Applications of a q Salagean type operator on multivalent functions

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R E S E A R C H

Open Access

Applications of a q-Salagean type

operator on multivalent functions

Saqib Hussain

1*

_{, Shahid Khan}

2

_{, Muhammad Asad Zaighum}

2

_{and Maslina Darus}

3

*_{Correspondence:} [email protected]

1_{Department of Mathematics,} COMSATS University Islamabad, Abbottabad Campus, Pakistan Full list of author information is available at the end of the article

Abstract

In this paper, we introduce a new classk-US(q,

γ

,m,p),

γ

∈C\{0}, of multivalent functions using a newly definedq-analogue of a Salagean type differential operator. We investigate the coefficient problem, Fekete–Szego inequality, and some other properties related to subordination. Relevant connections of the results presented here with those obtained in the earlier work are also pointed out.

MSC: Primary 30C45; secondary 30C50

Keywords: Multivalent functions;qderivative operator; Convolution; Conic domain

1 Introduction

For a positive integerp, letApdenote the set of all functionsf(z) which are analytic and

p-valent in the open unit diskE={z∈C:|z|< 1}and have series expansion of the form

f(z) =zp+

∞

n=p+1

anzn. (1.1)

Also, letf ∗gdenote the convolution (or Hadamard product) off,g∈Apdeﬁned as fol-lows:

(f ∗g)(z) =zp+

∞

n=p+1 anbnzn,

wheref(z) is given by (1.1) andg(z) =zp₊∞

n=p+1bnzn.

Quite recently,q-analysis has influenced the researchers a lot due to rapid applications in mathematics and related fields. In the last century many well-known researchers (for details, see [1,4,6–10,13,14,21,22,32]) did great work onq-calculus and found numerous applications. It is worth mentioning that convolution theory helps many researchers to investigate a number of properties of analytic univalent and multivalent functions. Several differential and integral operators were defined using ordinary derivative; for details, see [29].

Due to growing applications ofq-calculus, investigators are interested in studying prop-erties of functions usingq-operators instead of ordinary diﬀerential operators; for com-prehensive study, we refer to Kanas and Reducanu [15], Mahmood and Darus [19], and

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Mahmood and Sokol [20]. In this paper we deﬁne aq-analogue of a Salagean type opera-tor and study its eﬀect on multivalent functions in conic domains.

For any non-negative integern, theq-integer numberndenoted by [n]qis deﬁned by

[n]q= 1 –qn

1 –q , [0]q= 0.

For a non-negative integern, theq-number shift factorial is deﬁned as

[n]q! = [1]q[2]q[3]q...[n]q

[0]q! = 1

.

We note that whenq→1, [n]q! reduces to the classical deﬁnition of factorial. In general, [t]qis deﬁned as follows:

[t]q= 1 –qt

1 –q, [0]q= 0, q∈(0, 1).

Forf∈A, in [5], theq-derivative operator orq-diﬀerence operator is deﬁned as follows:

∂qf(z) =

f(qz) –f(z)

z(q– 1) .

It can easily be seen that

∂qzn= [n]qzn–1, ∂q _∞

n=1 anzn

=

∞

n=1

[n]qanzn–1.

Taking motivation from the above mentioned work, we deﬁne new convolution operators as follows.

Let

Φ(p,q,m,z) =zp+

∞

n=p+1

n+ (p– 1) m_qzn. (1.2)

Using the functionsΦ(p,q,m,z) and the definition ofq-derivative along with the idea of convolution, we now define the following differential operatorSm

q,pf(z) :Ap→Apfor mul-tivalent functions

Sm

q,pf(z) =Φ(p,q,m,z)∗f(z), m∈N∪ {0},

=zp+

∞

n=p+1

n+ (p– 1) m_qanzn,

=zp+

∞

n=p+1

ψnanzn, (1.3)

where

ψn=

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Forp= 1, the operatorSm

q,pf(z) reduces to the Salageanq-diﬀerential operator deﬁned by Govindaraj and Sivasubramanian [11], and forp= 1,q→1, the operatorSm

q,pf(z) reduces to the Salagean diﬀerential operator deﬁned by Salagean [26].

