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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 908104, 10 pages
http://dx.doi.org/10.1155/2012/908104
Research Article

On the Rational Approximation of Analytic Functions Having Generalized Types of Rate of Growth

Department of Mathematics, Research and Post Graduate Studies, M.M.H. College, Model Town, Ghaziabad 201001, UP, India

Received 21 March 2012; Accepted 2 July 2012

Academic Editor: NarendraΒ Govil

Copyright Β© 2012 Devendra Kumar. 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

The present paper is concerned with the rational approximation of functions holomorphic on a domain πΊβŠ‚πΆ, having generalized types of rates of growth. Moreover, we obtain the characterization of the rate of decay of product of the best approximation errors for functions f having fast and slow rates of growth of the maximum modulus.

1. Introduction

Let 𝐾 be a compact subset of the extended complex plane 𝐢 and let 𝐸𝑛 be the error in the best uniform approximation of a function 𝑓 (holomorphic on 𝐾) on 𝐾 in the class 𝑅𝑛 of all rational functions of order 𝑛: 𝐸𝑛=𝐸𝑛(𝑓,𝐾)=infπ‘Ÿβˆˆπ‘…π‘›β€–π‘“βˆ’π‘Ÿβ€–πΎ(1.1) for each nonnegative integer 𝑛, where ‖⋅‖𝐾 is the supremum norm on 𝐾.

In view of Walsh's inequality [1], if 𝑓 is holomorphic on 𝐢⧡𝑀, where 𝑀 is a compact set in 𝐢 and π‘€βˆ©πΎ=πœ™, then limsupπ‘›β†’βˆžπΈπ‘›1/𝑛≀1𝑑,(1.2) where 𝑑=exp(1/𝐢(𝐾,𝑀)) and 𝐢(𝐾,𝑀) is the capacity of the condenser (𝐾,𝑀), (see [2–4], for the definition and properties of the capacity).

The theory of Hankel operators permits one [5–7] to estimate the order of decrease of the product 𝐸1𝐸2⋯𝐸𝑛: limsupπ‘›β†’βˆžξ€·πΈ1𝐸2⋯𝐸𝑛1/𝑛2≀1𝑑.(1.3)

The last relation implies Walsh's inequality (1.2) and the following upper estimate for liminfπ‘›β†’βˆžπΈπ‘›1/𝑛: liminfπ‘›β†’βˆžπΈπ‘›1/𝑛≀1𝑑2.(1.4)

The present paper is concerned to results that make the inequalities (1.2), (1.3) and (1.4) more precise for analytic functions having generalized types of the rate of growth of the maximum modulus in the domain of analyticity of 𝑓.

The generalized order 𝜌(𝛼,𝛽,𝑓) of the rate of growth of entire functions 𝑓 was introduced by Ε eremeta [8], who obtained a characterization of 𝜌(𝛼,𝛽,𝑓) in terms of the coefficients of the power series of 𝑓. In [8], the relationship between the generalized order of entire functions 𝑓 and the degree of polynomial approximation of 𝑓 was studied. The coefficient characterization of a generalized order of the rate of growth of functions analytic in a disk has been discussed in several papers [9–12]. The degree of rational approximation of entire functions of a finite generalized order is investigated in [6].

Now let us consider the Dirichlet problem in the domain 𝐢⧡(𝐾βˆͺ𝑀) with boundary function equal to 1 on πœ•π‘€ and to 0 on πœ•πΎ. Here, 𝐾 and 𝑀 be disjoint compact sets with connected complements in the extended complex plane 𝐢 such that their boundaries consist of finitely many closed analytic Jordan curves. Since the domain 𝐢⧡(𝐾βˆͺ𝑀) is regular with respect to the Dirichlet problem, this problem is solvable. Let 𝑀(𝑧) be the solution which is extended by continuity to πΆβˆΆπ‘€(𝑧)=1 for π‘§βˆˆπ‘€ and 𝑀(𝑧)=0 for π‘§βˆˆπΎ. For 0<πœ€<1, let 𝛾(πœ€)={π‘§βˆΆπ‘€(𝑧)=πœ€}.

