The convergence of \((p,q)\) Bernstein operators for the Cauchy kernel with a pole via divided difference

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R E S E A R C H

Open Access

The convergence of

(p,

q)

-Bernstein

operators for the Cauchy kernel with a pole

via divided difference

Faisal Khan

1

_{, Mohd Saif}

2

_{, Aiman Mukheimer}

3

_{and M. Mursaleen}

1*

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Aligarh Muslim University, Aligarh, India

Full list of author information is available at the end of the article

Abstract

In this paper, some qualitative approximation results for the (p,q)-Bernstein operators Bn

p,q(f;x) are obtained for the Cauchy kernelx–1αwith a pole

α

∈[0, 1] forq>p> 1. The main focus lies in the study of behavior of operatorsBn

p,q(f;x) for the function fm(x) =_x_–_pm1_q–m,x=pmq–mandfm(pmq–m) =a,a∈Rand the extra parameterp provides ﬂexibility for the approximation.

MSC: 41A10; 41A25; 41A36

Keywords: (p,q)-integer; (p,q)-Bernstein operator; Convergence; Approximation of unbounded function; Cauchy kernel

1 Introduction and preliminaries

The uniform convergence of a sequence of operators to a continuous function was in-troduced by Bohman [9] and Korovkin [16]. Throughq-calculus various modiﬁcations of Bernstein operators [7] have been studied so far [10,18,31]. The (p,q)-integers are the generalization of theq-integers, which has an important role in the representation theory of quantum calculus in the physics literature. Recently, the approximation by the (p,q )-analog of a positive linear operator has become an active area of research. For the theory and numerical implementations of the (p,q)-analog of Bernstein operators introduced by Mursaleen et al. [22] and other (p,q)-analogs, the reader may refer to [1–5,11–15,19–21] and [32]. For most recent work on the (p,q)-approximation we refer to [8,24,26].

The (p,q)-integer, (p,q)-binomial expansion and the (p,q)-binomial coeﬃcients are de-ﬁned by

[m]p,q:=

pm_–_qm

p–q , m= 0, 1, 2, . . . ,p>q≥1,

(a+b)m_p,q:= (a+b)(pa+qb)p2a+q2b· · ·pm–1a+qm–1b

=

k

r=0

p(m–r)(m2–r–1)_q r(r–1)

2

m r

p,q

ar,

m r

p,q

:= [m]p,q! [r]p,q![m–r]p,q!

.

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It can easily be veriﬁed by induction that

(1 +a)(p+qa)p2+q2a· · ·pn–1+qn–1a=

k

r=0

p(m–r)(2m–r–1)_qr(r2–1)

m r

p,q

ar.

The (p,q)-analog of Euler’s identity is deﬁned by

m–1

s=0

ps–qsa:=

m

k=0

p(m–k)(2m–k–1)_qk(k2–1)

m k

p,q

ak.

Letf : [0, 1]−→Randq>p> 1. The (p,q)-Bernstein operators [22] off is deﬁned as

Bn_p,q(f;x) :=

n

k=0

f

[k]p,q

pk–n_[_n_] p,q

pn,k(p,q;x), n∈N, (1.1)

where the polynomialpn,k(p,q;x) is given by

pn,k(p,q;x) =

1

pn(n2–1)

n k

p,q

pk(k2–1)_xk

n–k–1

s=0

ps–qsx, x∈[0, 1], 0 <q<p< 1. (1.2)

Forp= 1,Bn

p,q(f;x) turns into theq-Bernstein operator. We have

Bn_p,q(f; 0) =f(0), B_p,qn (f; 1) =f(1), n∈N. (1.3)

The following (p,q)-diﬀerence form of Bernstein operators [25] is given by

Bn_p,q(f;x) :=

n

r=0

λn_p,qf

0,p

n–1_[1] p,q

[n]p,q

, . . . ,p

n–r_[_r_] p,q

[n]p,q

xr, (1.4)

wheref[x0,x1, . . . ,xn] indicates thenth order divided diﬀerence off with pairwise distinct

node, that is,

f[x0] =f(x0), f[x0,x1] =

f(x1) –f(x0)

x1–x0

,

f[x0,x1, . . . ,xn] =

f[x1, . . . ,xn] –f[x0, . . . ,xn–1]

[xn–x0]

andλn_p,qis given by

λn_p,q=

n r

p,q

[r]p,q!