Taking motivation from [12] and using (1.3), we deﬁne a new classk-US(q,γ,m,p) of multivalent functions as follows.

Throughout paper we shall assumek≥0,m∈N∪ {0},q∈(0, 1),γ ∈C\{0}, andp∈N.

Deﬁnition 1.1 A functionf(z)∈Apis in the classk-US(q,γ,m,p) if it satisﬁes the con-dition

Re

1 +1

γ

1 [p]q

_z∂ qSmq,pf(z)

Sm q,pf(z)

– 1

>k1

γ

1 [p]q

Sm q,pf(z)

– 1, z∈E.

By taking speciﬁc values of parameters, we obtain many important subclasses studied by various authors in earlier papers. Here we enlist some of them.

(i) Forp= 1, the classk-US(q,γ,m,p)reduces to the classk-US(q,γ,m)studied by Saqib et al. [12].

(ii) Forp= 1,m= 0,k= 0, andγ ∈C\{0}, the classk-US(q,γ,m,p)reduces to the classS_q∗(γ)studied by Seoudy and Aouf [27].

(iii) Forp= 1,m= 0,k= 0, andγ =_1–1_α, with0≤α< 1, the classk-US(q,γ,m,p) reduces to the classS_q∗(α)studied by Agrawal and Sahoo [2].

(iv) Forp= 1,m= 0,q→1, andγ =_1–1_α, with0≤α< 1, the classk-US(q,γ,m,p) reduces to the classSD(k,α)studied by Shams et al. [28].

(v) Forp= 1,m= 0,q→1, andγ = 2

1–α, with0≤α< 1, the classk-US(q,γ,m,p) reduces to the classKD(k,α)studied by Owa et al. [24].

(vi) Forp= 1,k= 1,m= 0,q→1, andγ =_1–1_α, with0≤α< 1, the class

k-US(q,γ,m,p)reduces to the classS(α)studied by Ali et al. [3]. (vii) Forp= 1,k= 1,m= 0,q→1, andγ =_1–2_α, with0≤α< 1, the class

k-US(q,γ,m,p)reduces to the classK(α)studied by Ali et al. [3].

(viii) Forp= 1,m= 0,q→1, the classk-US(q,γ,m,p)reduces to the classK-ST introduced by Kanas and Wisniowska [17].

(ix) Forp= 1,k= 0,m= 0,q→1, andγ =_1–1_α, with0≤α< 1, the class

k-US(q,γ,m,p)reduces to the classS∗(α), a well-known class of starlike functions of orderα, respectively.

Geometric interpretation. A functionf(z)∈Apis in the classk-US(q,γ,m,p) if and only if _[_p1_]

q(

z∂qSmq,pf(z)

Smq,pf(z) ) takes all the values in the conic domainΩk,γ=hk,γ(E) such that

Ωk,γ =γ Ωk+ (1 –α),

where

Ωk=

u+iv:u>k(u– 1)2₊_v2_.

Sincehk,γ(z) is convex univalent, so the above deﬁnition can be written as

1 [p]q

Sm q,pf(z)

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where

hk,γ(z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1+z

1–z, fork= 0,

1 +2_πγ2(log 1+√z

1–√z)

2_, _for_k_{= 1,}

1 +_1–2γ_k2sinh2{(_π2arccosk)arctanh √

z}, for 0 <k< 1,

1 +_k2γ_–1sin(

π

2R(t)

u_√(z)

t

0 √_1–_x2√1_1–(_tx₎2dx) +

γ

1–k2, fork> 1.

