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Abstract and Applied Analysis
Volume 2011, Article ID 101852, 13 pages
http://dx.doi.org/10.1155/2011/101852
Research Article

Direct and Inverse Approximation Theorems for Baskakov Operators with the Jacobi-Type Weight

1School of Mathematics and Information Engineering, Taizhou University, Zhejiang, Taizhou 317000, China
2School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 2 May 2011; Accepted 10 September 2011

Academic Editor: Ruediger Landes

Copyright © 2011 Guo Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new norm and a new K-functional 𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆. Using this K-functional, direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight are obtained in this paper.

1. Introduction and Main Results

Let 𝑓 be a function defined on the interval [0,). The operators 𝑉𝑛(𝑓;𝑥) are defined as follows:𝑉𝑛(𝑓;𝑥)=𝑘=0𝑓𝑘𝑛𝑣𝑛,𝑘(𝑥),(1.1) where 𝑣𝑛,𝑘𝑘𝑥(𝑥)=𝑛+𝑘1𝑘(1+𝑥)𝑛𝑘,(1.2) which were introduced by Baskakov in 1957 [1]. Becker [2] and Ditzian [3] had studied these operators and obtained direct and converse theorems. In [4, 5] Totik gave a result: if 𝑓𝐶𝐵[0,+),0<𝛼<1, then 𝑉𝑛(𝑓;𝑥)𝑓(𝑥)=𝑂(𝑛𝛼) if and only if 𝑥𝛼(1+𝑥)𝛼|Δ2(𝑓;𝑥)|𝑘2𝛼, where >0 and k is a positive constant. We may formulate the following question: do the Baskakov operators have similar property in the case of weighted approximation with the Jacobi weights? It is well known that the weighted approximation is not a simple extension, because the Baskakov operators are unbounded for the usual weighted norm 𝑓𝑤=𝑤𝑓. Xun and Zhou [6] introduced the norm 𝑓𝑤=𝑤𝑓+||||𝑓(0),𝑓𝐶𝐵[0,)(1.3) and have discussed the rate of convergence for the Baskakov operators with the Jacobi weights and obtained||𝑉𝑤(𝑥)𝑛||(𝑓;𝑥)𝑓(𝑥)=𝑂(𝑛𝛼)𝐾(𝑓;𝑡)𝑤=𝑂(𝑡𝛼),(1.4) where 𝑤(𝑥)=𝑥𝑎(1+𝑥)𝑏,0<𝑎<1,𝑏>0,0<𝛼<1, and 𝐶𝐵[0,) is the set of bounded continuous functions on [0,).

In this paper, we introduce a new norm and a new K-functional, using the K-functional, and we get direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight.

First, we introduce some useful definitions and notations.

Definition 1.1. Let 𝐶𝐵[0,) denote the set of bounded continuous functions on the interval [0,), and let 𝐶𝑎,𝑏,𝜆=𝑓𝑓𝐶𝐵[0,),𝜑2(2𝜆)𝑤𝑓𝐶𝐵[,𝐶0,)0𝑎,𝑏,𝜆=𝑓𝑓𝐶𝑎,𝑏,𝜆,,𝑓(0)=0(1.5) where 𝜑(𝑥)=𝑥(1+𝑥),𝑤(𝑥)=𝑥𝑎(1+𝑥)𝑏,𝑥[0,),0𝑎<𝜆1, and 𝑏0.
Moreover, the K-functional is given by 𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆=inf𝑔𝐷𝜑2(1𝜆)(𝑓𝑔)𝑤𝜑+𝑡2(2𝜆)𝑔𝑤,(1.6) where 𝐷={𝑔𝑔𝐶0𝑎,𝑏,𝜆,𝑔𝐴.𝐶.𝑙𝑜𝑐[0,),𝜑2(2𝜆)𝑔𝑤<}.
We are now in a position to state our main results.