[n]r p,q

p(n–r)(n2–r–1)_q r(r–1)

2

=

1 –p

n–1_[1] p,q

[n]p,q

1 –p

n–2_[2] p,q

[n]p,q

· · ·

1 –p

n–r+1_[_r_{– 1]} p,q

[n]p,q

, (1.5)

andλ0_p,q=λ1_p,q= 1, 0≤λn_p,q≤1,r= 0, 1, . . . ,n.

In this paper, some qualitative approximation results for the (p,q)-Bernstein operators

Bn

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The main focus lies in the study of behavior of operatorsBn

p,q(f;x) for the functionfm(x) = 1

x–pm_q–m,x=pmq–mandfm(pmq–m) =a,a∈Rand the extra parameterpprovides ﬂexibility

for the approximation.

The time scaleJp,qforq>p> 1 is denoted and deﬁned as

Jp,q={0} ∪

pkq–k∞_k=0. (1.6)

Here, we consider the (p,q)-Bernstein operators with the Cauchy kernel _x–1_α,α∈[0, 1]. The previously obtained results [27–30] lead to the following conclusions.

• Ifα= 0, that is,f(x) = 1_x,x= 0andf(0) =a, then, forq≥2,

lim

n→∞B

n p,q(f;x) =

⎧ ⎨ ⎩

f(x), x∈Jp,q,

∞, x∈/Jp,q.

(1.7)

• Ifα∈Jp,q\[0, 1]that isf(x) = _(x–1α) ifx=αandf(α) =a, then

lim

n→∞B n

p,q(f;x) =f(x), x∈Jp,q.

Furthermore, asn→ ∞,Bn

p,q(f;x)→f(x) uniformly on any compact subset of (–α,α)

andBn_p,q(f;x)→ ∞for|x|>α,x∈/Jp,q. Therefore, it is left to examine the caseα∈Jp,q\{0}

which is exactly the subject of the present paper. Let the functionfm:R→Rbe deﬁned

by

fm(x) = ⎧ ⎨ ⎩

1

(x–pm_q–m₎j, x∈R\ {pmq–m},

a, x=pm_q–m_, m∈N0,a∈R. (1.8)

2 Some auxiliary results

In this section, we prove some important lemmas.

Lemma 2.1 For the function fmdeﬁned by(1.8),we have

(a) form∈N,

lim

n→∞B n p,q

fm;pjq–j

=fm

pjq–j, j∈N0\ {m,m+ 1}.

Besides,

lim

n→∞B n p,q

fm;pmq–m

= –∞, and

lim

n→∞B n p,q

fm;p–(m+1)q–(m+1)

=fm

pm+1q–(m+1)–p

–m_qm_[_m_{+ 1]} p,q

p–1₍_q_{– 1)[}_m_] p,q

.

(b) Form= 0,

lim

n→∞B

n p,q

f0;pjq–j

=f0

pjq–j, j∈N0

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This describes the behavior ofBn

p,q(fm;·) on the time scaleJp,q.

Proof (a) From (1.2), we can easily see thatpn,n–k(p,q;pjq–j) = 0 fork>j, whence

Bn_p,qf;p–jqj=

min{k,j} k=0

f

[n–k]p,q

[n]p,q

pn,n–k

p,q;pjq–j. (2.1)

Besides

lim

n→∞pn,n–k

p,q;pjq–j=δk,j and lim n→∞

[n–k]p,q

[n]p,q

=pkq–k. (2.2)

Thus,lim_n_→∞fm( [n–k]p,q

[n]p,q )pn,n–k(p,q;p

j_q–j_{) =}_f

m(pkq–k)δj,kfor allk=m.

Now by easy calculation, we have

lim

n→∞f

[n–k]p,q

[n]p,q

pn,n–k

p,q;pjq–j=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

–∞, j=m,

–p_p––1m_(q–1)[m]qm[m+1]_pp_,,_qq, j=m+ 1,

0, ≥m+ 2,

and combining with (2.1) and (2.2) yields the result.