(1.6)

The boundary∂Ωk,γof the above set becomes an imaginary axis whenk= 0, and a hyper-bola when 0 <k< 1. Fork= 1, the boundary∂Ωk,γ becomes a parabola and it is an ellipse whenk> 1 and in this case where

u(z) = z–

√ t

1 –√tz, z∈E,

andt∈(0, 1) is chosen such thatk=cosh(πK(t)/(4K(t))). HereK(t) is Legendre’s com-plete elliptic integral of the ﬁrst kind andK(t) =K(√1 –t2_{), and}_K₍_t_{) is the}

complemen-tary integral ofK(t) (for details, see [16,17,23]). Moreover,hk,γ(E) is convex univalent inE, see [16,17]. All of these curves have the vertex at the point k_k+₊₁γ.

2 A set of lemmas

Each of the following lemmas will be needed in our present investigation.

Lemma 2.1([25]) Let h(z) =_n∞₌₁hnzn≺F(z) = _∞

n=1dnznin E.If F(z)is convex univalent

in E,then

|hn| ≤ |d1|, n≥1.

Lemma 2.2([31]) Let k∈[0,∞)and let hk,γ be deﬁned(1.6).If

hk,γ(z) = 1 +Q1z+Q2z2+· · ·, (2.1)

Q1=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2γA2

1–k2, 0≤k< 1, 8γ

π2, k= 1,

π2γ

4(1+t)√tK2₍_t₎₍_k2_–1), k> 1,

(2.2)

Q2=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

A2+2

3 Q1, 0≤k< 1, 2

3Q1, k= 1, 4K2₍_t₎₍_t2₊₆_t_+1)–_π2

24K2₍_t₎₍₁₊_t₎√_t Q1, k> 1,

(2.3)

where A=2cos_π–1k,and t∈(0, 1)is chosen such that k=cosh(π_KK₍_t(₎t)),K(t)is Legendre’s com-plete elliptic integral of the ﬁrst kind.

Lemma 2.3([18]) Let h(z) = 1 +∞_n₌₁cnznbe analytic in E and satisfyRe{h(z)}> 0for z

in E.Then the following sharp estimate holds:

c2–μc21≤2max

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3 Main results

In this section, we will prove our main results.

Theorem 3.1 Let f(z)∈k–US(q,γ,m,p).Then

Sm_q_,_pf(z)≺zexp z

0

p{hk,γ(w(z))}– 1

ζ dξ, (3.1)

where w(z)is analytic in E with w(0) = 0and|w(z)|< 1.Moreover,for|z|=ρ,we have

exp 1

0

p{hk,γ(–ρ)}– 1

ρ dρ

≤S

m q,pf(z)

z

≤exp

1

0

p{hk,γ(ρ)}– 1

ρ dρ

, (3.2)

where hk,γ(z)is deﬁned by(1.6).

Proof Iff(z)∈k–US(q,γ,m,p), then using identity (1.5), we obtain

1

p

Sm q,pf(z)

≺hk,γ(z),

1

p

Sm q,pf(z)

=hk,γ

w(z), (3.3)

∂qSmq,pf(z)

Sm q,pf(z)

–1

z =

p{hk,γ(w(z))}– 1

z .

For some functionw(z) is analytic inEwithw(0) = 0 and|w(z)|< 1. Integrating (3.3) and after some simpliﬁcation, we have

Sm_q_,_pf(z)≺zexp z

0

p{hk,γ(w(z))}– 1

ζ dξ. (3.4)

This proves (3.1). Noting that the univalent functionhk,γ(z) maps the disk|z|<ρ(0 <ρ≤ 1) onto a region which is convex and symmetric with respect to the real axis, we see

hk,γ

–ρ|z|≤Rehk,γ(w(ρz)

≤hk,γ

ρ|z| (0 <ρ≤1,z∈E). (3.5)

Using (3.4) and (3.5) gives

1

0

p{hk,γ(–ρ|z|)}– 1

ρ dρ≤Re

1

0

p{hk,γ(w(ρ(z))}– 1

ρ dρ

≤

1

0

p{hk,γ(ρ|z|)}– 1

ρ dρ

forz∈E. Consequently, subordination (3.4) leads to

1

0

p{hk,γ(–ρ|z|)}– 1

ρ dρ≤log

Smq,pf(z)

z

≤

1

0

p{hk,γ(ρ|z|)}– 1

ρ dρ,

hk,γ(–ρ)≤hk,γ

–ρ|z|, hk,γ

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implies that

exp 1

0

p{hk,γ(–ρ)}– 1

ρ dρ≤

Smq,pf(z)

z

≤exp 1

0

p{hk,γ(ρ)}– 1

ρ dρ.

