 4 αn3
(n+β)4Pn,υ(e4;x) + 6 α2n2
(n+β)4Pn,υ(e3;x)
+ 4α
3n
Lemma 3. For0≤x≤ 1
2n,we have
(1)◦◦ Sn∗,υ(e2;x) = α +1 n+β, (2)◦◦ Sn∗,υ(e3;x) = 2
+α+α2 (n+β)2 , (3)◦◦ Sn∗,υ(e4;x) = 6
+6α+3α2+α3 (n+β)3 , (4)◦◦ Sn∗,υ(e5;x) = 24
+4α+12α2+4α3+α4
Lemma 4. Supposeηj= (e2−x)jfor j=1, 2, 3, 4, where e2is defined in Lemma2. Then, for x≥ 21n we have
1∗ Sn∗,υ(η1;x) =
n n+β−1
x+ 1+2α
2(n+β),
2∗ Sn∗,υ(η2;x) =
n2
(n+β)2− 2n n+β+1
x2
+
n
(n+β)2(3+2υJn,υ(x)) + 2αn (n+β)2−
2α+1
n+β
x
+ α+α
2
(n+β)2− 1
4(n+β)2υJn,υ(x), 3∗ Sn∗,υ(η4;x) =
n4 (n+β)4−
4n3 (n+β)3+
6n2 (n+β)2−
4n n+β+1
x4
+
n3
(n+β)4(14+4α+4υJn,υ(x))
− 2n
2
(n+β)3
15+6α−4υeυ(−nχ(x))
eυ(nχ(x))
+ 6n
(n+β)2(3+2α+2υJn,υ(x))− 2+4α
n+β
x3
+
n2 2(n+β)4
99+60α+8υ2+4(17−4α)4υJn,υ(x)
− n
(n+β)3(39+16υ+12(3+α)(2+α)υJn,υ(x))
+ 1
2(n+β)2
12α+12α2+12α2n2−3υJn,υ(x)
x2
+
n 2(n+β)4
131+78α+16α3+31αυ+96υ2
+ (294+144α+16υ2)υJn,υ(x)
+ 1
2(n+β)3
7+16υ−12α2−8α3+ (68+6α)υJn,υ(x)
+ 6α
2n
(n+β)2(3+2υJn,υ(x))
x
+ 1
16(n+β)4
−367+432υ2+64υ3Jn,υ(x)−56α−128αυ
+ 16α2(2α+1) + (808−544α)υJn,υ(x)
− 3α 2
Lemma 5. Supposeηj= (e2−x)jfor j=1, 2, 3, 4, where e2defined in Lemma2. Then, for any0≤x≤ 21n we have
(1)∗∗ Sn∗,υ(η1;x) = α+1
n+β−x, (2)∗∗ Sn∗,υ(η2;x) = x2−2
(α+1) (n+β)x+
2+α+α2 (n+β)2 , (3)∗∗ Sn∗,υ(η4;x) = x4−4
(α+1) (n+β)x
3+6(2+α+α2) (n+β)2 x
2
− 4(6+6α+3α 2+α3)
(n+β)3 x+
24+24α+12α2+4α3+α4
(n+β)4 .
3. Korovkin type approximation
In the present section the results related to uniform convergence of the operators defined by (9) are given via well-known Korovkin’s and weighted Korovkin’s type theorems.
LetR+= [0,∞)andCB(R+)denote the linear normed space with the norm k f kCB(
R+ )=sup
x≥0
| f(x)|.
Let
H:={f : lim
x→∞ f(x)
1+x2 exists,x∈[0,∞)}.
Theorem 1. Let the function f ∈C[0,∞)∩ Hand the operatorsSn∗,υ(·;·)defined by(9). Then lim
n→∞S
∗
n,υ(f;x) = f(x) uniformly on U,where U is any compact subset of[0,∞).