(b) It can be obtained easily from (1.3) and (2.2) asf0is continuous at all pointspjq–j,

j∈N.

The next lemma is related to the coeﬃcient ofBn p,q(f0;·).

Lemma 2.2 Let fmbe a function as in(1.8).If Bnp,q(fm;x) = n

k=0C p,q k,nxkand

[k]p,q

[n]p,q =p

m_q–m

for k= 0, 1, 2, . . . ,n,then

C_k,np,q= – λ

p,q

k,np–m(k+1)qm(k+1)

(1 –pn–m–1qm[1]p,q

[n]p,q )(1 –

pn–m–2_qm_[2] p,q

[n]p,q )· · ·(1 –

pn–m–k_qm_[k] p,q

[n]p,q )

, (2.3)

whereλp,q_k,nare given by(1.5).

Proof Considerfm(z) =_z–pm1_q–m, which is analytic function inC\ {pmq–m}. It is well known that [17] thekth order divided diﬀerence off can be expressed as

f[x0,x1, . . . ,xk] =

1 2πi

L

f(ζ)dζ

(ζ–x0)(ζ–x1)· · ·(ζ–xk)

,

whereLis contour encirclingx0, . . . ,xkandf is assumed to be analytic on and withinL.

Hence, when the nodes 0,[1]p,q

[n]p,q,

[2]p,q

[n]p,q, . . . ,

[k]p,q

[n]p,q are insideLand the poleα=p

m_q–m_is

out-side, one has

f

0,p

n–1_[1] p,q

[n]p,q

, . . . ,p

n–r_[_r_] p,q

[n]p,q

= 1

2πi

L

fm(ζ)dζ

ζ(ζ–pn–1_[n][1]_p_,_qp,q)· · ·(ζ–pn_[n]–r[r]_p_,_qp,q)

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By the residue theorem

f

0,p

n–1_[1] p,q

[n]p,q

, . . . ,p

n–r_[_r_] p,q

[n]p,q

=

k

j=0

Res

z=pn–j[j]p,q [n]p,q

fm(z) k

j=0(z–pn–j [j]p,q

[n]p,q)

= –Resz=pm_q–m fm(z)

k

j=0(z–pn–j [j]p,q

[n]p,q)

= – p

–m(k+1)_qm(k+1) k

j=1(1 –pn–m–j [j]p,q

[n]p,q) .

Sincefm(z) =fm(x) forz=x∈[0, 1], the statement follows from the divided diﬀerence

rep-resentation (1.4).

Now, we ﬁnd the asymptotic estimates for the coeﬃcientC_k,np,qin the next lemma.

Lemma 2.3 We have

lim

n→∞ n–j

k=1

1 –pn–m–jqm[k]p,q

[n]p,q

=

p2j–m qj–m;

p q _∞ for j>m,q>p> 1.

Proof It is clear that

log

n–j

k=1

1 –pn–m–jqm[k]p,q

[n]p,q

=

n–1

k=j

log

1 –pn–m–jqm[n–k]p,q

[n]p,q

=

∞

k=j

ap,q_k,n,

where

ap,q_k,n=

⎧ ⎨ ⎩

log(1 –pn–m–jqm[n–k]_[n] p,q

p,q ), k<n,

0, k≥n.

Since

ap,q_k,n≤log

[n]p,q

≤ q

q–p

pn–m–k_qm_[_n_–_k_] p,q

[n]p,q ≤

q q–pp

n–k–m_qm_,

which gives∞_k=j|ap,q_k,n|<∞, and by the Lebesgue dominated convergence theorem, we have

lim

n→∞

n–1

k=j

log

[n]p,q

=

∞

k=j

lim

n→∞log

[n]p,q

=

∞

k=j

lim

n→∞

1 –pn–k–mq

m

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as a result

lim

n→∞log n–j

k=1

[n]p,q

=log ∞

k=j

1 –pn–k–mq

m

pk ,

which completes the proof.