Theorem 1.2. If 𝑓𝐶0𝑎,𝑏,𝜆, then 𝜑2(1𝜆)𝑉𝑛(𝑓)𝑓𝑤𝑀𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆.(1.7)

Theorem 1.3. Suppose 𝑓𝐶0𝑎,𝑏,𝜆,0<𝛼<1. Then the following statements are equivalent: (1)𝜑2(1𝜆)||𝑉(𝑥)𝑤(𝑥)𝑛||(𝑓(𝑥))𝑓(𝑥)=𝑂(𝑛𝛼),𝑛2;(2)𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆=𝑂(𝑡𝛼),0<𝑡<1.(1.8)

Throughout this paper, 𝑀 denotes a positive constant independent of 𝑥,𝑛, and 𝑓 which may be different in different places. It is worth mentioning that for 𝜆=1, we recover the results of [6].

2. Auxiliary Lemmas

To prove the theorems, we need some lemmas. By simple computation, we have 𝑉𝑛(𝑓;𝑥)=𝑛(𝑛+1)𝑘=0𝑣𝑛+2,𝑘(𝑓𝑥)𝑘+2𝑛2𝑓𝑘+1𝑛𝑘+𝑓𝑛(2.1) or 𝑉𝑛(𝑓;𝑥)=𝑘=0𝑣𝑛,𝑘(𝑘𝑥)𝑓𝑛𝑘(𝑘1)𝑥22𝑘(𝑛+𝑘)+𝑥(1+𝑥)(𝑛+𝑘)(𝑛+𝑘+1)(1+𝑥)2.(2.2)

Lemma 2.1. Let 𝑐0,𝑑. Then 𝑘=1𝑣𝑛,𝑘(𝑘𝑥)𝑛𝑐𝑘1+𝑛𝑑𝑀𝑥𝑐(1+𝑥)𝑑,for𝑥>0.(2.3)

Proof. We notice [7] 𝑘=1𝑣𝑛,𝑘(𝑛𝑥)𝑘𝑙𝑀𝑥𝑙,for𝑙,𝑘=0𝑣𝑛,𝑘𝑘(𝑥)1+𝑛𝑚𝑀(1+𝑥)𝑚,for𝑚.(2.4) For 𝑐=0,𝑑=0, the result of (2.3) is obvious. For 𝑐>0,𝑑0, there exists 𝑚, such that 0<2𝑑/𝑚<1. Using Hölder's inequality, we have 𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝑐𝑘1+𝑛𝑑𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛2𝑐1/2𝑘=1𝑣𝑛,𝑘𝑘(𝑥)1+𝑛2𝑑1/2𝑘=1𝑣𝑛,𝑘𝑛(𝑥)𝑘[2𝑐]+1𝑐/([2𝑐]+1)𝑘=1𝑣𝑛,𝑘𝑘(𝑥)1+𝑛𝑚𝑑/𝑚𝑥𝑀([2𝑐]+1)𝑐/([2𝑐]+1)((1+𝑥)𝑚)𝑑/𝑚𝑀𝑥𝑐(1+𝑥)𝑑.(2.5) For 𝑐>0,𝑑=0 or 𝑐=0,𝑑0, the proof is similar to that of (2.5). Thus, this proof is completed.

Lemma 2.2. Let 𝑓𝐶0𝑎,𝑏,𝜆,𝑛. Then ||𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛||𝜑(𝑓;𝑥)𝑀2(1𝜆)𝑓𝑤.(2.6)

Proof. By Lemma 2.1, we get ||𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛||=|||||(𝑓;𝑥)𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=1𝑓𝑘𝑛𝑣𝑛,𝑘|||||𝜑(𝑥)2(1𝜆)𝑓𝑤𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=1𝑣𝑛,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(𝜆1)𝑘𝑛𝜑𝑀2(1𝜆)𝑓𝑤.(2.7)

Lemma 2.3. Let 𝑓𝐶0𝑎,𝑏,𝜆,𝑛. Then 𝜑2(2𝜆)𝑉𝑛(𝑓)𝑤𝜑𝑀𝑛2(1𝜆)𝑓𝑤.(2.8)

