# Approximation by Bézier Variant of Baskakov-Durrmeyer-Type Hybrid Operators

Authors: Lahsen Aharouch; Khursheed J. Ansari; M. Mursaleen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We give a Bézier variant of Baskakov-Durrmeyer-type hybrid operators in the present article. First, we obtain the rate of convergence by using Ditzian-Totik modulus of smoothness and also for a class of Lipschitz function. Then, weighted modulus of continuity is investigated too. We study the rate of point-wise convergence for the functions having a derivative of bounded variation. Furthermore, we establish the quantitative Voronovskaja-type formula in terms of Ditzian-Totik modulus of smoothness at the end.


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## Body

## 1. Introduction

To approximate continuous functions, many approximating operators have been introduced under certain conditions and with different parameters too. Many researchers have later generalized and modified these introduced operators and discussed various approximating properties of these operators. In 1957, Baskakov [1] introduced and studied such a class of positive linear operators, called Baskakov operators defined on the positive semiaxis. For f∈C0,∞, the sequence of Baskakov operators is given as
(1)Bnf;y=∑k=0∞n+k−1kyk1+y−n−kfkn,for y∈0,∞ and n∈ℕ. Later on, many authors have been considering the Baskakov operators; for instance, Aral in [2] defines the parametric generalization of Baskakov operators as
(2)Bnvf;x=∑k=0∞Pn,kvxfkn,where
(3)Pn,kvx=xk−11+xn+k−1vx1+xn+k−1k−1−v1+xn+k−3k−2+1−vxn+k−1k,with n+k−1k−2=0 if k=0,1.Among interesting studies realized in this context, we cite those based on the Baskakov-Kantorovitch-type operators in the generalized form (the original operator given by Kantorovich in [3]) defined as, for f∈L10,1 (the class of Lebesgue integrable functions on 0,1),
(4)BKnf;x=∑k=0∞n+k−1kxk1+x−n−k∫01χn,ktftdt,where χn,k is the characteristic function of the interval k/n,k+1/n.It is well known that Bézier curves are the mathematically defined curves successively used in computer-aided geometric design (CAGD), image processing, and curve fitting. The miscellaneous Bézier variant of operators is crucial subject matter in approximation theory. In 1983, Chang [4] pioneered the Bernstein-Bézier operators. Afterwards, several researchers established the Bézier variant of various operators (c.f. [5, 6]). For more details on the approximation by Durrmeyer-type and Baskakov-Durrmeyer-type operators, one can refer to [7, 8], respectively. For more about Bézier variant of operators, one can refer to [9, 10].We will be mainly interested to the Bézier variant operator type based on those of Baskakov-Durrmeyer defined as follows:(5)Gn,ρv,θf;x=∑k=1∞Xn,kv,θx∫0∞Jn,kρtftdt+Xn,0v,θxf0,where
(6)Xn,kv,θx=ξn,kvxθ−ξn,k+1vxθ,ξn,kvx=∑j=k∞Pn,jvxk=0,1,2,⋯Jn,kρt=nρe−nρtnρtkρ−1Γkρ.If we takeθ=1, then operator (5) reduces to the following operator studied by [11].
(7)Bn,ρvf;x=∑k=1∞Pn,kvx∫0∞Jn,kρtftdt+Pn,0vxf0.Let us briefly summarize the outline of the paper. Next section is devoted to the computation of some auxiliary results which we need to prove our theorems in coming sections. In Section3, we will prove some approximations of functions using Ditzian-Totik modulus and then we will deal to functions lie in the Lipschitz spaces. We treat in Section 4 the rate of convergence in the context of suitable weighted spaces and functions having a derivative of bounded variation. Finally, in Section 5, we state and prove the quantitative Voronovskaja-type theorem.