The following lemma gives an upper bound forn–m– 1.

Lemma 2.4 If m∈N,k= 0, 1, 2, . . . ,n–m– 1,then

Cp,q_k,n≤Cm,p,qp–mnqmn,

whereCin RHS is a positive constant,whose value need not to be addressed.

Proof Forn>m+ 1 and from (2.3), we have

Cp,q_k,n≤ p

–m(k+1)_qm(k+1)

(1 –pn–m–1qm[1]p,q

[n]p,q )(1 –

pn–m–2_qm_[2] p,q

[n]p,q )· · ·(1 –

[n]p,q )

≤ p–m(n–m)qm(n–m) (1 –p2(n–_qmn–1–1)qm)(1 –

p2(n–m–2)_qm_[2] p,q

[n]p,q )· · ·(1 –

[n]p,q ) ,

Cp,q_k,n≤ p

–mn_qmn

p2m_q2m₍p q;

p q)∞

.

Further, we discuss the nature ofCn–m+1,n, . . . ,Cn,nas follows.

Lemma 2.5 For m∈N,q>p> 1,

Cp,q_n–m,n∼Cp,q,mp–(m+1)nq(m+1)n, n→ ∞.

Proof Using (2.3), we obtain the following

_Cp,q

n–m=λp,qn–m,n

p–m(n–m+1)_qm(n–m+1)

(1 –pn–m_[n]–1q_pm_,_q[1]p,q)(1 –pn–m–2_[n]q_pm_,_q[2]p,q)· · ·(1 –pn–m–(n–_[n]m)_pqm_,_q[n–m]p,q) .

From Lemma2.3, we have

Cp,q_n–m∼

(p_qmm+1+1;p_q)∞qm(n–m+1)p–m(n–m+1)pm(pn–qn) (p_q;p_q)∞pn(pm–qm)

∼ (

pm+1

qm+1;

p q)∞q

mn_p–mn_pm₍_pn_–_qn₎

(p_q;p_q)∞qm(m–1)pm(m–1)(pm–qm),

C_n–m,np,q =Cp,q_m p–n(m+1)qn(m+1).

(2.5)

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Lemma 2.6 For j= 1, 2, . . . ,m,we have

lim

n→∞

C_n–m+j,np,q Cp,qn–m,n

= (–1)j

m j

p,q

p(n–j)(n2–j–1)_q j(j–1)

2 _.

Proof Using (2.3) and (1.5), we get

C_n–m+j,np,q =C_n–m,np,q

(1 –pm[n–m]_[n] p,q

p,q )· · ·(1 –p

m–j+1 [n–m+j–1]p,q

[n]p,q )

(1 –p–1_qm[n–m+1]

[n]p,q )· · ·(1 –p

–j_qm[n–m+j]p,q

[n]p,q ) ,

lim

n→∞

C_n–m+j,np,q Cp,qn–m,n

=

(1 –p_qmm)· · ·(1 –p m–j+1

qm–j+1)

(1 –q_p)· · ·(1 –q_pjj) ,

lim

n→∞

C_n–m+j,np,q

Cp,qn–m,n

= (–1)j

m j

p,q

p(n–j)(n2–j–1)_q j(j–1)

2 _.

Corollary 2.7 The following estimate holds:

Cp,q_k,n≤Cp,q,mp–(m+1)nq(m+1)n, k= 0, 1, 2, . . . ,n, (2.6)

andCp,q,mis independent of both k and n.

Corollary 2.8 We have the following:

lim

n→∞

Cn–m,n+· · ·+Cn–m+j,nxj+· · ·+Cn,nxn

Cn–m,n

= (x;p,q)m. (2.7)

Proof The statement follows from Rothe’s identity [6],

(x;p,q)m= m

j=0

(–1)j

m j

p,q

p(n–j)(n2–j–1)_q j(j–1)

2 _.

3 Main results

• First we consider the case when poleα∈Jp,q\ {0, 1}.

Now, we obtain the results that concern with the uniform approximation offm(x),m∈N

by its (p,q)-Bernstein operators. It may be noted that, while the case whenα∈[0, 1]\ Jp,qcan easily be examined by using the result and method of [27], the conditionα∈Jp,q

requires a diﬀerent approach.

