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Evaluation of Norm of (p, q)-Bernstein Operators

  • Nabiullah Khan , Mohd Saif and Talha Usman EMAIL logo
Published/Copyright: March 31, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 2

Abstract

In this paper, we aim to study about the estimation of norm of (p, q)-Bernstein operators Bp,qn in C[0,1] for the case q > p > 1 by applying (p, q)-calculus and divided difference analogue of (p, q)-Bernstein operators. Some basic theorem and related results are also discussed in this paper. Here, the extra parameter p shows more flexibility by choosing the value of p.

2020 Mathematics Subject Classification: Primary 41A10; 41A25; 41A36
Keywords: (p, q)-integers; (p, q)-Bernstein operators; convergence

1. Introduction and definition

In recent years, the application of quantum calculus and post quantum calculus plays an important role in the area of approximation theory number theory theoretical physics and many other branches of Mathematics and Physics. G. M. Phillips [28] initiated the convergence of Bernstein polynomial based on q-calculus. Many results and properties were obtained by S. Ostrovska and some researchers (see, for more details, [24–27]). The Bernstein operators, q-Bernstein operators [5,30] and (p, q)-Bernstein operators [3,6,12], their Kantorovich form [4], bivariate form [1,18] and others were introduced, and their approximation properties made them an area of intensive research. For in-depth knowledge and details see [2]–[15], [17,31]. Initially, by using the concept of (p, q)-calculus Mursaleen et al. (see [19]–[23]) first applied the concept of (p, q)-calculus in approximation theory and introduced the (p, q)-analogue of Bernstein operators. For the present work, we recall the following definition and notations:

The q-integers and (p, q)-integers are defined, respectively, as:

[n]q:=1−qn1−qand[n]p,q:=pn−qnp−q,n∈N∪{0}.

The (p, q)-binomial expansion is given as:

(x+y)p,qn:=∏ν=0n−1(pνx−qνy)and(p,q;x):=∏ν=0n−1(pν−qνx).

It can be seen by induction that

(1+x)(p+qx)(p2+q2x)⋯(pn−1+qn−1x)=∑ν=0np(n−ν)(n−ν−1)2qν(ν−1)2nνp,qxν,

wherein (p, q)-binomial coefficients are defined by

nνp,q:=[n]p,q![ν]p,q![n−ν]p,q!.

The Euler’s identity based on (p, q)-analogue is given by

∏ν=0n−1(pν−qνx):=∑ν=0np(n−ν)(n−ν−1)2qν(ν−1)2nνp,qxν.

The (p, q)-analogue of Bernstein operator of a bounded function g introduced by Mursaleen et al. (see [19]) is given as follows:

Bp,qn(g;x):=∑ν=0ngpn−ν[k]p,q[n]p,qQn,ν(p,q;x),n∈N,

where polynomial 𝒬n,ν(p, q; x) is given as:

1.1 Qn,ν(p,q;x)=1pn(n−1)2nνp,qpν(ν−1)2xν∏ν=0n−ν−1(pν−qνx),x∈[0,1].

For p = 1, Bp,qn(g;x) reduces into q-Bernstein operators which are studied by several authors (see [9,15,16,29]). Here we mark out end-point interpolation properties of the (p, q)-Bernstein operator Bp,qn(g,x), which are valid for the case q ≠ 1:

Bp,qn(g;0)=g(0),Bp,kqn(g;1)=g(1).

2. Statement of main results

In this section we mainly talk about some auxiliary results and statements of our main theorem, which will be provided in the next section. We start with a simple statement for the case q > p > 1, which gives the better approximation of Bernstein operators Bp,qn and it is useful to understand the asymptomatic behavior as n → ∞. The following result involves the function (1z;pq)∞ analytic in ℂ \ {0} and has an essential singularity at 0.

Theorem 2.1

  1. The following results whose convergence is uniform on any compact set M ⊂ ℂ \ {0} for n ∈ ℕ, z ≠ 0, q > p > 1,

    limn→∞pn(n−1)2Qn,k(p,q;z)(−1)nqn(n−1)2zn=(−1)k(1z;pq)∞qk(k−1)2(pq;pq)k.

  2. The following results whose convergence is uniform on any compact set M ⊂ ℂ \ {0}

    limn→∞pnkQn,n−k(p,q;z)qnkzn=(−1)k(1z;pq)kqk(k+1)2(pq;pq)k.

The next corollary shows that outside the set Tp,q={0}∪{pkq−k}k=0∞, the polynomial 𝒬n,k(p, q; x) given in (1.1) tends to infinity.