## 2. Preliminary Results

Lemma 1.
ξn,kvx satisfies the following important properties:
(1)
ξn,kvx−ξn,k+1vx=Pn,kvxk=0,1,2,⋯(2)
ξn,0vx>ξn,1vx>.⋯ξn,kvx>ξn,k+1vx>⋯(3)
ξn,kvxθ−ξn,k+1vxθ≤θPn,kvxifθ≥1Pn,kvxθifθ≤1Proof.
Since (1) and (2) are evident, we prove only the assertion (3).
Ifθ≥1, it suffices to remark that by the mean value theorem, we have
(8)bθ−aθ≤θb−aforevery0<a<b<1.
Ifθ<1, we shall prove that
(9)bθ−aθ≤b−aθforevery0<a<b.
Dividing this inequality byaθ, it is equivalent to prove that
(10)fr=r−1θ−rθ+1≥0foreveryr>1.
We havef′r=θ/r−1eθlnr−1−θ/reθlnr; then,
(11)f′r>0ifandonlyiflnrr−1>lneθlnreθlnr−1,and this is true as θ<1.
We proved thenf is increasing, so fr>fs for all r>s>1, letting s to 1, and we deduce that fr≥0.Remark 2.
The operatorsGn,ρv,θf;x have the integral representation
(12)Gn,ρv,θf;x=∫0∞Kn,kv,θx,ufudu,where Kn,kv,θx,u is the kernal defined by
(13)Kn,kv,θx,u=∑k=1∞Xn,kv,θxJn,kρu+Xn,0v,θxδu.
δu is the Dirac-delta function.Lemma 3.
Letemt=tm and φt=1/1+tn+2. For the operator Bn,ρvf;x, we have
(1)
Bn,ρve0;x=∑k=0∞Pn,kvx=∑k=0∞Pn,k0x=∑k=0∞−1kφkx/k!=φ0=1(2)
Bn,ρvem;x=∑k=1∞Pn,kvx.kρ+m−1.kρ+m−2⋯kρ/nρm,m=1,2,3,⋯As an easy consequence of last lemma, we will prove the following result.Lemma 4.
We have the following moments:(1)
Bn,ρvt;x=x+2xv−1/n(2)
Bn,ρvt2;x=x2+x24v−3/n+x−2+n+2v+nρ+4ρv−1/n2ρ(3)
Bn,ρvt−x;x=2xv−1/n(4)
Bn,ρvt−x2;x=x2/n+x/n2ρn1+ρ+2v−11+2ρ(5)
n2Bn,ρvt−x4;x=x4/n3α1+6x3/n3ρα2+x21+ρ/n3ρ2α3+x1+ρ/n4ρ3α4where
(14)α1=3n+16v−10,α2=n+6v−4+ρn+8v−6,α3=3n1+ρ+4v7+8ρ−25ρ−17,(15)α4=n1+ρ3+ρ+4v−13+4ρ2+ρ.Remark 5.
We have(1)
limn⟶∞nBn,ρvt−x;x=2xv−1(2)
limn⟶∞nBn,ρvt−x2;x=x1+ρ+ρx/ρ(3)
limn⟶∞n2Bn,ρvt−x4;x=3x1+ρ+ρx2/ρ2Remark 6.
Forn large enough, we have the following inequalities:
(1)
Bn,ρvt−x2;x≤C1x1+ρ+ρx/nρ(2)
Bn,ρvt−x4;x≤C2x1+ρ+ρx2/nρ2Throughout this article, letCBℝ0+ denote the space of all functions f on ℝ0+ which are bounded and continuous. We endowed it by the norm ∥f∥=supx∈ℝ0+∣fx∣.Lemma 7.
Letf∈CBℝ0+, and we have
(1)
∥Gn,ρv,θf;x∥≤Gn,ρv,θe0;x∥f∥ and Gn,ρv,θe0;x=1(2)
Gn,ρv,θf;x≤θBn,ρvf;x≤θ∥f∥Proof.
(1)
On the one hand, we have(16)Gn,ρv,θf;x=∑k=1∞Xn,kv,θx∫0∞Jn,kρtftdt+Xn,0v,θxf0≤∑k=1∞Xn,kv,θx∫0∞Jn,kρtdt+Xn,0v,θx.∥f∥≤Gn,ρv,θe0;x∥f∥.
On the other hand,(17)Gn,ρv,θe0;x=∑k=0∞Xn,kv,θx=ξn,0vxθ=∑k=0∞Pn,kvxθ=1θ=1.(2)
We have(18)Gn,ρv,θf;x=∑k=1∞Xn,kv,θx∫0∞Jn,kρtftdt+Xn,0v,θxf0=∑k=1∞ξn,kvxθ−ξn,k+1vxθ∫0∞Jn,kρtftdt+ξn,0vxθ−ξn,1vxθf0.
Using Lemma1, it is easy to see that
(19)Gn,ρv,θf;x≤θ∑k=1∞Pn,kvx∫0∞Jn,kρtftdt+θPn,0vxf0≤θBn,ρvf;x.