This completes the proof.

Whenp= 1, we have the following known result proved by Saqib et al. in [12].

Corollary 3.2 Let f(z)∈k–US(q,γ,m).Then

Sm_qf(z)≺zexp z

0

hk,γ(w(ξ)) – 1

ζ dξ,

where w(z)is analytic in E with w(0) = 0and|w(z)|< 1.Moreover,for|z|=ρ,we have

exp 1

0

hk,γ(–ρ) – 1

ρ dρ

≤S

m qf(z)

z

≤exp 1

0

hk,γ(ρ) – 1

ρ dρ

,

where hk,γ(z)is deﬁned by(1.6).

Theorem 3.3 If f(z)∈k–US(q,γ,m,p),then

|ap+1| ≤

δ {[p+ 1]q–p}ψp+1

(3.6)

and

|an+p–1| ≤

δ

{[n+p– 1]q–p}ψn+p–1

n–2

j=1

1 + δ

{[j+p]q–p}

for n= 3, 4, . . . , (3.7)

whereδ=p|Q1|with Q1given by(2.2).

Proof Let

1 [p]q

Sm q,pf(z)

=h(z),

z∂qSmq,pf(z) = [p]qSqm,pf(z)h(z),

(3.8)

whereh(z) is analytic inEandh(0) = 1. Leth(z) = 1 +_n∞₌₁cnznandSmq,pf(z) be given by (1.3). Then (3.8) becomes

zp+

∞

n=p+1

[n]qψnanzn=p _∞

n=0 cnzn

zp+

∞

n=p+1 ψnanzn

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Now comparing the coeﬃcients ofzn+p–1_{, we obtain}

[n+p– 1]qψn+p–1an+p–1=pψn+p–1an+p–1+p{c1ψn+p–2an+p–2+· · ·+cn–1},

[n+p– 1]q–p

ψn+p–1an+p–1=p{c1ψn+p–2an+p–2+· · ·+cn–1}.

Taking the absolute on both sides and then applying the coeﬃcient estimates|cn| ≤ |Q1|,

see in [23], we have

|an+p–1| ≤

p|Q1|

{[n+p– 1]q–p}ψn+p–1

1 +ψp+1|ap+1|+· · ·+ψn+p–2|an+p–2|

.

Let us takeδ=p|Q1|, then we have

|an+p–1| ≤

δ

{[n+p– 1]q–p}ψn+p–1

1 +ψp+1|ap+1|+· · ·+ψn+p–2|an+p–2|

. (3.9)

We apply mathematical induction on (3.9), so forn= 2 in (3.9), we have

|ap+1| ≤

δ {[p+ 1]q–p}ψp+1

, (3.10)

which shows that (3.7) holds forn= 2. Now consider the casen= 3 in (3.9), we have

|ap+2| ≤

δ {[p+ 2]q–p}ψp+2

1 +ψp+1|ap+1|

.

Using (3.10), we have

|ap+2| ≤

δ {[p+ 2]q–p}ψp+2

1 + δ

[p+ 1]q–p

,

which shows that (3.7) holds forn= 3. Let us assume that (3.7) is true forn≤t, that is,

|at+p–1| ≤

δ

{[t+p– 1]q–p}ψt+p–1

t–2

j=1

1 + δ [j+p]q–p

forn= 3, 4, . . . .

Consider

|at+p| ≤

δ {[t+p]q–p}ψt+p

1 +ψp+1|ap+1|+· · ·ψt+p–1|at+p–1|

≤ δ

{[t+p]q–p}ψt+p

1 +_[_p_+1]δ_q_–_p+_[_p_+2]δ_q_–_p(1 +_[_p_+1]δ_q_–_p) +· · ·

+ δ

{[t+p–1]q–p}

t–2

j=1(1 +[j+pδ]q–p)

= δ

{[t+p]q–p}ψt+p t–1

j=1

1 + δ [j+p]q–p

,

which proves the assertion of theoremn=t+ 1. Hence (3.7) holds for alln,n≥3.