Theorem 3.1 If m∈N,then Bn

p,q(fm;x)→fm(x)as n→ ∞uniformly on any compact

sub-set of(–p(m+1)_q(m+1)_,_p(m+1)_q(m+1)_).

Proof We consider the complex (p,q)-Bernstein operators given by

Bn_p,q(f;x) =

n

k=0

f

[k]p,q

pk–n_[_n_] p,q

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and the functionfm(z) =_(z–pm1_q–m₎,z∈C. Letnbe large enough to satisfy the condition

[k]p,q

[n]p,q =p

m_q–m_{. Then}

Bn_p,q(fm;z) = n

k=0

Cp,q_k,nzk,

whereC_k,np,qis given by (2.3). Letρ∈(0,p(m+1)_q–(m+1)_{). Therefore for}_|_z_{| ≤}_ρ_{the following}

estimate is valid by Corollary2.7:

Bn_p,q(fm;z)≤ n

k=0

C_k,np,qρk≤Cp,q,m n

k=0

p–(m+1)q(m+1)ρk≤Cp,q,m

1

(1 –p–(m+1)_q(m+1)_ρ₎.

Hence it follows that the operators{Bn

p,q(fm,z)} are uniformly bounded in the disk{z:

|z| ≤ρ}and convergent on the sequence {pj_q–j_}∞

j=m+2 having an accumulation point at

0 to the function fm(z) analytic in this disc. Using Vitali’s convergence theorem, we

have Bn_p,q(fm;z)→fm(z) (n→ ∞) uniformly on any compact set in{z:|z| ≤ρ} asρ∈

(0,p(m+1)_q–(m+1)_{) was arbitrary. This completes the proof.}

Next we demonstrate that, outside of the interval, operators diverge everywhere except a ﬁnite number of points.

Theorem 3.2 If m∈N,thenlimn→∞Bnp,q(fm;x) =∞for|x|>p(m+1)q–(m+1),x=p(m+1)×

q–(m+1)_,_x₌_p(m–1)_q–(m–1)_,_x₌_p(m–2)_q–(m–2)_{, . . . , 1.}

Proof For exceptional pointsp(m–1)_q–(m–1)_,_p(m–2)_q–(m–2)_{, . . . , 1, the situation has been}

ana-lyzed in Lemma2.1(a). We takexsatisfying|x|>p(m–1)_q–(m–1)_{diﬀerent from these values.}

Letn>mbe suﬃciently large such that (2.3) holds. By Lemma2.4, we obtain

∞

k=0

Cp,q_k,nxk

≤Cm,p,q

n–m–1

k=0

p–mkqmkxk=Cm,p,q

(p–m_qm_x₎n–m_{– 1}

p–m_qm_x_{– 1}

=op–(m+1)q(m+1)xn, n→ ∞.

Hence

Bn_p,q(fm;x) = n

k=n–m

C_k,np,qxk+op–(m+1)q(m+1)xn

=Cp,q

n–mxn–mgn(x) +o

p–(m+1)_q(m+1)_xn_, _n_{→ ∞}_.

By Lemma2.5,|Cn–mp,q | ∼Cp,q,m(x)(p–(m+1)q(m+1)x)nasn→ ∞whenever|x|>p(m+1)q–(m+1),

sincelimn→∞gn(x) = (x;p,q)m= 0, whenx∈ {/ p(m+1)q–(m+1), . . . , 1}.

Lemma 3.1 Let f0be given by putting m= 0in(1.8).If Bnp,q(f0;x) = n

k=0C p,q k,nxkthen

C_k,np,q= –1 (1 –pn–k[k]p,q

[n]p,q)

, k= 0, 1, 2, . . . ,n– 1, Cp,q_n,n=a+

n–1

k=0

1

(1 –pn–k[k]p,q

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Proof Fork= 0, 1, . . . ,n– 1, on a speciﬁc choice of the contourL, such that the nodes 0,[1]_[n]p,q

p,q, . . . ,

[k]p,q

[n]p,q are insideLwhile the poleα= 1 is outside, formula (2.4) implies

C_k,np,q= –λ

p,q k,n k

j=0(1 –pn–j [j]p,q

[n] )

= –1

(1 –pn–k[k]p,q

[n]p,q) ,

since by (1.3),Bn

p,q(f0; 1) =f0(1) =aand the statement is proved.