Proof of Theorem 1. We apply the well-known Korovkin’s theorem to prove the uniform convergence of operatorsSn∗,υ(·;·). Therefore, for`=1, 2, 3, we see limn→∞Sn∗,υ(e`;x) =x
`−1uniformly. Therefore, we have
lim
n→∞S
∗
n,υ(e1;x) =1; nlim→∞S
∗
n,υ(e2;x) =x; nlim→∞S
∗
n,υ(e3;x) =x 2.
Hence proved.
We recall the weighted spaces defined by Gadžiev [8]. We writeBσ(R+)for the set of all functions such that
| f(x)|≤mfσ(x)
with||f||σ=supx≥0|σf((xx))|, wheremf is a constant depends onf , andx→φ(x)is a continuous and strictly
increasing function such asσ(x) =1+φ2(x)with limx→∞σ(x) =∞. LetCσ(R+) = Bσ(R+)∩C(R+). Note that [8] the sequence of positive linear operators{Ln}n≥1mapsCσ(R+)intoBσ(R+)if and only if
withσ(x) =1+φ2(x),x∈R+andKis a positive constant. LetC0σ(R
+)be a subset ofC
σ(R+)such that
lim
x→∞ f(x)
σ(x) =kf <∞. Theorem 2. LetS∗
n,υbe the sequence of positive linear operators acting from Cσ(R+)into Bσ(R+)such that lim
n→∞k S
∗
n,υ(ϕ
k(t);x)−
ϕk(x)kσ=0 k=0, 1, 2. Then for all f ∈C0σ(R+),we have
lim
n→∞k S
∗
n,υ(f(t);x)−f(x)kσ=0. Proof of Theorem 2. Considerϕ(x) =x,σ(x) =1+x2and
S ∗
n,υ(e`;x)−x `−1
σ =sup
x≥0 S
∗
n,υ(e`;x)−x `−1
1+x2 .
Then by Korovkin’s theorem, it is easily proved that limn→∞ S ∗
n,υ(e`;x)−x `−1
σ
=0, for` = 1, 2, 3.
Hence for any f ∈Cσ0(R+), we get lim
n→∞
Sn∗,υ(f(t);x)− f(x)
σ =0.
Theorem 3. LetSn∗,υ(·;·)be the operators defined by(9). Then for every f ∈C0
σ(R+),we have lim
n→∞k S
∗
n,υ(f;x)− f kσ=0.
Proof of Theorem 3. We prove this theorem in the light of2. Take f(t) =e`defined by Lemma2. Then, for anyf(t)∈C0σ(R+),Sn∗,υ(e`;x)→x
`−1(`=1, 2, 3) uniformly asn→∞. For`=1, by applying Lemma 2, we getSn∗,υ(e1;x) =1, so that
lim
n→∞
Sn∗,υ(e1;x)−1
σ =0. (9)
Take`=2 andx ≥ 1
2n, we get
Sn∗,υ(e2;x)−x σ =sup
x≥0
Sn∗,υ(e2;x)−x 1+x2
=sup
x≥0
n
n+βPn,υ(e2;x)−x+ α
n+β 1+x2
≤
n n+β−1
sup
x≥0 x 1+x2 +
1+2α
2(n+β)supx≥0
In case of 0≤x≤ 1
2n, we get
Sn∗,υ(e2;x)−x σ
= max
0≤x≤1 2n
Sn∗,υ(e2;x)−x 1+x2 ≤ max
0≤x≤1 2n
Sn∗,υ(e2;x)−x
≤ max 0≤x≤1
2n
α+1
n+β−x
= 1
n+βmax
α+1, α− β 2n . Then lim
n→∞
Sn∗,υ(e2;x)−x
σ =0. (10)
In similar way if take`=3 andx≥ 1
2n, we get
S ∗
n,υ(e3;x)−x 2 σ =sup
x≥0
Sn∗,υ(e3;x)−x2 1+x2
=sup
x≥0
n2
(n+β)2Pn,υ(e3;x) + 2αn
(n+β)2Pn,υ(e2;x) + α2 (n+β)2 −x
2 1+x2
≤
n2
(n+β)2 −1
sup
x≥0 x2 1+x2+
n
(n+β)2(2α+3+2υJn,υ(x))sup
x≥0 x 1+x2
+ 1
4(n+β)2
4α+4α2− Jn,υ(x)
sup
x≥0 1 1+x2, In case of 0≤x≤ 1
2n, we get
Sn∗,υ(e3;x)−x σ = max
0≤x≤1 2n
Sn∗,υ(e2;x)−x 1+x2 ≤ max
0≤x≤1 2n
Sn∗,υ(e2;x)−x
≤ max 0≤x≤1
2n
2+α+α2 (n+β)2 −x
2
= 2+α+α 2
(n+β)2 . lim
n→∞ S ∗
n,υ(e3;x)−x 2 σ
=0. (11)
4. Order of approximation ofSn∗,υ(·;·)
We denote the set of all uniformly continuous functions by ˜C[0,∞). Let ˜ω(f; ˜δ)denote the modulus of continuity off ∈C˜[0,∞), i.e.