Proof. For 𝑥𝐸𝑐𝑛=[0,1/𝑛],𝑥0,(𝑛+1)𝑥(𝑥+1)2𝑛2𝑥4; using (2.1) and Lemma 2.1, we have ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||(𝑓;𝑥)𝑤(𝑥)𝜑2(1𝜆)×(𝑥)𝑛(𝑛+1)𝑥(1+𝑥)𝑘=0𝑣𝑛+2,𝑘(𝑥)𝑤1𝑘+2𝑛𝜑2(1𝜆)𝑘+2𝑛+2𝑘=0𝑣𝑛+2,𝑘(𝑥)𝑤1𝑘+1𝑛𝜑2(1𝜆)𝑘+1𝑛+𝑘=1𝑣𝑛+2,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(1𝜆)𝑘𝑛𝜑2(1𝜆)𝑓𝑤𝑀𝑛𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑤1(𝑥)𝜑2(1𝜆)𝜑(𝑥)2(1𝜆)𝑓𝑤𝜑𝑀𝑛2(1𝜆)𝑓𝑤.(2.9) For 𝑥𝐸𝑛=(1/𝑛,), by (2.2), we get ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||=|||||𝑛(𝑓;𝑥)2𝑤(𝑥)𝜑2𝜆(𝑥)𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑓𝑛𝑘𝑛𝑥21+2𝑥𝑛𝑘𝑛𝑥𝑥(1+𝑥)𝑛|||||𝑛2𝑤(𝑥)𝜑2𝜆𝜑(𝑥)2(1𝜆)𝑓𝑤𝑘=1𝑣𝑛,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(1𝜆)𝑘𝑛𝑘𝑛𝑥2+1+2𝑥𝑛|||𝑘𝑛|||+𝑥𝑥(1+𝑥)𝑛=𝑛2𝑤(𝑥)𝜑2𝜆𝜑(𝑥)2(1𝜆)𝑓𝑤𝐼1(𝑛,𝑥)+𝐼2(𝑛,𝑥)+𝐼3.(𝑛,𝑥)(2.10) Note that for 𝑥𝐸𝑛, one has the following inequality [7] 𝑛2𝑚𝑉𝑛(𝑡𝑥)2𝑚;𝑥𝑀𝑛𝑚(𝜑(𝑥))2𝑚,𝑚.(2.11) Applying Hölder’s inequality and Lemma 2.1, we have 𝐼1(𝑛,𝑥)=𝑘=1𝑣𝑛,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(1𝜆)𝑘𝑛𝑘𝑛𝑥2𝑘=1𝑣𝑛,𝑘(𝑥)𝑤2𝑘𝑛𝜑4(1𝜆)𝑘𝑛1/2𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝑥41/2𝑀𝑥𝑎1+𝜆(1+𝑥)𝑏+𝜆1𝑥(1+𝑥)𝑛𝑀𝑛1𝑤1(𝑥)𝜑2𝜆𝐼(𝑥),(2.12)2(𝑛,𝑥)=𝑘=1𝑣𝑛,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(1𝜆)𝑘𝑛|||𝑘𝑛|||𝑥1+2𝑥𝑛1+2𝑥𝑛𝑘=1𝑣𝑛,𝑘(𝑥)𝑤2𝑘𝑛𝜑4(1𝜆)𝑘𝑛1/2𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝑥21/2𝑀𝑤1(𝑥)𝜑2𝜆(𝑥)𝑛3/211+𝑥1/2.(2.13) Note that for 𝑥>1/𝑛, one has 1+1/𝑥<2𝑛. Hence, 𝐼2(𝑛,𝑥)𝑀𝑛1𝑤1(𝑥)𝜑2𝜆𝐼(𝑥),(2.14)3(𝑛,𝑥)=𝑘=1𝑣𝑛,𝑘(𝑥)𝑤1𝑘𝑛𝜑2(1𝜆)𝑘𝑛𝑥(1+𝑥)𝑛𝑀𝑛1𝑥(1+𝑥)𝑥𝑎1+𝜆(1+𝑥)𝑏1+𝜆=𝑀𝑛1𝑤1(𝑥)𝜑2𝜆(𝑥).(2.15) Combining (2.9)–(2.14), we get ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||(𝑓;𝑥)𝑤𝜑𝑀𝑛2(1𝜆)𝑓𝑤.(2.16) Thus, 𝜑2(2𝜆)𝑉𝑛(𝑓)𝑤𝜑𝑀𝑛2(1𝜆)𝑓𝑤.(2.17) The proof is completed.