Let 𝛼 and 𝛽 be continuous positive functions on [π‘Ž,∞) satisfying the following properties:(i)limπ‘₯β†’βˆžπ›Ό(π‘₯)=+∞, and limπ‘₯β†’βˆžπ›½(π‘₯)=+∞;(ii)limπ‘₯β†’βˆž(𝛽(π‘₯+π‘œ(π‘₯))/𝛽(π‘₯))=1;(iii)π›Όβˆ’1(log(1/℧𝛽(π‘₯)))/π›Όβˆ’1(log(1/β„§ξ…žπ›½(π‘₯)))=π‘œ(π‘₯) as π‘₯β†’0 for all β„§ξ…ž>℧>0.

Let 𝑓 be holomorphic on 𝐺=𝐢⧡𝑀. We define the generalized order 𝜌(𝛼,𝛽,𝑓) and generalized type 𝑇(𝛼,𝛽,𝑓) of 𝑓 in the domain 𝐺 by the formulae:(a)𝜌(𝛼,𝛽,𝑓)=limsupπœ€β†’1(𝛼(log‖𝑓‖𝛾(πœ€))/𝛽(log(1/(1βˆ’πœ€)))), (b)𝑇(𝛼,𝛽,𝑓)=limsupπœ€β†’1(𝛼(‖𝑓‖𝛾(πœ€))/[(1/(1βˆ’πœ€))]𝜌(𝛼,𝛽,𝑓)), where ‖𝑓‖𝛾(πœ€)=maxπ‘§βˆˆπ›Ύ(πœ€)|𝑓(𝑧)|.

It is easy to see that for the functions 𝛼(π‘₯)=log𝑝π‘₯,𝑝β‰₯2, and 𝛽(π‘₯)=π‘₯ properties (i)–(iii) will hold. The following theorem gives the characterization of the rate of decay of product 𝐸0𝐸1⋯𝐸𝑛 for functions 𝑓 having fast rates of growth of the maximum modulus. So to avoid some trivial cases, we will assume that limπœ€β†’1‖𝑓‖𝛾(πœ€)=∞.

Theorem 1.1. Suppose that 𝑓 is holomorphic on 𝐺,𝛼 and 𝛽 satisfy conditions (i)–(iii), and 𝑓 has generalized order 𝜌(𝛼,𝛽,𝑓)>0 and generalized type 𝑇(𝛼,𝛽,𝑓) in the domain 𝐺. Then, limsupπ‘›β†’βˆžξ‚ƒπ›½ξ‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚ξ‚ξ‚ξ‚„πœŒβ‰€π‘‡(𝛼,𝛽,𝑓),(1.5) where log+π‘₯=max(0,logπ‘₯) for π‘₯β‰₯0.