Corollary 2.1.1

The undermentioned result holds true:

limx→∞Q(p;q;z){0ifz∈ Tp,q\{0}0ifz=0,k≠0,1ifz=0,k=0,∞otherwise.

Theorem 2.1 suggests that it is interesting to investigate the behavior of polynomial (1.1) in Banach space of analytic function

2.1 nνp,q:=[n]p,q![k]p,q![n−k]p,q!=qn(n+1)2pq,pqnqk(k+1)2q(n−k)(n−k−1)2pq,pqkpq,pqn−k∼qk(n−k)pk(n−k).

Theorem 2.2

Let gn,k(p,q;x)=xk∏j=0n−k−1(pj−qjx). Then

‖gn,n(p, q; x)‖=‖gn,n(p, q; x)‖0=1

and

∥gn,k(p,q;x)∥=∥gn,k(p,q;x)∥0=q(n−k)(n−k−1)2n.p(n−k)(n−k−1)2,k=0,1,…,n−1.

Corollary 2.2.1

For q > p > 1 and k = 0,1, 2 ... n – 1, the following asymptotic estimate holds true:

∥Qn,k(p,q;x)∥=∥Qn,k(p,q;x)∥0=q(n−k)(n−k−1)2n.p(n−k)(n−k−1)2,k=0,1,…,n−1.

Corollary 2.2.2

The following estimate holds:

∥Bp,qn∥=q(n−1)n2n.p(n−1)n2,n→∞.

Theorem 2.3

The following asymptotic estimate holds:

‖Bp,qn‖∼(p2q2;p2q2)∞qn(n−1)2n⋅e pn(n−1)2,n→∞.

3. Main results

In this section, we prove some results related to approximation of analytic function of (p, q)-Bernstein operators and we handle estimation of norm of the operator 𝒬n,k(p, q; z).

Proof of Theorem 2.1.

  1. For z ≠ 0, we have

    Qn,k(p,q;z)=nνp,qzn(−1)n−kq(n−k)(n−k−1)2pz;pzn−k,

    which by using (2.1) becomes

    3.1 Qn,k(z)(−1)nqn(n−1)2zn=(−1k)pq;pqnpq;pqn−kpq;pqkpq;pqn−kqk(k−1)2.

    As we know that

    3.2 limn→∞pq;pqn=limn→∞pq;pqn−k=pq;pq∞,

    while pq;pqn−k→pq;pq∞ as n → ∞ converges uniformly on any compact set M ⊂ ℂ \ {0}.

    The statement follows from (3.1).

  2. As we know that

    nn−kp,q=nkp,q,

    we can write

    Qn,n−k(p,q;z)=nkp,qzn(−1)kqk(k−1)2pq;pqk.

    Again, by using (2.1), we have

    Qn,n−k(z)qnkzn=(−1)kpq;pqkpq;pqnpq;pqkpq;pqn−kqk(k+1)2.

    By using (3.2), we get the desired result. □

The following lemma tells us about the norm estimation of hk(x).

Lemma 3.1

Let hk(x) = xk(1 – x)(qx – p), x ∈ [0,1], k ∈ ℕ. Then we have the following result:

∥hk∥∼∥hk∥0∼1k.

Proof. It can be easily seen that

3.3 max{xk(1−x)}=kk+11−kk+1=1k(k+1)k+1∼1k.

Therefore,

∥hk∥≤∥hk∥0≤1k.

Suppose that k≥2pq−p, then k(k+1)∈pq;1 and consequently we have

∥hk∥≥∥hk∥0≥hkkk+1=1k(k+1)k+1k(q−p)−1(k+1).p≥1k.

Hence, we get

‖hk‖∼‖hk‖0∼1k.

Proof of Theorem 2.2. Since gn,n = xn, the equality

∥gn,n(p,q;x)∥=∥gn,n(p,q;x)∥0=1

is obvious. The asymptotic relations for k = n – 1 and k = n – 2,

∥gn,n−1(p,q;x)∥∼1nand∥gn,n−2(p,q;x)∥∼1n,

are easily derivable from (3.3) and Lemma 3.1. Also, for points k = 0, 1, ... , n–3, we discuss norm of gn,k(p, q; x) as:

3.4 ∥gn,k(p,q;x)∥=max{∥gn,k∥C[0,p(n−k−1)q(n−k−1)],∥gn,k(q,q;x)∥s,s=0,1,2,…,n−k−2}.

