The present paper is concerned with the rational approximation of functions holomorphic on a domain 𝐺⊂𝐶, 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 𝐶 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)=inf𝑟∈𝑅𝑛‖𝑓−𝑟‖𝐾(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 𝐶⧵𝑀, where 𝑀 is a compact set in 𝐶 and 𝑀∩𝐾=𝜙, then limsupğ‘›â†’âˆžğ¸ğ‘›1/𝑛≤1𝑑,(1.2) where 𝑑=exp(1/𝐶(𝐾,𝑀)) and 𝐶(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: limsupğ‘›â†’âˆžî€·ğ¸1𝐸2⋯𝐸𝑛1/𝑛2≤1𝑑.(1.3)

The last relation implies Walsh's inequality (1.2) and the following upper estimate for liminfğ‘›â†’âˆžğ¸ğ‘›1/𝑛: liminfğ‘›â†’âˆžğ¸ğ‘›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 Å 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 𝐶⧵(𝐾∪𝑀) with boundary function equal to 1 on 𝜕𝑀 and to 0 on 𝜕𝐾. Here, 𝐾 and 𝑀 be disjoint compact sets with connected complements in the extended complex plane 𝐶 such that their boundaries consist of finitely many closed analytic Jordan curves. Since the domain 𝐶⧵(𝐾∪𝑀) is regular with respect to the Dirichlet problem, this problem is solvable. Let 𝑤(𝑧) be the solution which is extended by continuity to 𝐶∶𝑤(𝑧)=1 for 𝑧∈𝑀 and 𝑤(𝑧)=0 for 𝑧∈𝐾. For 0<𝜀<1, let 𝛾(𝜀)={𝑧∶𝑤(𝑧)=𝜀}.

Let 𝛼 and 𝛽 be continuous positive functions on [ğ‘Ž,∞) satisfying the following properties:(i)limğ‘¥â†’âˆžğ›¼(𝑥)=+∞, and limğ‘¥â†’âˆžğ›½(𝑥)=+∞;(ii)limğ‘¥â†’âˆž(𝛽(𝑥+𝑜(𝑥))/𝛽(𝑥))=1;(iii)𝛼−1(log(1/℧𝛽(𝑥)))/𝛼−1(log(1/â„§î…žğ›½(𝑥)))=𝑜(𝑥) as 𝑥→0 for all ℧>℧>0.

Let 𝑓 be holomorphic on 𝐺=𝐶⧵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)=limsup𝜀→1(𝛼(log‖𝑓‖𝛾(𝜀))/𝛽(log(1/(1−𝜀)))), (b)𝑇(𝛼,𝛽,𝑓)=limsup𝜀→1(𝛼(‖𝑓‖𝛾(𝜀))/[(1/(1−𝜀))]𝜌(𝛼,𝛽,𝑓)), where ‖𝑓‖𝛾(𝜀)=max𝑧∈𝛾(𝜀)|𝑓(𝑧)|.

It is easy to see that for the functions 𝛼(𝑥)=log𝑝𝑥,𝑝≥2, and 𝛽(𝑥)=𝑥 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 lim𝜀→1‖𝑓‖𝛾(𝜀)=∞.

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, limsupğ‘›â†’âˆžî‚ƒğ›½î‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌≤𝑇(𝛼,𝛽,𝑓),(1.5) where log+𝑥=max(0,log𝑥) for 𝑥≥0.