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Corollary 3.4([12]) If f(z)∈k–US(q,γ,m),then

|a2| ≤ δ {[2]q– 1}[2]mq

and

|an| ≤

δ {[n]q– 1}[n]mq

n–2

j=1

1 + δ [j+ 1]q– 1

for n= 3, 4, . . . ,

whereδ=|Q1|with Q1given by(2.2).

Theorem 3.5 Let0≤k<∞be ﬁxed and let f(z)∈k–US(q,γ,m,p)with the form(1.1).

Then,for a complex numberμ,

_a_p₊₂_–_μa2

p+1≤

pQ1

2[2p+ 1]m

q{[p+ 2]q–p}

max1,|2v– 1| , (3.11)

where

v=1 2

1 –Q2

Q1

–Q1

4p {[p+ 1]q–p}

–μ4p[2p+ 1]

m

q{[p+ 2]q–p} ([2p]m

q)2{[p+ 1]q–p}

, (3.12)

andδ=p|Q1|,with Q1and Q2given by(2.2)and(2.3).

Proof Letf(z)∈k–US(q,γ,m,p), then there exists a Schwarz functionw(z), withw(0) = 0 and|w(z)|< 1, such that

1 [p]q

Sm q,pf(z)

≺hk,γ(z), z∈E,

1 [p]q

Sm q,pf(z)

=hk,γ

w(z).

(3.13)

Leth(z)∈Pbe a function deﬁned as

h(z) = 1 +w(z) 1 –w(z),

which gives

w(z) = c1 2z+

1 2

c2– c2₁

2

z2+· · ·

and

hk,γ

w(z)= 1 +Q1c1 2 z+

Q2c21

4 + 1 2

c2– c2

1

2

Q1

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Using (3.14) in (3.13) and along with (1.3), we obtain

ap+1=

pQ1c1

[2p]m

q{[p+ 1]q–p}

and

ap+2=

p

[2p+ 1]m

q{[p+ 2]q–p}

Q1c2

2 +

c2 1

4

Q2–Q1+

4pQ2 1 {[p+ 1]q–p}

.

Using any complex numberμand the above coeﬃcients, we have

ap+2–μa2p+1=

pQ1

2[2p+ 1]m

q{[p+ 2]q–p}

c2–vc21

. (3.15)

Using Lemma 2.3 on (3.15), we have

ap+2–μa2p+1≤

pQ1

2[2p+ 1]m

q{[p+ 2]q–p}

max1,|2v– 1| ,

where

v=1 2

1 –Q2

Q1

–Q1

4p {[p+ 1]q–p}

–μ4p[2p+ 1]

m

q{[p+ 2]q–p} ([2p]m

q)2{[p+ 1]q–p}

.

This is our required result (3.11).

Corollary 3.6([12]) Let0≤k<∞be ﬁxed and let f(z)∈k–US(q,γ,m)with the form

(1.1).Then,for a complex numberμ,

a3–μa22≤

Q1

2[3]m

q{[3]q– 1}

max1,|2v– 1| ,

where

v=1 2

1 –Q2

Q1

–Q1

4 [2]q– 1

–μ 4[3]

m

q{[3]q– 1} ([2]m

q)2{[2]q– 1}

,

and Q1and Q2are given by(2.2)and(2.3).

Theorem 3.7 If a function f(z)∈Aphas the form(1.1)and satisﬁes the condition

∞

n=p+1

[n]q–p(k+ 1) +p|γ|

|ψn||an| ≤ |γ||p|, (3.16)

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Proof Let

1_pz∂qSqm,pf(z)

Sm q,pf(z)

– 1=z∂qS m

q,pf(z) –pSmq,pf(z)

pSm q,pf(z)

= ∞

n=p+1ψn{[n]q–p}anzn

pzp₊_p∞

n=p+1ψnanzn

≤

∞

n=p+1|ψn{[n]q–p}||an|

|p|–∞_n₌_p₊₁p|ψn||an|

. (3.17)

From (3.16), it follows that

p–

∞

n=p+1

p|ψn||an|> 0.