Corollary 3.2 For k= 0, 1, 2, . . . ,n– 1with q>p> 1we have the following result:

Cp,q_k,n≤ q q–p.

• Now, we consider the case when poleα= 1.

Here the point of singularityx= 1 is one of the nodes[k]_[n]p_p,_,q_q. Consider the functionf0

f0(x) = ⎧ ⎨ ⎩ 1

x–1, x∈R\ {1},

a, x= 1. (3.2)

Theorem 3.3 If f0is given by(3.2),then the following holds:

(1) For allx∈(–1, 1],

lim

n→∞B n

p,q(f0;x) =f0(x)

and the convergence is uniform on any compact subset of(–1, 1). (2) For allx∈R\(–1, 1],

lim

n→∞B n

p,q(f0;x) =∞.

Proof (1) SinceBn

p,q(f0; 1) =f0(1), we need to prove only the uniform convergence of the

compact subset of (–1, 1). For anyρ∈(0, 1) and|z| ≤ρ. From Corollary3.2, we have

n–1

k=0

Cp,q_k,nzk

≤

Cp,q

1 –ρ.

Apart from that,

Cp,q_n,nzk≤ |a|+

n–1

k=0

1

1 –pn–k[k]p,q

[n]p,q

≤ |a|+nCp,q_k,n≤ |a|+ nq

q–p,

whence

Cp,q_n,nzn≤ |a|ρn+ρn nq

q–p≤Cp,q,ρ.

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(2) Given thatxsatisﬁes|x|> 1, by Able’s inequality, we have

n–1

k=0

Cp,q_k,nxk

≤|

x|n_{– 1}

|x|– 1C

p,q 0,n+ 2C

p,q n–1,n≤

|x|n

|x|– 1

1 + 2p

p–q =Cp,q,x|x|

n_.

Meanwhile,

Cp,q_n,nxn≥

_n–1

k=0

1

1 –pn–k[k]p,q

[n]p,q

|x|n–|a| · |x|n≥n–|a||x|n.

Thus,|Bn

p,q(f0;x)| ≥n|x|n– (Cp,q,x+|a|)|x|n→ ∞asn→ ∞.

Atx= –1, we have

Bn_p,q(f0; –1) = n–1

k=0

C_k,np,q(–1)k+

a+

n–1

k=0

1

1 –pn–k[k]p,q

[n]p,q

(–1)n,

and again applying Able’s inequality,

n–1

k=0

Cp,q_k,n,(–1)k

≤ |C0,n|+ 2|Cn–1,n| ≤1 +

2p p–q.

On the other hand

a+

n–1

k=0

1

1 –pn–k[k]p,q

[n]p,q

(–1)n

≥n–|a|,

which implies that

Bn_p,q(f0; –1)≥n–|a|–

1 + 2p

p–q → ∞, n→ ∞.

Remark For justiﬁcation of the statement that the extra parameterpprovides ﬂexibility for approximation, one can see Remark 1 of [23].

Moreover, since for q > p = 1 we recapture the q-Bernstein operators studied in [30], it is clear that the interval of uniform convergence forBn_p,q in Theorem3.1, i.e. (–pm+1qm+1,pm+1qm+1), is larger than the interval of uniform convergence (–qm+1,qm+1), obtained by Theorem 2.1 in [30].

Funding

The third author would like to thank “Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM)” group number RG-DES-2017-01-17.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors contributed equally and signiﬁcantly in writing this paper. All of them read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Aligarh Muslim University, Aligarh, India.}2_{Department of Applied Mathematics, Aligarh}

Muslim University, Aligarh, India.3_{Department of Mathematics and General Sciences, Prince Sultan University, Riyadh,}

(11)

Publisher’s Note

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Received: 4 February 2019 Accepted: 3 May 2019

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