˜
ω(f; ˜δ) = sup
|x1−x2|≤δ
| f(x1)− f(x2)|; x1,x2∈[0,∞), ˜δ>0. (12)
which satisfies that limδ˜→0+ω(˜ f; ˜δ) =0, and
| f(x1)−f(x2)|≤ |
x1−x2| ˜
δ
+1
˜
ω(f; ˜δ). (13)
In the light of Lemma4and5we use the notation
q S∗
n,υ(η2;x) =δ˜n,υ(x), (14) where
˜
δn,υ(x) 2
=
x2−2(α+1) (n+β)x+
2+α+α2
(n+β)2 ; if 0≤x ≤ 1 2n,
n2
(n+β)2 −n2+nβ+1
x2 +
n
(n+β)2(3+2υJn,υ(x)) + 2αn (n+β)2 −
2α+1
n+β
x
+ α+α2 (n+β)2 −
1
4(n+β)2υJn,υ(x); if x≥ 1 2n
(15)
and
Jn,υ(x) =J
∗
n,υ(nχn(x)) =
1; if 0≤x≤ 1
2n,
J∗
n,υ
nx−1 2
; if x≥ 1 2n.
(16)
Theorem 4. For any f ∈C˜[0,∞),
| Sn∗,υ(f;x)− f(x)|≤2 ˜ω f; ˜δn,υ(x)
,
whereδ˜n,υ(x)is defined by(15).
Proof of Theorem 4. Using (12) and (13), we get
Sn∗,υ(f;x)−f(x) = Sn∗,υ(f;x)− f(x)Sn∗,υ(e1;x) = Sn∗,υ(f(t)−f(x);x) ≤ Sn∗,υ(|f(t)−f(x)|;x)
SinceS∗
n,υ(e1;x) =1, by (13) we get
Sn∗,υ(f;x)−f(x)
≤ Sn∗,υ
1+|η1|
˜
δ ;x
˜
ω(f; ˜δ) =
1+1
˜
δ
Sn∗,υ(|η1|;x)
˜
From the Cauchy-Schwarz inequality we conclude that
Sn∗,υ(|η1|;x) ≤ Sn∗,υ(e1;x)
1 2 S∗
n,υ(η2;x)
1 2
= Sn∗,υ(η2;x)
1 2.
Therefore,
Sn∗,υ(f;x)−f(x) ≤
1+1
˜
δ
Sn∗,υ(η2;x)
1 2
˜
ω(f; ˜δ).
Choose ˜δ=δ˜n,υ(x) = q
S∗
n,υ(η2;x), then we get the result.
Here we use the usual class of Lipschitz functions and obtain the rate of convergence of the sequence of positive linear operatorsS∗
n,υ(·;·)(9). ForL>0, 0<$≤1 and for the continuous functions f on
[0,∞), the class of Lipschitz functionsLipL,$(f)is
LipL,$(f) ={f :| f(ς1)− f(ς2)|≤ L |ς1−ς2|$; L>0, 0<$≤1 (ς1,ς2∈[0,∞))} (17) Theorem 5. For any f ∈LipL,$,we have
| Sn∗,υ(f;x)− f(x)|≤ L δ˜n,υ(x) $
whereδ˜n,υ(x)is defined by(15).