Proof. Let us assume that 𝑇(𝛼,𝛽,𝑓)<∞. Fix arbitrary numbers π‘‡ξ…žξ…ž>π‘‡ξ…ž>𝑇(𝛼,𝛽,𝑓). For 𝑛=1,2,…, we set 𝛿𝑛1=min4,π›½βˆ’1ξ€Ίπ‘‡ξ…žξ…žξ€»exp(βˆ’π›Ό(𝑛))1/πœŒξ‚.(1.6)
We have 𝛿𝑛→0 as π‘›β†’βˆž. Using (1.5) for all sufficiently large values of 𝑛,𝑛β‰₯𝑛1, we set log‖𝑓‖𝛾2,π‘›β‰€π›Όβˆ’1𝑇logξ…žξ€Ίπ›½ξ€·π›Ώπ‘›ξ€Έξ€»βˆ’πœŒξ€Έξ€Ύ=π›Όβˆ’11𝜌logξ€·π‘‡ξ…ž1/πœŒπ›½ξ€·π›Ώπ‘›.ξ€Έξ€Έξƒͺξƒ°(1.7)
From (1.6), we have 𝑛=π›Όβˆ’1𝑇logξ…žξ…žξ€Ίπ›½ξ€·π›Ώπ‘›ξ€Έξ€»βˆ’πœŒ.ξ€Έξ€Ύ(1.8)
In (1.7), 𝛾2,𝑛 defined as subsets of the extended complex plane 𝐢: π›Ύπ‘˜,𝑛=ξ€½π‘§βˆΆπ‘€(𝑧)=πœ€π‘˜,𝑛,𝐷𝑛=ξ€½π‘§βˆΆπ‘€(𝑧)>πœ€0,𝑛,(1.9) where πœ€0,𝑛=π‘˜/2𝑛,πœ€1,𝑛=π‘˜/𝑛,πœ€2,𝑛=1βˆ’π›Ώπ‘›,π‘˜=0,1,2, and 𝑛=1,2,…. It is given [13] that 𝛾0,𝑛,𝛾1,𝑛, and 𝛾2,𝑛,𝑛=1,2,…, consist of finitely many closed analytic curves whose lengths are bounded from above by a positive quantity not depending on 𝑛. It is assumed that 𝛾0,𝑛 and 𝛾2,𝑛 are positively oriented with respect to 𝐷𝑛 and {π‘§βˆΆπ‘€(𝑧)>πœ€2,𝑛}, respectively.
In view of (1.8) for 𝑛β‰₯max(𝑛0,𝑛1), we may use the inequality 3.1 of [13] in the form: 𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)ξ€·πΆπ‘‘β‰€π‘›π‘š(𝑛+1)!𝑛8𝑛1/𝑛(𝑛+1)𝛼×expβˆ’1ξ€·ξ€·πœŒlog1/π‘‡ξ…ž1/πœŒπ›½ξ€·π›Ώπ‘›ξ€Έξ€Έξ€Έπ›Όβˆ’1ξ€·ξ€·πœŒlog1/π‘‡ξ…žξ…ž1/πœŒπ›½ξ€·π›Ώπ‘›+𝛿𝑛ξƒͺ.𝐢(𝐾,𝑀)(1.10)
Now, using property (iii), we get log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑≀𝛿𝑛𝛿𝐢(𝐾,𝑀)+π‘œπ‘›ξ€Έ.(1.11)
It gives limsupπ‘›β†’βˆžξ‚ƒπ›½ξ‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚ξ‚„πœŒβ‰€π‘‡ξ…žξ…ž.(1.12)
On letting π‘‡ξ…žξ…žβ†’π‘‡(𝛼,𝛽,𝑓), the proof is complete.

In the consequence of Theorem 1.1, we have the following.

Corollary 1.2. With the assumption of Theorem 1.1, the following inequalities are valid: limsupπ‘›β†’βˆžξ€Ίπ›½ξ€·exp(𝛼(𝑛))log+𝐸𝑛1/π‘›π‘‘ξ€Έξ€»πœŒβ‰€π‘‡(𝛼,𝛽,𝑓),(1.13)limsupπ‘›β†’βˆžξ€Ίπ›½ξ€·exp(𝛼(𝑛))log+𝐸𝑛1/𝑛𝑑2ξ€Έξ€»πœŒβ‰€π‘‡(𝛼,𝛽,𝑓).(1.14)

Proof. Using the fact πΈπ‘›β‰€πΈπ‘›βˆ’1≀⋯≀𝐸0, we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that liminfπ‘›β†’βˆžξ€Ίπ›½ξ€·exp𝛼(𝑛)log+𝐸𝑛1/𝑛𝑑2ξ€Έξ€Έξ€»πœŒ>π‘‡ξ…ž>𝑇(𝛼,𝛽,𝑓).(1.15)
Then, for sufficiently large values of 𝑛, we get 𝛽log+𝐸𝑛1/𝑛𝑑2>ξ‚Έπ‘‡ξ€Έξ€»ξ…žξ‚Ήexp𝛼(𝑛)1/𝜌(1.16) or log+𝐸𝑛𝑑2𝑛β‰₯π‘›π›½βˆ’1ξ‚Έπ‘‡ξ…žξ‚Ήexp(𝛼(𝑛))1/𝜌.(1.17)
Since the functions 𝛼 and 𝛽 are increasing, (1.17) gives log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑β‰₯ξ‚€βˆ‘π‘›π‘˜=0π‘˜π›½βˆ’1ξ‚†ξ€Ίπ‘‡ξ…žξ€»/exp𝛼(𝑛)1/𝜌+𝑐𝑛(𝑛+1)β‰₯π›½βˆ’1ξƒ―ξ‚Έπ‘‡ξ…žξ‚Ήexp𝛼(𝑛)1/𝜌+𝑐,𝑛(𝑛+1)(1.18) where 𝑐 is a constant. Using (ii), we get 𝛽log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚ξ‚„πœŒβ‰₯ξ‚Έπ‘‡ξ…žξ‚Ήexp𝛼(𝑛)(1.19) or liminfπ‘›β†’βˆžξ‚ƒπ›½ξ‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)ξ‚ξ‚„πœŒβ‰₯π‘‡ξ…ž>𝑇(𝛼,𝛽,𝑓)(1.20) which contradicts (1.5). Thus, (1.14) is valid.