Lemma 2.4. Let 𝑓𝐷,𝑛, and 𝑛2. Then 𝜑2(2𝜆)𝑉𝑛(𝑓)𝑤𝜑𝑀2(2𝜆)𝑓𝑤.(2.18)

Proof. (1) For the case 𝜆1 or 𝑎0, if 𝜆2+𝑏0, using (2.1) and Lemma 2.1, we have ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||=|||||𝑤(𝑓;𝑥)(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑘=0𝑣𝑛+2,𝑘(𝑥)01/𝑛01/𝑛𝑓𝑘𝑛|||||𝜑+𝑢+𝑣𝑑𝑢𝑑𝑣2(2𝜆)𝑓𝑤|||||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑘=0𝑣𝑛+2,𝑘(𝑥)01/𝑛01/𝑛𝑘𝑛+𝑢+𝑣𝜆𝑎2𝑘1+𝑛+𝑢+𝑣𝜆+𝑏2||||𝜑𝑑𝑢𝑑𝑣2(2𝜆)𝑓𝑤|||||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑘=1𝑣𝑛+2,𝑘(𝑥)01/𝑛01/𝑛𝑘𝑛𝜆𝑎21+𝑘+2𝑛𝜆+𝑏2|||||+𝜑𝑑𝑢𝑑𝑣2(2𝜆)𝑓𝑤||||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑣𝑛+2,0(𝑥)01/𝑛01/𝑛(𝑢+𝑣)𝜆𝑎2(1+𝑢+𝑣)𝜆+𝑏2||||𝜑𝑑𝑢𝑑𝑣2(2𝜆)𝑓𝑤𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑘=1𝑣𝑛+2,𝑘(𝑥)𝑛2𝑘𝑛𝜆𝑎21+𝑘+2𝑛𝜆+𝑏2+3𝜆+𝑏2𝜑2(2𝜆)𝑓𝑤𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑣𝑛+2,0(𝑥)01/𝑛1𝑢1+𝑎𝜆𝜆𝑎1𝑑𝑢23𝜆+𝑏2𝜑2(2𝜆)𝑓𝑤𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑘=1𝑣𝑛+2,𝑘𝑘(𝑥)𝑛𝜆𝑎2𝑘1+𝑛𝜆+𝑏2+23𝜆+𝑏2𝜑2(2𝜆)𝑓𝑤𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑣𝑛+2,01(𝑥)1(1+𝑎𝜆)(𝜆𝑎)𝑛𝜆𝑎23𝜆+𝑏2𝜑2(2𝜆)𝑓𝑤𝑥1+2+𝑎𝜆𝑛(𝑛+1)(1+𝑎𝜆)(𝜆𝑎)(1+𝑥)𝑛1𝑛𝜆𝑎.(2.19)
(i)If 𝑥𝐸𝑐𝑛, 𝑥2+𝑎𝜆𝑛(𝑛+1)(1+𝑎𝜆)(𝜆𝑎)(1+𝑥)𝑛1𝑛𝜆𝑎𝑛(𝑛+1)(11+𝑎𝜆)(𝜆𝑎)𝑛22(1+𝑎𝜆)(𝜆𝑎).(2.20)(ii)If 𝑥𝐸𝑛,𝑛2, 𝑥2+𝑎𝜆𝑛(𝑛+1)(1+𝑎𝜆)(𝜆𝑎)(1+𝑥)𝑛1𝑛𝜆𝑎𝑛𝜆𝑎𝑥2𝑛(𝑛+1)(1+𝑎𝜆)(𝜆𝑎)(1+𝑥)𝑛1𝑛𝜆𝑎𝑥2𝑛(𝑛+1)(1+𝑎𝜆)(𝜆𝑎)𝑛(𝑛1)𝑥23.(1+𝑎𝜆)(𝜆𝑎)(2.21) Combining (2.19)–(2.21), we have ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||𝜑(𝑓;𝑥)𝑀2(2𝜆)𝑓𝑤.(2.22) Thus, 𝜑2(2𝜆)(𝑥)𝑉𝑛(𝑓)𝑤𝜑𝑀2(2𝜆)𝑓𝑤.(2.23) If 𝜆2+𝑏<0, we have ||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑉𝑛||𝜑(𝑓;𝑥)2(2𝜆)𝑓𝑤|||||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑘=1𝑣𝑛+2,𝑘(𝑥)01/𝑛01/𝑛𝑘𝑛𝜆𝑎2𝑘1+𝑛𝜆+𝑏2|||||+𝜑𝑑𝑢𝑑𝑣2(2𝜆)𝑓𝑤||||𝑤(𝑥)𝜑2(2𝜆)(𝑥)𝑛(𝑛+1)𝑣𝑛+2,0(𝑥)01/𝑛01/𝑛(𝑢+𝑣)𝜆𝑎2||||.𝑑𝑢𝑑𝑣(2.24) By using the method similar to that of (2.19)–(2.23), it is not difficult to obtain the same inequality as (2.23).
(2) For the case 𝜆=1,𝑎=0, the proof is similar to that of case (1) and even simpler. Therefore the proof is completed.