To show thatf(z)∈k–US(q,γ,m,p), it is enough to prove that

_γk1_pz∂qSmq,pf(z)

Sm q,pf(z)

– 1–Re

1

γ

1

p

Sm q,pf(z)

– 1

≤1.

From (3.17), we have

_γk1_pz∂qSmq,pf(z)

Sm q,pf(z)

– 1–Re

1

γ

1

p

Sm q,pf(z)

– 1

≤ k

|γ|

1_pz∂qSmq,pf(z)

Sm q,pf(z)

– 1+ 1

|γ|

1_pz∂qSmq,pf(z)

Sm q,pf(z)

– 1

≤(k+ 1) |γ|

1_pz∂qSmq,pf(z)

Sm q,pf(z)

– 1

=(k+ 1)

|γ|

z∂qSmq,pf(z) –pSmq,pf(z)

pSm q,pf(z)

≤(k+ 1) |γ|

∞

n=p+1|ψn{[n]q–p}||an|

|p|–∞_n₌_p₊₁p|ψn||an|

≤1.

Whenp= 1, we have the following known result proved by Hussain et al. in [12].

Corollary 3.8([12]) If a function f(z)∈Ahas the form(1.1)and satisﬁes the condition

∞

n=2

[n]q– 1(k+ 1) +|γ|

[n]m_q|an| ≤ |γ|,

then f(z)∈k–US(q,γ,m).

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Corollary 3.9 A function f ∈A of the form(1.1)is in the classSD(k,α)if it satisﬁes the condition

∞

n=2

n(k+ 1) – (k+α)|an| ≤1 –α,

where0≤α< 1and k≥0.

Whenq→1,p= 1,m= 0,γ = 1 –α, with 0≤α< 1 andk= 0, we have the following known result proved by Silverman in [30].

Corollary 3.10 A function f ∈A of the form(1.1)is in the classSD(α)if it satisﬁes the condition

∞

n=2

{n–α}|an| ≤1 –α.

Theorem 3.11 Let f(z)∈k–US(q,γ,m,p).Then f(E)contains an open disk of radius

r= {[p+ 1]q–p}ψp+1 (p+ 1){[p+ 1]q–p}ψp+1+δ

,

whereδ=p|Q1|with Q1given by(2.2).

Proof Letw0= 0 be a complex number such thatf(z)=w0forz∈E. Then

f1(z) =

w0f(z) w0–f(z)

=z+

ap+1+

1

w0

zp+1+· · ·.

Sincef1(z) is univalent, so

ap+1+

1

w0

≤p+ 1.

Now, by using (3.6), we have

_w1₀≤(p+ 1){[p+ 1]q–p}ψp+1+δ {[p+ 1]q–p}ψp+1

.

Hence we have

|w0| ≥ {

[p+ 1]q–p}ψp+1

(p+ 1){[p+ 1]q–p}ψp+1+δ

.

Corollary 3.12([12]) Let f(z)∈k–US(q,γ,m).Then f(E)contains an open disk of radius

r= {[2]q– 1}[2] m q 2[2]m

q{[2]q– 1}+Q1

,

(12)

Funding

The work here is supported by GUP-2017-064 and UKM Grant Nos. DPP-2015-FST.

Competing interests

The authors declare that there are no competing interests.

Authors’ contributions

SH came with the main thoughts and helped to draft the manuscript, SK and MAZ proved the main theorems, MD revised the paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan.}2_{Department of}

Mathematics, Riphah International University, Islamabad, Pakistan. 3_{Faculty of Science and Technology, School of}

Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi, Malaysia.

Publisher’s Note

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Received: 17 July 2018 Accepted: 23 October 2018

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