2. Rational Approximation of Analytic Functions Having Slow Rates of Growth

For a function 𝑓 analytic in a domain 𝐺, the type of 𝑓 in 𝐺 can be defined by (b) for 𝛼(π‘₯)=logπ‘₯ and 𝛽(π‘₯)=π‘₯: 𝑇=limsupπœ€β†’1log‖𝑓‖𝛾(πœ€)(1/1βˆ’πœ€)𝜌.(2.1)

For 𝛼(π‘₯)=logπ‘₯ and 𝛽(π‘₯)=π‘₯, the property (iii) fails to hold. However, we have the following: π›Όβˆ’1(𝑐log(1/𝛽(π‘₯)))π›Όβˆ’1((𝑐+1)log(1/𝛽(π‘₯)))=π‘₯,(2.2) and we may repeat the arguments involving (1.10), we get 𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑐𝑑≀𝑛(𝑛+1)!𝑛8𝑛1/𝑛(𝑛+1)𝛼×expβˆ’1ξ€·ξ€Ίlog1/π‘‡ξ…ž1/πœŒπ›½ξ€·π›Ώπ‘›ξ€Έξ€»πœŒξ€Έπ›Όβˆ’1ξ€·ξ€Ίlog1/π‘‡ξ…žξ…ž1/πœŒπ›½ξ€·π›Ώπ‘›ξ€Έξ€»πœŒξ€Έ+𝛿𝑛ξƒͺ.𝐢(𝐾,𝑀)(2.3)

Taking π‘‡ξ…žξ…ž=π‘‡ξ…ž+1, and π‘₯=π‘‡ξ…žξ…ž1/πœŒπ›Ώπ‘› in (2.2), for sufficiently large values of 𝑛 we have π‘›ξ‚€ξ€·π‘‡ξ…žξ€Έ+11/𝜌log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚πœŒβ‰€π‘‡ξ…ž(2.4) or limsupπ‘›β†’βˆžπ‘›ξ‚ƒlog+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚„πœŒβ‰€π‘‡.𝑇+1(2.5)

We summarize the above facts in the following.

Theorem 2.1. Let 𝑓 have an order 𝜌>0 and generalized type 𝑇 in the domain 𝐺. Then, limsupπ‘›β†’βˆžπ‘›ξ‚ƒlog+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚„πœŒβ‰€π‘‡.𝑇+1(2.6)

By the inequality πΈπ‘›β‰€πΈπ‘›βˆ’1≀⋯≀𝐸0, one gets the following.

Corollary 2.2. With the assumption of Theorem 2.1limsupπ‘›β†’βˆžπ‘›ξ€Ίlog+𝐸𝑛1/π‘›π‘‘ξ€Έξ€»πœŒβ‰€π‘‡.𝑇+1(2.7)

Theorem 2.1 also gives us the following corollary.

Corollary 2.3. With the assumption of Theorem 2.1, liminfπ‘›β†’βˆžπ‘›ξ€Ίlog+𝐸𝑛1/𝑛𝑑2ξ€Έξ€»πœŒβ‰€π‘‡.𝑇+1(2.8)

Proof. Let liminfπ‘›β†’βˆžπ‘›ξ€Ίlog+𝐸𝑛1/𝑛𝑑2ξ€Έξ€»πœŒ>𝑇1>𝑇.𝑇+1(2.9)
Then, from the relation limπ‘›β†’βˆžβˆ‘π‘›π‘˜=0π‘˜1βˆ’1/𝑇1𝑛2βˆ’1/𝑇1=12βˆ’1/𝑇1,(2.10) we obtain liminfπ‘›β†’βˆžπ‘›ξ‚ƒlog+𝐸1𝐸2⋯𝐸𝑛1/𝑛(𝑛+1)π‘‘ξ‚ξ‚„πœŒβ‰₯𝑇1>𝑇,𝑇+1(2.11) which contradicts the inequality (2.6).

Now, we define 𝛼-type of 𝑓 to classify functions having slow rates of growth.