Lemma 2.5 (see [8, page 200]). Let Ω(𝑡) be an increasing positive function on (0,𝑎), the inequality (𝑟>𝛼)𝑡Ω()𝑀𝛼+𝑡𝑟Ω(𝑡)(2.25) holds true for ,𝑡(0,𝑎). Then one has Ω(𝑡)=𝑂(𝑡𝛼).(2.26)

3. Proofs of Theorems

3.1. Proof of Theorem 1.2

Proof. First, we prove it as follows.(i)If 𝑥𝐸𝑐𝑛, then 𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=0𝑣𝑛,𝑘(||||𝑥)𝑥𝑘/𝑛|||𝑘𝑛|||𝑤𝑢1(𝑢)𝜑2(2𝜆)(||||𝑢)𝑑𝑢𝑀𝑛1.(3.1)(ii)If 𝑥𝐸𝑛, then 𝑉𝑛(𝑡𝑥)2(1+𝑡)𝑏2+𝜆;𝑥𝑀𝑛1𝜑2(𝑥)(1+𝑥)𝑏2+𝜆.(3.2)
The Proof of (3.1)
In fact, (i) for 𝑘=0, since 𝑥𝐸𝑐𝑛, we have 𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑣𝑛,0(𝑥)𝑥0𝑢𝑤1(𝑢)𝜑2(2𝜆)(𝑢)𝑑𝑢=𝑤(𝑥)𝜑2(1𝜆)(𝑥)(1+𝑥)𝑛𝑥0𝑢𝜆𝑎1(1+𝑢)𝑏2+𝜆𝑑𝑢.(3.3) If 𝑏2+𝜆0, we get 𝑤(𝑥)𝜑2(1𝜆)(𝑥)(1+𝑥)𝑛𝑥0𝑢𝜆𝑎1(1+𝑢)𝑏2+𝜆𝑑𝑢𝑀𝑤(𝑥)𝜑2(1𝜆)(𝑥)(1+𝑥)𝑛𝑥𝜆𝑎𝑀𝑛1.(3.4) If 𝑏2+𝜆>0, we have 𝑤(𝑥)𝜑2(1𝜆)(𝑥)(1+𝑥)𝑛𝑥0𝑢𝜆𝑎1(1+𝑢)𝑏2+𝜆𝑑𝑢𝑀𝑤(𝑥)𝜑2(1𝜆)(𝑥)(1+𝑥)𝑛+𝑏2+𝜆𝑥0𝑢𝜆𝑎1𝑑𝑢𝑀𝑛1.(3.5)
(ii) If 𝑘1, since 𝑥𝐸𝑐𝑛, we have 𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=1𝑣𝑛,𝑘(||||𝑥)𝑥𝑘/𝑛|||𝑘𝑛|||𝑤𝑢1(𝑢)𝜑2(2𝜆)(||||𝑢)𝑑𝑢𝑀𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝜑𝑥2(2𝜆)𝑘(𝑥)1+𝑛𝑏||||𝑥𝑘/𝑛𝑢𝑎||||𝑑𝑢𝑀𝑤(𝑥)𝜑2(𝑥)𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝑘𝑥1+𝑛𝑏𝑘𝑛1𝑎𝑥1𝑎𝑀𝑤(𝑥)𝜑2(𝑥)𝑘=1𝑣𝑛,𝑘𝑘(𝑥)𝑛𝑥2𝑎𝑘1+𝑛𝑏𝑀𝑛1.(3.6) Combining (3.4), (3.5) and (3.6), we obtain (3.1).
The proof of (3.2)
If 𝑏2+𝜆0, by (9.5.10) and (9.6.3) of [7], using the Cauchy-Schwarz inequality and the Hölder inequality, we obtain 𝑉𝑛(𝑡𝑥)2(1+𝑡)𝑏2+𝜆𝑉;𝑥𝑛(𝑡𝑥)4;𝑥1/2𝑉𝑛(1+𝑡)2(𝑏2+𝜆);𝑥1/2𝑉𝑛(𝑡𝑥)4;𝑥1/2𝑉𝑛(1+𝑡)2;𝑥(2𝑏𝜆)/2𝑀𝑛1𝜑2(𝑥)(1+𝑥)𝑏2+𝜆.(3.7)
If 𝑏2+𝜆>0, by (2.3), we get 𝑉𝑛((1+𝑡)𝑏2+𝜆;𝑥)𝑀(1+𝑥)𝑏2+𝜆, and using the Cauchy-Schwarz inequality and the Hölder inequality, we have 𝑉𝑛(𝑡𝑥)2(1+𝑡)𝑏2+𝜆𝑉;𝑥𝑛(𝑡𝑥)4;𝑥1/2𝑉𝑛(1+𝑡)2(𝑏2+𝜆);𝑥1/2𝑀𝑛1𝜑2(𝑥)(1+𝑥)𝑏2+𝜆.(3.8) Combining (3.7) and (3.8), we obtain (3.2).