If x∈[0,p(n−k−1)q(n−k−1)] ,then |gn,k(p, q; x)| ≤ 1. Wherefrom ∥gn,k(p,q;x)∥C[0,p(n−k−1)q(n−k−1)]≤1.

Taking s0 = n – k – 2; then for x∈ps0+1qs0+1;ps0qs0 we get

gn,k(p,q;x)=xk∏j=0s0(pj−qjx)(qs0+1x−p)≤qs0+1x−1≤q−p≤1,

which shows ∥gn,k(p, q; x)∥n–k–2 ≤ 1.

For estimating ∥gn,k(p, q; x)∥0 consider the norm on the small interval on the right, and choosing x∈(pq;1), we have

|gn,k(p, q; x)|=xk(1−x)(qx−p)∏j=2n−k−1(qjx−pj)=xn−2(1−x)(qx−p)q(n−k)(n−k−1)2−1∏j=2n−k−1(1−pjqjx),

as

1≥∏j=2n−k−1(1−pjqjx)=∏j=1n−k−1(1−q−jpjqx)≥∏j=1n−k−2(1−pjqj)≥(pq;pq)∞.

Now, by applying Lemma 3.1,

∥gn,k(p,q;x)∥0∼q(n−k)(n−k−1)2p(n−k)(n−k−1)2∥hn−2(p,q;x)∥0∼q(n−k)(n−k−1)2n.p(n−k)(n−k−1)2.

For s0 = 1, 2,..., n – k – 3, we take x∈ps0+1qs0+1;ps0qs0 so that y=qs0ps0x∈(pq,1).

After calculating, we get

|gn,k(p,q,x)|=xk(ps0−qs0x)(qs0+1x−ps0+1)∏s=0s0−1(ps−qsx)∏s=s0+2n−k−1(psx−qs)=q(s0+2)+⋯+(n−k−1)xn−k−s0−2xk(1−qs0x)(1−qs0+1x−1)∏s=0s0−1(ps−qsx)∏s=s0n−k−1(1−psqsx)×q(n−k)(n−k−1)2−s0(2n−1−s0)/2−1hn−s0−2(y)∏s=0s0(ps−qsx)∏s=s0+2n−k−1(1−psqsx).

Therefore,

1≥∏s=0s0−1(ps−qsx)=∏t=0s0(1−q−t(qs0x))≥∏s=1s0(1−q−s)≥(pq;pq)∞.

and

1≥∏s=s0+2n−k−1(1−psqsx)=∏t=1n−k−s0−2(1−q−t1qs0+1x)≥∏t=1n−k−s0−2(1−q−t)≥(1−q−t)≥(pq;pq)∞.

We have

∥gn,k(p,q;x)∥s0∼q(n−k−1)(n−k)/2−s0(2n−k−1)/2/(n−s0−2).

Lastly, by using (3.4), we get the required result

∥gn,k(p,q;x)∥∼∥gn,k(p,q;x)∥0∼q(n−k−1)(n−k)2n.p(n−k−1)(n−k)2.

□

Proof of Theorem 2.3. Since we have

∥Bp,qn(x)∥≥∥Qn,0(p,q;.)∥,

using Corollary 2.2.1 we get

∥Bp,qn(x)∥≥qn(n−1)22.

Again,

‖Bp,qn(x)‖≤∑k=0n‖Qn,k(p,q;.)‖≤1+∑k=0n−1q(n−1)(n+k−1)2n≤1+qn(n−1)2n∑k=0n−1q(n−1)(n+k−1)2.

As ∑k=0∞q−k(k−1)2<∞, we deduce that ∑k=0n−1q−k(k−1)2<∞ and hence ∥Bp,qn(x)∥≤qn(n−1)2n.

Thus,

∥Bp,qn(x)∥∼qn(n−1)2n.

□

Finally, we estimate the strong asymptotic order of the norm of ∥Bp,qn(x)∥.

Now for k ≤ n – 2, x∈(pq;1) and by (3.1), we have

3.5 Qn,k(p,q;x)=q(n−k)(n+k−1)2xn−1(1−x)pq;pqnpq;pqxn−k−1pq;pqkpq;pqn−k.

Therefore, for large value of n, it is seen that 1−1n∈(pq,1) and for k ≤ n – 2 we obtain:

3.6 |Qn,k(p,q;(1−1n))|=q(n−k)(n+k−1)2(1−1n)n−11n(nk)1/q(pnq(n−1);pq)n−k−1≥q(n−k)(n+k−1)21e⋅n(nk)1/q(p⋅nq(n−1);pq)∞.