## 3. Direct Approximation

Before we discuss the different approximations, we need some definitions. First, we recall the definition of the well-known Ditizian-Totik modulus of smoothnesswφτ.,. and Peetre’s K-functional [12].Definition 8.
Letφx=x and f∈CBℝ0+. For 0≤τ≤1, we define
(20)wφτf,δ=sup0≤h≤δsupx±hφτx/2∈ℝ0+fx+hφτx2−fx−hφτx2,and the K-functional
(21)Kφτf,δ=infg∈Wτf−g+δφτg′, where
(22)Wτ=g∈ACloc:φτg′<∞,with ACloc is the set of all absolutely continuous function on every finite subinterval of ℝ0+.Remark 9.
wφτf,δ and Kφτf,δ are equivalent, that is, there exists a constant C>0 such that
(23)C−1wφτf,δ≤Kφτf,δ≤Cwφτf,δ.In the next definition, we cite Lipschitz-type functions:Definition 10 [13].
Fora≥0,b>0 to be fixed, the class of two parametric Lipschitz-type functions is defined as
(24)LipMa,bβ=g∈CBℝ0+:∣fy−fx∣≤My−xβy+ax2+bxβ/2,x,y>0,where M is any positive constant and 0<β≤1.
The spaceLipM0,1β is the space LipM∗β given by Szász [14].
We now proceed with the approximation results.Theorem 11.
Forf∈CBℝ0+, we have
(25)Gn,ρv,θf;x−fx≤Cwφτf,φ2−τ1+xn,where wφτ is given by (20) and C is a constant free from the choice of n and x.
For the proof of this theorem, we use the following lemma proved in [15].Lemma 12.
Letφx=x and 0≤τ≤1; then, for f∈Wτ and x,y>0, we have
(26)∫xyf′udu≤2τx−τ/2x−yφτf′.Proof (Theorem11).
Letg∈Wτ . Using Lemma 7, we have
(27)Gn,ρv,θf;x−fx=Gn,ρv,θf−g;x+fx−gx+Gn,ρv,θg;x−gx≤1+θfx−gx+Gn,ρv,θg;x−gx.
Sincegy=gx+∫xyg′udu and Gn,ρv,θ1;x=1, we conclude that
(28)Gn,ρv,θg;x−gx=Gn,ρv,θ∫xyg′udu;x.Therefore, Lemma 12 implies
(29)Gn,ρv,θg;x−gx≤2τx−τ/2φτg′Gn,ρv,θ∣x−y∣;x.
By Cauchy-Schwarz inequality and Remark6, it is easy to check that
(30)Gn,ρv,θ∣x−y∣;x=Gn,ρv,θ1;xGn,ρv,θx−y2;x≤C1xθ1+ρ+ρxnρ.
Combining (27)-(30), we get
(31)Gn,ρv,θf;x−fx≤1+θfx−gx+C3φτg′φ2−τ1+xn.
Let now taking the infimum overg∈Wτ, and we have
(32)Gn,ρv,θf;x−fx≤C4Kφτf,φ2−τ1+xn. We thank to (26).
(33)Gn,ρv,θf;x−fx≤Cwφτf,φ2−τ1+xn.Theorem 13.
Forf∈LipMa,bβ, then for every n∈ℕ,ρ>0,θ≥1 and x∈0,+∞, we have
(34)Gn,ρv,θf;x−fx≤MθBn,ρvy−x2;xax2+bxβ/2,where Bn,ρvy−x2;x is given in Lemma 4.Proof.
Letf∈LipMa,bβ and x∈0+∞, and we have
(35)Gn,ρv,θf;x−fx=Gn,ρv,θfy−fx;x≤Gn,ρv,θ∣fy−fx∣;x≤Gn,ρv,θMy−xβy+ax2+bxβ/2;x.
Let us consider the caseβ=1. By the Cauchy-Schwarz inequality and the fact Gn,ρv,θ1;x=1, we have immediately that
(36)Gn,ρv,θf;x−fx≤Max2+bxGn,ρv,θy−x2;x1/2≤Max2+bxθBn,ρvy−x2;x1/2≤MθBn,ρvy−x2;xax2+bx1/2.
This proves the result forβ=1.
If0<β<1, Holder’s inequality with exponents p=1/β and p′=1/1−β, we get
(37)Gn,ρv,θf;x−fx≤Max2+bxβ/2Gn,ρv,θ∣y−x∣;xβ.
Using again the Cauchy-Schwarz inequality, we obtain(38)Gn,ρv,θf;x−fx≤Max2+bxβ/2Gn,ρv,θy−x2;xβ/2≤Max2+bxβ/2θBn,ρvy−x2;xβ/2≤MθBn,ρv,θy−x2;xax2+bxβ/2,and this gives the result.