Proof of Theorem 5. By Hölder inequality and (17), we get
| Sn∗,υ(f;x)−f(x)| ≤ | Sn∗,υ(f(t)− f(x);x)| ≤ Sn∗,υ(| f(t)−f(x)|;x) ≤ LSn∗,υ(|η1|$;x) ≤ L Sn∗,υ(e1;x)
2−2$
Sn∗,υ(|η1|2;x) $2
= L Sn∗,υ(η2;x)
$ 2 .
The space of all that continuous and bounded functions onR+is denoted byCB(R+)and
C2B(R+) ={ψ∈CB(R+):ψ0,ψ00∈CB(R+)}. (18) The norm onC2B(R+)is given by
kψkC2
B(R+)=kψ 00k
CB(R+)+kψ0kCB(R+)+kψkCB(R+), (19)
where the norm forCB(R+)is
kψkCB(R+)=sup
x≥0
|ψ(x)|. (20)
Theorem 6. Letψ∈CB2(R+).Then
whereµn,υ(x) =δ˜n,υ(x) +
(δ˜n,υ(x))2
2 .
Proof of Theorem 6. By Taylor series expansion forψ∈C2B(R+)we obtain ψ(t) =ψ(x) +ψ0(x)η1+ψ00(ϕ) η2
2 whereη1,η2are given by (4), (5)
|ψ(t)−ψ(x)| ≤ U1|η1|+ 1 2U2η2, where
U1=sup
x≥0 ψ0(x)
=||ψ0||C
B(R+)≤ ||ψ||C2 B(R+),
U2=sup
x≥0 ψ00(x)
=||ψ00||C
B(R+)≤ ||ψ||C2 B(R+).
Therefore,
|ψ(t)−ψ(x)| ≤
|η1|+ 1 2η2
||ψ||C2
B(R+).
By using linearity ofS∗
n,υwe get
Sn∗,υ(ψ,x)−ψ(x) ≤
Sn∗,υ(|η1|;x) +1 2S
∗
n,υ η2;x
||ψ||C2
B(R+).
So that
Sn∗,υ(ψ,x)−ψ(x) =
Sn∗,υ(ψ(t)−ψ(x);x)
≤ Sn∗,υ(|ψ(t)−ψ(x)|;x) From Cauchy-Schwarz inequality,
Sn∗,υ(|η1|;x)≤
Sn∗,υ(η2;x) 12
.
Thus, we have
Sn∗,υ(ψ,x)−ψ(x) ≤
˜
δn,υ(x) +
(δ˜n,υ(x))2 2
||ψ||C2
B(R+).
5. Peetre’s K-functional and Direct theorem ofS∗
n,υ(·;·)
The Peetre’s K-functional is an influence of potential research work on approximation process has given by J. Peetre in 1968. The Peetre was enable to to investigate the interpolation spaces between two Banach spaces and an interactions to the real interpolation on K-functional. For any f ∈ CB(R+), the Peetre’s, well-knownK-functional property defined as:
K2(f, ˘δ) =inf n
k f−ψkCB(R+)+δ˘kψ 00k
C2B(R+)
:ψ∈ W2 o
, (21)
where
W2=
For any ˘δ>0 and a positive constantC one hasK2(f; ˘δ)≤ Cω2(f; ˘δ12), where
ω2(f; ˘δ12) = sup
0<h<δ˘
1 2
sup
t≥0
| f(t+2h)−2f(t+h) + f(t)|. (23)
Theorem 7. Let f ∈CB(R+). Then there exists a positive constantDsuch as
Sn∗,υ(f;x)− f(x)
≤2D (
ω2 f; r
λn,υ(x) 2
! +min
1;λn,υ(x) 2
||f||CB(R+)
) ,
whereλn,υ(x)is given by6andω2
f;λn,υ(x)
2
is given by(15).