Next, we prove Theorem 1.2. For 𝑔𝐷, if 𝑥𝐸𝑐𝑛, by (3.1), we have ||𝑤(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||=||||(𝑔;𝑥)𝑔(𝑥)𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛𝑡𝑥(𝑡𝑢)𝑔||||(𝑢)𝑑𝑢;𝑥𝑤(𝑥)𝜑2(1𝜆)𝜑(𝑥)2(2𝜆)𝑔𝑤𝑉𝑛||||𝑡𝑥|𝑡𝑢|𝑤1(𝑢)𝜑2(2𝜆)||||𝜑(𝑢)𝑑𝑢;𝑥𝑀2(2𝜆)𝑔𝑤𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑘=0𝑣𝑛,𝑘||||(𝑥)𝑥𝑘/𝑛|||𝑘𝑛|||𝑤𝑢1(𝑢)𝜑2(2𝜆)||||(𝑢)𝑑𝑢𝑀𝑛1𝜑2(2𝜆)𝑔𝑤.(3.9) If 𝑥𝐸𝑛, by (3.2), we get ||𝑤(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||=||||(𝑔;𝑥)𝑔(𝑥)𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛𝑡𝑥(𝑡𝑢)𝑔||||𝜑(𝑢)𝑑𝑢;𝑥𝑀2(2𝜆)𝑔𝑤𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛||||𝑡𝑥|𝑡𝑢|𝑤1(𝑢)𝜑2(2𝜆)||||𝜑(𝑢)𝑑𝑢;𝑥𝑀2(2𝜆)𝑔𝑤||𝜑2(𝑥)𝑉𝑛(𝑡𝑥)2;𝑥+𝑥2𝑎+𝜆𝑤(𝑥)𝜑2(1𝜆)(𝑥)𝑉𝑛(𝑡𝑥)2(1+𝑡)𝑏2+𝜆||;𝑥𝑀𝑛1𝜑2(2𝜆)𝑔𝑤.(3.10) Therefore, for 𝑓𝐶0𝑎,𝑏,𝜆,𝑔𝐷, by Lemma 2.2 and (3.9), (3.10), and the definition of 𝐾𝜑𝜆(𝑓;𝑛1)𝑤,𝜆, we obtain ||𝑤(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||||𝑤(𝑓;𝑥)𝑓(𝑥)(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||+||𝑤(𝑓𝑔;𝑥)(𝑥)𝜑2(1𝜆)||+||(𝑥)(𝑓(𝑥)𝑔(𝑥))𝑤(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||𝜑(𝑔;𝑥)𝑔(𝑥)𝑀2(1𝜆)(𝑓𝑔)𝑤+||𝑤(𝑥)𝜑2(1𝜆)(||+||𝑤𝑥)(𝑓(𝑥)𝑔(𝑥))(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||𝜑(𝑔;𝑥)𝑔(𝑥)𝑀2(1𝜆)(𝑓𝑔)𝑤+𝑛1𝜑2(2𝜆)𝑔𝑤.(3.11) Taking the infimum on the right-hand side over all 𝑔𝐷, we get ||𝑤(𝑥)𝜑2(1𝜆)𝑉(𝑥)𝑛||(𝑓;𝑥)𝑓(𝑥)𝑀𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆.(3.12) This completes the proof of Theorem 1.2.