## 4. Rate of Convergence in Weighted Spaces

In this section, we focus about the rate of convergence of operators (5) in the context of suitable weighted function spaces and functions having a derivative of bounded variation. We will use the following spaces:
(39)B2ℝ0+=f:fx≤Mf1+x2,Mfisaconstantdependonf.Introduce also(40)C2ℝ0+=f∈B2ℝ0+:fiscontinuous,C2∗ℝ0+=f∈C2ℝ0+:∃limx⟶∞∣fx∣1+x2<∞.These spaces are endowed with the norm(41)f2=supx∈ℝ0+∣fx∣1+x2.The weighted modulus of continuity is defined as (see [16])
(42)Ωf,δ=supx≥0sup∣t∣<δ∣fx+t−fx∣1+x+t2.Theorem 14.
Letf∈C2∗ℝ0+. Then, for x∈ℝ0+,ρ,δ>0,θ≥1 and for large enough n, we have
(43)Gn,ρv,θf;x−fx≤21+x2Ωf,1n×1+θC1x1+ρ+xρnρ+θC1x1+ρ+xρρ1/2×1+θC2x1+ρ+xρnρ,where C1,C2>1 are constants independent of x and n.Proof.
Letu,x∈ℝ0+,δ>0. An immediate consequence of the definition of weighted modulus of continuity is
(44)fu−fx≤21+x21+u−x21+∣u−x∣δΩf,δ.
SinceGn,ρv,θf;x is linear and increasing, we have from (44)
(45)Gn,ρv,θ∣fu−fx∣;x≤21+x2Ωf,δGn,ρv,θ1+u−x2;x+Gn,ρv,θ1+u−x2∣u−x∣δ;x.
Cauchy-Schwarz inequality was applied in the last term, and it gives us(46)Gn,ρv,θ∣fu−fx∣;x≤21+x2Ωf,δ1+Gn,ρv,θu−x2;x1δGn,ρv,θu−x4;x1/2Gn,ρv,θu−x2;x1/2+1δGn,ρv,θu−x2;x1/2.
Choosingδ=1/n, we get the required result in virtue of Remark 6.