A continuous positive function β„Ž on [π‘Ž,+∞) belongs to the class Ξ›, if this function satisfies the following.

β„Ž is strictly increasing on [π‘Ž,+∞), limπ‘₯β†’βˆžβ„Ž(π‘₯)=+∞,(2.12)limπ‘₯β†’+βˆžβ„Ž(𝑐π‘₯)β„Ž(π‘₯)=1,(2.13) for any 𝑐>0.

Let π›ΌβˆˆΞ›. We define 𝛼-order and 𝛼-type of 𝑓 in 𝐺 by the formulae: 𝜌(𝛼,𝑓)=limsupπœ€β†’1𝛼log‖𝑓‖𝛾(πœ€)ξ€Έ,𝛼(log(1/(1βˆ’πœ€)))(2.14)𝑇(𝛼,𝑓)=limsupπœ€β†’1𝛼log‖𝑓‖𝛾(πœ€)ξ€Έ[]𝛼(1/(1βˆ’πœ€))𝜌.(2.15)

The following results are concerned with the degree of rational approximation of functions having 𝛼-type 𝑇(𝛼,𝑓). The functions 𝛼(π‘₯)=log𝑝π‘₯,𝑝β‰₯1, and 𝛼(π‘₯)=exp(logπ‘₯)𝛿,0<𝛿<1, satisfy the condition π›Όπœ–Ξ›. For 𝛼(π‘₯)=logπ‘₯, the parameter 𝑇(𝛼,𝑓) is called the logarithmic type of 𝑓 in 𝐺 [14].

Theorem 2.4. Let 𝑓, analytic in 𝐺, be of 𝛼-order 𝜌(𝛼,𝑓)β‰₯1, and 𝛼-type 𝑇(𝛼,𝑓),π›ΌβˆˆΞ›. Then, limsupπ‘›β†’βˆžπ›Όξ€·πΈξ‚ƒξ‚€0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑𝑛[]𝛼(𝑛)𝜌(𝛼,𝑓)≀𝑇(𝛼,𝑓).(2.16)

Proof. The inequality (2.16) holds for 𝑇(𝛼,𝑓)=∞ obviously. Now, let 𝑇(𝛼,𝑓)<∞ and ‖𝑓‖𝛾(πœ€)β†’βˆž as πœ€β†’1. Fix π‘‡ξ…ž>𝑇(𝛼,𝑓). Then, for πœ€ sufficiently close to 1, from (2.15), we have ‖𝑓‖𝛾(πœ€)β‰€π›Όβˆ’1ξ‚Έπ‘‡ξ…žξ‚ƒπ›Όξ‚€11βˆ’πœ€ξ‚ξ‚„πœŒξ‚Ή,𝜌(𝛼,𝑓)β‰‘πœŒ.(2.17)
Define 𝛿𝑛=min(1/4,1/𝑛),𝑛=1,2,…. Using [13, Equation (3.1)] with (2.17), for all sufficiently large values of 𝑛,𝑛β‰₯𝑛0, we have 𝐸0𝐸1⋯𝐸𝑛𝑑𝑛(𝑛+1)≀(𝑛+1)!𝑐𝑛𝑛8𝑛𝛼exp(𝑛+1)logβˆ’1ξ€·π‘‡ξ…ž[]𝛼(𝑛)𝜌+1𝐢(𝐾,𝑀).(2.18)
Since 𝛼 is strictly increasing, for 𝑛β‰₯𝑛0, we get 𝐸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛≀𝑐1π›Όβˆ’1ξ€Ίπ‘‡ξ…ž[]𝛼(𝑛)πœŒξ€».(2.19)
In view of (2.13), (2.19) gives limsupπ‘›β†’βˆžπ›Όξ‚ƒξ€·πΈ0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)πœŒβ‰€π‘‡ξ…ž.(2.20)
In order to complete the proof, it remains to let π‘‡ξ…ž tend to 𝑇(𝛼,𝑓).

Now, we have the following corollaries.

Corollary 2.5. With assumption of Theorem 2.4, limsupπ‘›β†’βˆžπ›Όξ€·πΈπ‘›π‘‘π‘›ξ€Έ[]𝛼(𝑛)𝜌(𝛼,𝑓)≀𝑇(𝛼,𝑓).(2.21)

The proof is immediate in view of πΈπ‘›β‰€πΈπ‘›βˆ’1≀⋯𝐸0.