Proof. Let us assume that 𝑇(𝛼,𝛽,𝑓)<∞. Fix arbitrary numbers ğ‘‡î…žî…ž>ğ‘‡î…ž>𝑇(𝛼,𝛽,𝑓). For 𝑛=1,2,…, we set 𝛿𝑛1=min4,𝛽−1î€ºğ‘‡î…žî…žî€»exp(−𝛼(𝑛))1/𝜌.(1.6)
We have 𝛿𝑛→0 as ğ‘›â†’âˆž. Using (1.5) for all sufficiently large values of 𝑛,𝑛≥𝑛1, we set log‖𝑓‖𝛾2,𝑛≤𝛼−1𝑇logî…žî€ºğ›½î€·ğ›¿ğ‘›î€¸î€»âˆ’ğœŒî€¸î€¾=𝛼−11𝜌logî€·ğ‘‡î…ž1/𝜌𝛽𝛿𝑛.(1.7)
From (1.6), we have 𝑛=𝛼−1𝑇logî…žî…žî€ºğ›½î€·ğ›¿ğ‘›î€¸î€»âˆ’ğœŒ.(1.8)
In (1.7), 𝛾2,𝑛 defined as subsets of the extended complex plane 𝐶: 𝛾𝑘,𝑛=𝑧∶𝑤(𝑧)=𝜀𝑘,𝑛,𝐷𝑛=𝑧∶𝑤(𝑧)>𝜀0,𝑛,(1.9) where 𝜀0,𝑛=𝑘/2𝑛,𝜀1,𝑛=𝑘/𝑛,𝜀2,𝑛=1−𝛿𝑛,𝑘=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 {𝑧∶𝑤(𝑧)>𝜀2,𝑛}, respectively.
In view of (1.8) for 𝑛≥max(𝑛0,𝑛1), we may use the inequality 3.1 of [13] in the form: 𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝐶𝑑≤𝑛𝑚(𝑛+1)!𝑛8𝑛1/𝑛(𝑛+1)𝛼×exp−1𝜌log1/ğ‘‡î…ž1/𝜌𝛽𝛿𝑛𝛼−1𝜌log1/ğ‘‡î…žî…ž1/𝜌𝛽𝛿𝑛+𝛿𝑛.𝐶(𝐾,𝑀)(1.10)
Now, using property (iii), we get log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑≤𝛿𝑛𝛿𝐶(𝐾,𝑀)+𝑜𝑛.(1.11)
It gives limsupğ‘›â†’âˆžî‚ƒğ›½î‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)ğ‘‘î‚î‚„ğœŒâ‰¤ğ‘‡î…žî…ž.(1.12)
On letting ğ‘‡î…žî…žâ†’ğ‘‡(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: limsupğ‘›â†’âˆžî€ºğ›½î€·exp(𝛼(𝑛))log+𝐸𝑛1/𝑛𝑑𝜌≤𝑇(𝛼,𝛽,𝑓),(1.13)limsupğ‘›â†’âˆžî€ºğ›½î€·exp(𝛼(𝑛))log+𝐸𝑛1/𝑛𝑑2𝜌≤𝑇(𝛼,𝛽,𝑓).(1.14)

Proof. Using the fact 𝐸𝑛≤𝐸𝑛−1≤⋯≤𝐸0, we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that liminfğ‘›â†’âˆžî€ºğ›½î€·exp𝛼(𝑛)log+𝐸𝑛1/𝑛𝑑2𝜌>ğ‘‡î…ž>𝑇(𝛼,𝛽,𝑓).(1.15)
Then, for sufficiently large values of 𝑛, we get 𝛽log+𝐸𝑛1/𝑛𝑑2>î‚¸ğ‘‡î€¸î€»î…žî‚¹exp𝛼(𝑛)1/𝜌(1.16) or log+𝐸𝑛𝑑2𝑛≥𝑛𝛽−1î‚¸ğ‘‡î…žî‚¹exp(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≥∑𝑛𝑘=0𝑘𝛽−1î‚†î€ºğ‘‡î…žî€»/exp𝛼(𝑛)1/𝜌+𝑐𝑛(𝑛+1)≥𝛽−1îƒ¯î‚¸ğ‘‡î…žî‚¹exp𝛼(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)ğ‘‘î‚î‚„ğœŒâ‰¥î‚¸ğ‘‡î…žî‚¹exp𝛼(𝑛)(1.19) or liminfğ‘›â†’âˆžî‚ƒğ›½î‚€exp𝛼(𝑛)log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)î‚î‚„ğœŒâ‰¥ğ‘‡î…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(𝑥)=log𝑥 and 𝛽(𝑥)=𝑥: 𝑇=limsup𝜀→1log‖𝑓‖𝛾(𝜀)(1/1−𝜀)𝜌.(2.1)

For 𝛼(𝑥)=log𝑥 and 𝛽(𝑥)=𝑥, the property (iii) fails to hold. However, we have the following: 𝛼−1(𝑐log(1/𝛽(𝑥)))𝛼−1((𝑐+1)log(1/𝛽(𝑥)))=𝑥,(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×exp−1log1/ğ‘‡î…ž1/𝜌𝛽𝛿𝑛𝜌𝛼−1log1/ğ‘‡î…žî…ž1/𝜌𝛽𝛿𝑛𝜌+𝛿𝑛.𝐶(𝐾,𝑀)(2.3)