Putting n = k + 1, we have

Qn,n−1p,q;1−1n=[n]p,q1−1nn−11n≥qpn−1[n]p,q1n⋅e≥qpn−1[n]1/q1/n⋅ep.nq(n−1);pq∞,

which agree with (3.6).

For k = n and n ≥ 2

Qn,n1−1n=1−1nn>1e

and (3.6) is also true. It follows that

∥Bp,qn∥ = maxx∈[0,1] (∑k=0n|Qn,k|)≥∑k=0n|Qn,k(1−1n)|.

By using (3.6), we can write

q−n(n−1)2n. e ‖Bp,qn‖≥(p.nq(n−1);pq)∞∑k=0nq−k(k−1)2(nk)1/q.

Now by using the Rothe’s Identity, we get

q−n(n−1)2n. e ‖Bp,qn‖ ≥ (n.pq(n−1);pq)∞(−1;qp)n.

Since g(x)=pqx;pq is continuous at x = 1, the limit of right-hand side as n → ∞ exists and equals

pq;pq−1;pq∞=2p2q2;p2q2.

As a result we obtain

lim infn→∞q−n(n−1)2n. e ‖Bp,qn‖  ≥ 2  (p2q2;p2q2)∞.

From above, we now estimate ∥Bp,qn∥ in conjunction with the result (3.5) and Theorem 2.3 implies that for k + 2 ≤ n and n large enough, we have

3.7 ∥Qn,k(p,q;x)∥≤q(n−k)(n+k−1)2pq;pqnpq;pqxn−k−1pq;pqkpq;pqn−k maxx∈[p/q;1]xn−1(1−x),∥Qn,k(p,q;x)∥=q(n−k)(n+k−1)2pq;pqnpq;pqkpq;pqn−k1−1nn−11n.

Also, we have ∥𝒬n,n(p, q; x)∥ = 1 and ∥Qn,n−1(p,q;x)∥≤qn−12, that implies

limn→∞‖Qn,n(p, q; x)‖ n⋅ e q−n(n−1)2=limn→∞q−n(n−1)2n ⋅ e‖Qn,n−1(p, q; x)‖  = 0.

Consequently, inadequate to estimate limn→∞∑k=0n−2‖Qn,n(p, q; x)‖ n⋅ e q−n(n−1)2.

Using (3.7), we have

n⋅e q−n(n−1)2∑k=0n−2‖Qn,n(p, q; x)‖ ≤ e⋅(1−1n)−1(pq;pq)∑k=0n−2q−k(k−1)/2(pq;pq)(1−(pq)−k).

For k, n ∈ ℤ+ we fix

cp,qk,n=pk(k−1)2q−k(k−1)2pq;pqk1−(pq)n−k,k<n−20,otherwise.

Evidently,

∑k=0n−2q−k(k−1)2(pq;pq(1−(1q)n−k)=∑k=0∞ck,np,q

and

ck,np,q2≤q−k(k−1)2pq2;pq2(1−(1q2)2)=dk.

Because we have ∑k=0∞dk<∞, by applying the Lebesgue Dominated Convergence Theorem, we obtain

limn→∞∑k=0∞ck,np,q2=∑k=0∞(limn→∞ck,np,q2)=∑k=0∞−k(k−1)/2(1q2;1q2)k=(−1;pq2)∞

By using Euler’s identity, we get

3.8 lim sup n→∞q−n(n−1)/22n. e ‖Bp,q2n‖  ≤   (pq2;pq2)∞(−1;qp2)∞=2(p2q2;p2q2)∞.

Now the required result easily follows from (3.7) and (3.8).

(Communicated by Marek Balcerzak)

References

[1] ACAR, T.: (p, q)-generalization of Szász-Mirakyan operators, Math. Methods Appl. Sci. 39(10) (2016), 2685–2695.10.1002/mma.3721Search in Google Scholar

[2] ACAR, T.—MOHIUDDIN, A.: On Kantorovich modification of (p, q)-Baskakov operators, J. Inequal. Appl. 2016 (2016), Art. No. 98.10.1186/s13660-016-1045-9Search in Google Scholar

[3] ACAR, T.—ARAL, A.: On pointwise convergence of q-Bernstein operators and their q-derivatives, Numer. Funct. Anal. Optim. 36(3) (2015), 287–304.10.1080/01630563.2014.970646Search in Google Scholar