For 𝑐>0, let 𝐹[]𝛼π‘₯,𝑐,𝜌=logβˆ’1[](𝑐𝛼(π‘₯)𝜌)ξ€Έ.(2.22)

Corollary 2.6. Let a function 𝑓, analytic in 𝐺, be of 𝛼-order 𝜌(𝛼,𝑓)β‰₯1, and 𝛼-type 𝑇(𝛼,𝑓) where π›ΌβˆˆΞ› is continuously differentiable on [π‘Ž,+∞) and for all 1<𝑐<∞ the function π‘₯(𝐹(π‘₯,𝑐,𝜌))ξ…ž=𝑂(1) as π‘₯β†’βˆž or is increasing and limπ‘₯β†’βˆžπ‘₯(𝐹(π‘₯,𝑐,𝜌))ξ…žπΉ(π‘₯,𝑐,𝜌)=0.(2.23)
Then, liminfπ‘›β†’βˆžπ›Όξ€·πΈπ‘›π‘‘2𝑛[]𝛼(𝑛)𝜌(𝛼,𝑓)≀𝑇(𝛼,𝑓).(2.24)

Proof. We may assume that 𝑇(𝛼,𝑓)<∞. Let liminfπ‘›β†’βˆžπ›Όξ€·πΈπ‘›π‘‘2𝑛[]𝛼(𝑛)𝜌>π‘‡ξ…ž>𝑇(𝛼,𝑓).(2.25)
For sufficiently large values of 𝑛, 𝛼𝐸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌β‰₯π›Όξ€·ξ€Ίξ€·βˆ‘exp(1/(𝑛+1))π‘›π‘˜=1πΉξ€Ίπ‘˜,π‘‡ξ…žξ€»,𝜌+𝑐[]𝛼(𝑛)𝜌.(2.26)
Since 𝐹[π‘₯,π‘‡ξ…ž,𝜌] is increasing, we get π‘›βˆ’1ξ“π‘˜=1πΉξ€Ίπ‘˜,π‘‡ξ…žξ€»β‰€ξ€œ,πœŒπ‘›1𝐹π‘₯,π‘‡ξ…žξ€»,πœŒπ‘‘π‘₯β‰€π‘›ξ“π‘˜=2πΉξ€Ίπ‘˜,π‘‡ξ…žξ€»,ξ€œ,πœŒπ‘›1𝐹π‘₯,π‘‡ξ…žξ€»ξ€Ί,πœŒπ‘‘π‘₯=𝑛𝐹𝑛,π‘‡ξ…žξ€»ξ€Ί,πœŒβˆ’πΉ1,π‘‡ξ…žξ€»βˆ’ξ€œ,πœŒπ‘›1π‘₯𝐹π‘₯,π‘‡ξ…ž,πœŒξ€»ξ€Έξ…žπ‘‘π‘₯.(2.27)
We see that 1𝑛𝐹𝑛,π‘‡ξ…žξ€»ξ€œ,πœŒπ‘›1π‘₯𝐹π‘₯,π‘‡ξ…ž,πœŒξ€»ξ€Έξ…žπ‘‘π‘₯⟢0asπ‘›βŸΆβˆž.(2.28)
Thus, βˆ‘(1/(𝑛+1))π‘›π‘˜=1πΉξ€Ίπ‘˜,π‘‡ξ…žξ€»,πœŒπΉξ€Ίπ‘›,π‘‡ξ…žξ€»,𝜌⟢1asπ‘›βŸΆβˆž.(2.29)
From this and (2.26), we get liminfπ‘›β†’βˆžπ›Όξ‚€ξ€·πΈ0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌β‰₯𝛼exp𝐹𝑛,π‘‡ξ…ž,πœŒξ€»ξ€Έ[]𝛼(𝑛)𝜌β‰₯π‘‡ξ…ž>𝑇(𝛼,𝐹)(2.30) which contradicts (2.16). Hence the proof is complete.

Remark 2.7. The function 𝛼(π‘₯)=log𝑝π‘₯,𝑝β‰₯1, and 𝛼(π‘₯)=exp(logπ‘₯)𝜌,0<𝛿<1, satisfy the assumptions of Corollary 2.6.

Acknowledgment

The author is extremely thankful to the reviewers for giving fruitful comments to improve the paper.

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