3.2. Proof of Theorem 1.3

Proof. By Theorem 1.2, we know (2)(1). Now,we will prove (1)(2). In view of (1), we get 𝜑2(1𝜆)𝑉𝑛(𝑓)𝑓𝑤𝑀𝑛𝛼.(3.13) By the definition of K-functional, we may choose 𝑔𝐷 to satisfy 𝜑2(1𝜆)(𝑓𝑔)𝑤+𝑛1𝜑2(2𝜆)𝑔𝑤2𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆.(3.14) Using Lemma 2.2 and Lemma 2.3, we have 𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆𝜑2(1𝜆)𝑉𝑛(𝑓)𝑓𝑤𝜑+𝑡2(2𝜆)𝑉𝑛(𝑓)𝑤𝑀𝑛𝛼𝜑+𝑡2(2𝜆)𝑉𝑛(𝑓𝑔)𝑤+𝜑2(2𝜆)𝑉𝑛(𝑔)𝑤𝑀𝑛𝛼𝜑+𝑡𝑛𝑀2(1𝜆)(𝑓𝑔)𝑤𝜑+𝑀2(2𝜆)𝑔𝑤𝑀𝑛𝛼𝜑+𝑡𝑛𝑀2(1𝜆)(𝑓𝑔)𝑤+𝑛1𝜑2(2𝜆)𝑔𝑤.(3.15) Taking the infimum on the right-hand side over all 𝑔𝐷, we get 𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆𝑛𝑀𝛼+𝑡𝑛1𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆.(3.16) By Lemma 2.4, we get 𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆𝑀(𝑛𝛼).(3.17) Leting (𝑛+1)1<𝑡𝑛1, we get 𝐾𝜑𝜆(𝑓;𝑡)𝑤,𝜆𝑀𝐾𝜑𝜆𝑓;𝑛1𝑤,𝜆𝑛𝑀𝑛+1𝛼(𝑛+1)𝛼𝑀(𝑛+1)𝛼𝑀𝑡𝛼.(3.18) This completes the proof of Theorem 1.3.

Acknowledgments

The author would like to thank Professor Ruediger Landes and the anonymous referees for their valuable comments, remarks, and suggestions which greatly help us to improve the presentation of this paper and make it more readable. The project is supported by the Natural Science Foundation of China (Grant no. 10671019).

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