[4] ACAR, T.—ARAL, A.—MOHIUDDINE, S. A.: Approximation by bivariate (p, q)-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci. 42(2) (2018), 655–662.10.1007/s40995-016-0045-4Search in Google Scholar

[5] ACAR, T.—MOHIUDDINE, S. A.—MURSALEEN, M.: Approximation by (p, q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory 12(6) (2018), 1453–1468.10.1007/s11785-016-0633-5Search in Google Scholar

[6] ACAR, T.—ARAL, A.—MOHIUDDINE, S. A.: On Kantorovich modification of (p, q)-Bernstein operators, Iran. J. Sci. Technol. Trans. A Sci. Science 42(3) (2018), 1459–1464.10.1007/s40995-017-0154-8Search in Google Scholar

[7] BOHMAN, H.: On approximation of continuous and analytic functions, Arkiv för Matematik 2(3) (1951), 44–56.10.1007/BF02591381Search in Google Scholar

[8] BURBAN, I.: Two-parameter deformation of the oscillator algebra and (p, q)-analogue of two dimensional conformal field theory, J. Nonlinear Math. Phys. 2(3–4) (1995), 384–391.10.2991/jnmp.1995.2.3-4.18Search in Google Scholar

[9] BERNSTEIN, S. N.: Démonstration du théoréme de Weierstrass fondeé sue la calcul des probabilités, Commun. Soc. Math. Charkow 213 (1992), 1–2.Search in Google Scholar

[10] GAL, S. G.: Approximation by complex q-Lorentz polynomial q > 1, Mathematica (Cluj) 77(1) (2012).Search in Google Scholar

[11] HOUNKONNOU, M. N.—KYEMBA, J. D. B.: R(p, q)-calculus: differentiation and integration, SUT J. Math. 49(2) (2013), 145–167.10.55937/sut/1394548362Search in Google Scholar

[12] HATICE, H. G. İ.—ACAR, T.: Approximation by bivariate (p, q)-Baskakov-Kantorovich operators, Georgian Math. J. 25(3) (2018), 397–407.10.1515/gmj-2016-0057Search in Google Scholar

[13] KHAN, F.—SAIF, M.—MUKHEIMER, A.—MURSALEEN, M.: The convergence of (p, q)-Bernstein operators for the Cauchy kernel with a pole via divided difference, J. Inequal. Appl. 2019(1) (2019), Art. No. 132.10.1186/s13660-019-2090-ySearch in Google Scholar

[14] KHAN, N. U.—AMAN, M.—USMAN T.—CHOI, J.: Legendre-Gould Hopper-based Sheffer polynomials and operational methods, Symmetry 12 (2020), Art. ID 2051.10.3390/sym12122051Search in Google Scholar

[15] KOROVKIN, P. P.: Convergence of linear positive operator in the space of contineous function, Dokl. Akad. Nauk. Russian. SSSR (N.S.) 90 (1953), 961–964.Search in Google Scholar

[16] MAHMUDOV, N. I.: Convergence properties and iterations for q-Stancu polynomials in compact disks, Comput. Math. Appl. 59(12) (2010), 3763–3769.10.1016/j.camwa.2010.04.010Search in Google Scholar

[17] MAHMUDOV N. I.—KARA, M.: Approximation theorems for generalized complex Kantorovich-type operators, J. Appl. Math. 2012 (2012), Art. ID 454579.10.1155/2012/454579Search in Google Scholar

[18] MOHIUDDINE, S. A.—ACAR, T.—ALOTAIBI, A.: Durrmeyer type (p, q)-Baskakov operators preserving linear functions, J. Math. Inequal 12(4) (2018), 961–973.10.7153/jmi-2018-12-73Search in Google Scholar

[19] MURSALEEN, M.—ANSARI, K. J.—KHAN, A.: Some approximation results of (p, q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput. 264 (2015), 392–402.10.1016/j.amc.2015.03.135Search in Google Scholar

[20] MURSALEEN, M.—ANSARI, K. J.—KHAN, A.: On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874–882. [Erratum: Appl. Math. Comput. 278 (2016), 70–71].10.1016/j.amc.2015.04.090Search in Google Scholar

[21] MURSALEEN, M.—NASIRUZZAMAN, M.—KHAN, F.—KHAN, A.: (p, q)-analogue of divided difference and Bernstein operators, J. Nonlinear. Funct. Anal. 2017 (2017), Art. No. 25.10.23952/jnfa.2017.25Search in Google Scholar

[22] MURSALEEN, M.—KHAN, F.—KHAN, A.: Approximation by (p, q)-Lorentz polynomial on a compact disk, Complex Anal. Oper. Theory 16(7) (2022).Search in Google Scholar