## 5. Rate of Convergence for Functions of Bounded Variation

LetDBVℝ0+ be the space of functions on ℝ0+ having a derivative of bounded variation on every finite subinterval of ℝ0+. Consider the space
(47)DBV2ℝ0+=f∈DBVℝ0+:fx≤Mf1+x2forsomeconstantMf>0.It is known that every functionf in DBV2ℝ0+ has a representation of the form
(48)fx=∫0xgudu+f0,where g is a function of bounded variation on each finite subinterval of ℝ0+.Lemma 15.
Letx∈ℝ0+, and let Kn,ρv,θx,u be the kernel defined by (13). Then, for C1>1 and for n large enough, we have
(1)
ξn,ρv,θx;y=∫0yKn,ρv,θx,udu≤θC1x1+ρ+xρ/nρ1/x−y2,0≤y<x(2)
1−ξn,ρv,θx;z=∫z∞Kn,ρv,θx,udu≤θC1x1+ρ+xρ/nρ1/z−x2,x<z<∞Proof.
Using Remark6, we get
(49)ξn,ρv,θx;y=∫0yKn,ρv,θx,udu≤∫0yu−yx−y2Kn,ρv,θx,udu≤1x−y2Gn,ρv,θu−x2,x≤θC1x1+ρ+xρnρ1x−y2.
Similarly, we can show the second part; hence, the proof is omitted.Theorem 16.
Letf∈DBV2ℝ0+, and for every x∈0,∞, consider the function fx′ defined by
(50)fx′u=f′u−f′x−,if0≤u<x,0,ifu=x,f′u−f′x+,ifx<u<∞.
Let us denote by∨cdfx′ the total variation of fx′ on c,d⊂ℝ0+. Then, for every x∈0,∞ and large n,
(51)Gn,ρv,θf;x−fx≤θ1+θf′x++θf′x−C1x1+ρ+xρnρ1/2+θ3/21+θf′x++θf′x−C1x1+ρ+xρnρ1/2+θC11+ρ+xρnρ∑k=1n∨x−xkxfx′+xn∨x−xnxfx′+xn∨xx+xnfx′+θC11+ρ+xρnρ∑0n∨xx+xkfx′.Proof.
For anyf∈DBV2ℝ0+, from the definition of fx′u, we can write
(52)f′u=11+θf′x++θf′x−+δxuf′x−12f′x++f′x−+fx′u+12f′x+−f′x−sgnu−x+θ−11+θ, where
(53)δxu=1,ifu=x,0,ifu≠x.
By the fact thatGn,ρv,θ1;x=1, we have
(54)Gn,ρv,θf;x−fx=Gn,ρv,θfu−fx;x=∫0∞Kn,ρv,θx,ufu−fxdu=∫0∞Kn,ρv,θx,u∫xuf′vdvdu.
From (52), we obtain
(55)Gn,ρv,θf;x−fx=∫0∞Kn,ρv,θx,u∫xu11+θf′x++θf′x−dvdu+∫0∞Kn,ρv,θx,u∫xu12f′x+−f′x−×sgnv−x+θ−11+θdvdu+∫0∞Kn,ρv,θx,u∫xuδxvf′x−12f′x++f′x−dvdu+∫0∞Kn,ρv,θx,u∫xufx′vdvdu.
From the definition ofδxv, it is clear that
(56)∫0∞Kn,ρv,θx,u∫xuδxvf′x−12f′x++f′x−dvdu=0.
The first integral on the right hand side of (55) can be estimated as follows:
(57)∫0∞Kn,ρv,θx,u∫xu11+θf′x++θf′x−dvdu≤11+θf′x++θf′x−∫0∞Kn,ρv,θx,uu−xdu.
Applying the Cauchy-Schwarz inequality and Remark 2, we have, forn large enough,
(58)∫0∞Kn,ρv,θx,u∫xu11+θf′x++θf′x−dvdu≤11+θf′x++θf′x−Gn,ρv,θu−x2;x≤11+θf′x++θf′x−C1θx1+ρ+xρnρ1/2≤θ1+θf′x++θf′x−C1x1+ρ+xρnρ1/2.
Similarly, it is easy to find(59)∫0∞Kn,ρv,θx,u∫xu12f′x+−f′x−sgnv−x+θ−11+θdvdu≤θ3/21+θf′x++θf′x−C1x1+ρ+xρnρ1/2.
Write the last term of (55) as
(60)∫0∞Kn,ρv,θx,u∫xufx′vdvdu=An,ρv,θfx′;x+Bn,ρv,θfx′;x, where
(61)An,ρv,θfx′;x=∫0xKn,ρv,θx,u∫xufx′vdvdu,Bn,ρv,θfx′;x=∫x∞Kn,ρv,θx,u∫xufx′vdvdu.
Now, we estimate the termsAn,ρv,θfx′;x and Bn,ρv,θfx′;x.
Using the definition ofξn,ρv,θ.;. given in Lemma 15 and integrating by parts, we can write
(62)An,ρv,θfx′;x=∫0x∫xufx′vdv∂ξn,ρv,θx;u∂udu=∫0xfx′uξn,ρv,θx;udu.
Thus,(63)An,ρv,θfx′;x≤∫0x−x/nfx′uξn,ρv,θx;udu+∫x−x/nxfx′uξn,ρv,θx;udu.
Sincefx′x=0 and ξn,ρv,θx;u≤1, we get
(64)∫x−x/nxfx′uξn,ρv,θx;udu=∫x−x/nxfx′u−fx′xξn,ρv,θx;udu≤∫x−x/nxfx′u−fx′xdu≤∫x−x/nx∨uxfx′du≤xn∨x−xnxfx′.
Concerning the first integral on the right hand side of (63), using Lemma 15, we have
(65)∫0x−x/nfx′uξn,ρv,θx;udu≤θC1x1+ρ+xρnρ∫0x−x/nfx′ux−y2du=θC1x1+ρ+xρnρ∫0x−x/nfx′u−fx′xx−y2du=θC1x1+ρ+xρnρ∫0x−x/n∨uxfx′dux−y2.
By changing of variableu=x−x/v, we deduce that
(66)∫0x−x/nfx′uξn,ρv,θx;udu≤θC11+ρ+xρnρ∫1n∨x−xvxfx′dv≤θC11+ρ+xρnρ∑k=1n∨x−xkxfx′.
Therefore,(67)An,ρv,θfx′;x≤θC11+ρ+xρnρ∑k=1n∨x−xkxfx′+xn∨x−xnxfx′.
What concerns the second term of the right hand side of (60), integrating by parts and Lemma 15 with z=x+x/n, we can write
(68)Bn,ρv,θfx′;x≤∫xzfx′u1−ξn,ρv,θx;udu+∫z∞fx′u1−ξn,ρv,θx;udu≤∫xz∨xufx′du+θC1x1+ρ+xρnρ∫z∞∨xufx′1u−x2du≤xn∨xx+xnfx′+θC1x1+ρ+xρnρ∫x+x/n∞∨xufx′u−x−2du.
Puttingu=x+x/v, we get
(69)θC1x1+ρ+xρnρ∫x+x/n∞∨xufx′u−x−2du≤θC11+ρ+xρnρ∫0n∨xx+xvfx′dv≤θC11+ρ+xρnρ∑0n∨xx+xkfx′.
Combining (68) and (69), we have
(70)Bn,ρv,θfx′;x≤xn∨xx+xnfx′+θC11+ρ+xρnρ∑0n∨xx+xkfx′.
Finally, by combining (52)-(70), we get (51).