Proof of Theorem 7. Takeψ∈C2B(R+). Thus we get
Sn∗,υ(f;x)−f(x) ≤
Sn∗,υ(f−ψ;x) +
Sn∗,υ(ψ;x)−ψ(x)
+|f(x)−ψ(x)| ≤2||f−ψ||CB(R+)+λn,υ(x)||ψ||C2
B(R+) =2
||f−ψ||CB(R+)+λn,υ
(x)
2 ||ψ||C2B(R+)
.
By taking the infimum over allψ∈C2B(R+)and by using (21), we get
Sn∗,υ(f;x)− f(x) ≤2K2
f;λn,υ(x) 2
.
Now, for an absolute constantD>0 in [5], we use the following relation:
K2(f; ˘δ)≤ D{ω2(f; p
˘
δ) +min(1; ˘δ)||f||CB(R+)}
where ˘δ= λn,υ2(x). This completes the proof.
For an arbitrary f ∈Cσ0(R+)the following weighted modulus of continuity was defined in [3] ¯
Ω(f; ˆδ) = sup
|h|≤δˆ, x≥0
| f(x+h)− f(x)|
(1+x2)(1+h2) , (24)
satisfying
lim ˆ δ→0
¯
Ω(f; ˆδ) =0, (25)
and
| f(t)−f(x)|≤2
|t−x| ˆ
δ +1
(1+δˆ2)(1+x2)
(t−x)2+1Ω¯(f; ˆδ). (26)
Theorem 8. For any f ∈C0
σ(R+),we have sup
x∈[0,An,υ)
Sn∗,υ(f;x)−f(x) 1+x2 ≤ A
1+O An,υ
Ω
f;O pAn,υ
whereAis a positive constant and for x≥ 1 2n
An,υ = max
n2
(n+β)2 − 2n n+β+1,
α+α2 (n+β)2−
1
4(n+β)2υJn,υ(x), n
(n+β)2(3+2υJn,υ(x)) + 2αn (n+β)2−
2α+1
n+β
,
and for0≤x≤ 21n,
An,υ = max
1,2(α+1)
(n+β),
2+α+α2 (n+β)2
.
Proof of Theorem 8. We prove it by using (24), (26) and Cauchy-Schwarz inequality. We have
Sn∗,υ(f;x)−f(x)
≤2(1+δˆ2)(1+x2)Ω(f; ˆδ) 1+Sn∗,υ η2;x
+Sn∗,υ (1+η2) |η1|
ˆ
δ
;x !
. (27)
From the Lemma4,5we easily conclude that,
Sn∗,υ η2;x ≤ A1O
An,υ
(1+x+x2) ≤ A2(1+x+x2)
whereA1andA2are positive constants, and forx≥ 21n
An,υ=max
n2
(n+β)2− 2n n+β+1,
α+α2 (n+β)2−
1
4(n+β)2υJn,υ(x), n
(n+β)2(3+2υJn,υ(x)) + 2αn (n+β)2 −
2α+1
n+β
, (28)
and for 0≤x≤ 1 2n,
An,υ=max
1,2(α+1)
(n+β),
2+α+α2 (n+β)2
.
By apply the Cauchy-Schwarz inequality we easily see that
Sn∗,υ
(1+η2) |η1|
δ ;x
≤2
Sn∗,υ 1+η4;x 12
Sn∗,υ
η2 ˆ
δ2;x 12
. (29)
Similarly for the constantsA3>0 andA4>0, we have
Sn∗,υ 1+η4;x
12 ≤ A3
Sn∗,υ
η2
ˆ
δ2;x 12
≤ 1
ˆ
δ A4O
An,υ
12
1+x+x2 12
Finally, in view of (27), (28) and (29), and choosing ˆδ=O
p An,υ
, andA =2(1+A2+2A3A4),
we easily led to the desire result.
6. Conflicts of Interest
The author declares no conflict of interest.
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