[23] MURSALEEN, M.—KHAN, F.—SAIF M.—KHAN, A. H.: On estimation of uniform convergence of analytic functions by (p, q)-Bernstein operators, Korean J. Math. 27(2) (2019), 505–514.Search in Google Scholar

[24] OSTROVSKA, S.: q-Bernstein polynomial of the Cauchy kernel, Appl. Math. Comput. 198(1) (2008), 261–270.10.1016/j.amc.2007.08.066Search in Google Scholar

[25] OSTROVSKA, S.: q-Bernstein polynomial and their iterates, J. Approx. Theory 123 (2003), 232–255.10.1016/S0021-9045(03)00104-7Search in Google Scholar

[26] OSTROVSKA, S.—ÖZBAN, A. Y.: On the q-Bernstein polynomial of unbounded function with q > 1, Abstr. Appl. Anal. 13 (2013), Art. Id. 349156, 7p.10.1155/2013/349156Search in Google Scholar

[27] OSTROVSKA, S.: The first decade of the q-Bernstein polynomial, result and properties, J. Math. Anal. Approx. Theory 2(1) (2007), 35–51.Search in Google Scholar

[28] PHILLIPS, G. M.: Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997), 511–518.Search in Google Scholar

[29] SADJANG, P. N.: On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, Results Math. 73(1) (2018), Art. No. 39.Search in Google Scholar

[30] RAMAPATI, M.—SHARMA, H.—GUPTA, C.: Approximation properties of Kantorovich type modifications of (p, q)-Meyer-König-Zeller operators, Constr. Math. Anal. 1(1) (2018), 58–72.10.33205/cma.436071Search in Google Scholar

[31] USMAN, T.—SAIF, M.—CHOI, J.: Certain identities associated with (p, q)-binomial coefficients and (p, q)-Stirling polynomials of the second kind, Symmetry 12 (2020), Art. ID 1436.10.3390/sym12091436Search in Google Scholar
Received: 2021-02-11
Accepted: 2021-11-19
Published Online: 2023-03-31
Published in Print: 2023-04-01

© 2023 Mathematical Institute Slovak Academy of Sciences

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  5. D(4)-Triples with Two Largest Elements in Common
  6. Hardy-Leindler Type Inequalities for Multiple Integrals on Time Scales
  7. A New Version of q-Hermite-Hadamard’s Midpoint and Trapezoid Type Inequalities for Convex Functions
  8. Existence and Finite-Time Stability Results for Impulsive Caputo-Type Fractional Stochastic Differential Equations with Time Delays
  9. On Oblique Domains of Janowski Functions
  10. Sharp Approximations for the Generalized Elliptic Integral of the First Kind
  11. Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces
  12. Evaluation of Norm of (p, q)-Bernstein Operators
  13. Weyl ρ-Almost Periodic Functions in General Metric
  14. Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity
  15. Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields
  16. On the Bivariate Generalized Gamma-Lindley Distribution
  17. New Insights on the Multivariate Skew Exponential Power Distribution
  18. On Locally Finite Orthomodular Lattices
https://doi.org/10.1515/ms-2023-0034
Keywords for this article
(p, q)-integers; (p, q)-Bernstein operators; convergence
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Articles in the same Issue

  1. Voltage Lifts of Graphs from a Category Theory Viewpoint
  2. Extensions and Congruences of Partial Lattices
  3. On Recurrences in Generalized Arithmetic Triangle
  4. The Asymptotics of the Geometric Polynomials
  5. D(4)-Triples with Two Largest Elements in Common
  6. Hardy-Leindler Type Inequalities for Multiple Integrals on Time Scales
  7. A New Version of q-Hermite-Hadamard’s Midpoint and Trapezoid Type Inequalities for Convex Functions
  8. Existence and Finite-Time Stability Results for Impulsive Caputo-Type Fractional Stochastic Differential Equations with Time Delays
  9. On Oblique Domains of Janowski Functions
  10. Sharp Approximations for the Generalized Elliptic Integral of the First Kind
  11. Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces
  12. Evaluation of Norm of (p, q)-Bernstein Operators
  13. Weyl ρ-Almost Periodic Functions in General Metric
  14. Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity
  15. Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields
  16. On the Bivariate Generalized Gamma-Lindley Distribution
  17. New Insights on the Multivariate Skew Exponential Power Distribution
  18. On Locally Finite Orthomodular Lattices
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