## 6. Quantitative Voronovskaja-Type Asymptotic Formula

In this last section, we deal with the Voronovskaja-type asymptotic theorem forGn,ρv,θ. More precisely we will prove the following result:Theorem 17.
Forf∈CBℝ0+ such that f′,f′′∈CBℝ0+. Then,
(71)nGn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x≤Cθx1+ρ+xρρwφτf,φ2−τ1+xn,where C is independent of n and x.Proof.
By Taylor’s formula, we write(72)fu=fx+u−xf′x+∫xuu−vf′′vdv.
It is clear that(73)fu−fx−u−xf′x−12u−x2f′′x=∫xuu−vf′′v−f′′xdv.
On the one hand, we applyGn,ρv,θ.;x to both sides of the above equality, and we get
(74)Gn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x=Gn,ρv,θ∫xuu−vf′′v−f′′xdv;x≤Gn,ρv,θ∫xuu−vf′′v−f′′xdv;x.
On the other hand, forg∈Wτ, we have
(75)∫xuu−vf′′v−f′′xdv≤f′′−gu−x2+2τφ−τφτg′u−x3,which implies, by (74),
(76)Gn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x≤f′′−gGn,ρv,θu−x2;x+2τφ−τφτg′Gn,ρv,θu−x3;x.
After using the Cauchy-Schwarz inequality in the last term, we obtain(77)Gn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x≤2τφ−τφτg′Gn,ρv,θu−x4;x1/2Gn,ρv,θu−x2;x1/2+f′′−gGn,ρv,θu−x2;x.
In view of Remark6, we have
(78)Gn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x≤2τφ−τφτg′C1θx1+ρ+xρnρ1/2C2θx1+ρ+xρnρ21/2+f′′−gC1θx1+ρ+xρnρ≤C1θx1+ρ+xρnρf′′−g+M∗φ2−τ1+xnφτg′.
Taking the infimum on the right-hand side of the above inequality overg∈Wτ, we get
(79)nGn,ρv,θf;x−fx−f′xGn,ρv,θu−x;x−12f′′xGn,ρv,θu−x2;x≤Cθx1+ρ+xρρKφτf,M∗φ2−τ1+xn.
Recalling (23), the theorem is proved.



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*Source: 6673663-2021-09-13.xml*