Taking ğ‘‡î…žî…ž=ğ‘‡î…ž+1, and 𝑥=ğ‘‡î…žî…ž1/𝜌𝛿𝑛 in (2.2), for sufficiently large values of 𝑛 we have ğ‘›î‚€î€·ğ‘‡î…žî€¸+11/𝜌log+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)ğ‘‘î‚ğœŒâ‰¤ğ‘‡î…ž(2.4) or limsupğ‘›â†’âˆžğ‘›î‚ƒlog+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌≤𝑇.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, limsupğ‘›â†’âˆžğ‘›î‚ƒlog+𝐸0𝐸1⋯𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌≤𝑇.𝑇+1(2.6)

By the inequality 𝐸𝑛≤𝐸𝑛−1≤⋯≤𝐸0, one gets the following.

Corollary 2.2. With the assumption of Theorem 2.1limsupğ‘›â†’âˆžğ‘›î€ºlog+𝐸𝑛1/𝑛𝑑𝜌≤𝑇.𝑇+1(2.7)

Theorem 2.1 also gives us the following corollary.

Corollary 2.3. With the assumption of Theorem 2.1, liminfğ‘›â†’âˆžğ‘›î€ºlog+𝐸𝑛1/𝑛𝑑2𝜌≤𝑇.𝑇+1(2.8)

Proof. Let liminfğ‘›â†’âˆžğ‘›î€ºlog+𝐸𝑛1/𝑛𝑑2𝜌>𝑇1>𝑇.𝑇+1(2.9)
Then, from the relation limğ‘›â†’âˆžâˆ‘ğ‘›ğ‘˜=0𝑘1−1/𝑇1𝑛2−1/𝑇1=12−1/𝑇1,(2.10) we obtain liminfğ‘›â†’âˆžğ‘›î‚ƒlog+𝐸1𝐸2⋯𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌≥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 ℎ on [ğ‘Ž,+∞) belongs to the class Λ, if this function satisfies the following.

ℎ is strictly increasing on [ğ‘Ž,+∞), limğ‘¥â†’âˆžâ„Ž(𝑥)=+∞,(2.12)lim𝑥→+∞ℎ(𝑐𝑥)ℎ(𝑥)=1,(2.13) for any 𝑐>0.

Let 𝛼∈Λ. We define 𝛼-order and 𝛼-type of 𝑓 in 𝐺 by the formulae: 𝜌(𝛼,𝑓)=limsup𝜀→1𝛼log‖𝑓‖𝛾(𝜀),𝛼(log(1/(1−𝜀)))(2.14)𝑇(𝛼,𝑓)=limsup𝜀→1𝛼log‖𝑓‖𝛾(𝜀)[]𝛼(1/(1−𝜀))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(𝑥)=log𝑝𝑥,𝑝≥1, and 𝛼(𝑥)=exp(log𝑥)𝛿,0<𝛿<1, satisfy the condition 𝛼𝜖Λ. For 𝛼(𝑥)=log𝑥, 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)≥1, and 𝛼-type 𝑇(𝛼,𝑓),𝛼∈Λ. Then, limsupğ‘›â†’âˆžğ›¼î€·ğ¸î‚ƒî‚€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(𝜀)→∞ as 𝜀→1. Fix ğ‘‡î…ž>𝑇(𝛼,𝑓). Then, for 𝜀 sufficiently close to 1, from (2.15), we have ‖𝑓‖𝛾(𝜀)≤𝛼−1î‚¸ğ‘‡î…žî‚ƒğ›¼î‚€11−𝜀𝜌,𝜌(𝛼,𝑓)≡𝜌.(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,𝑛≥𝑛0, we have 𝐸0𝐸1⋯𝐸𝑛𝑑𝑛(𝑛+1)≤(𝑛+1)!𝑐𝑛𝑛8𝑛𝛼exp(𝑛+1)log−1î€·ğ‘‡î…ž[]𝛼(𝑛)𝜌+1𝐶(𝐾,𝑀).(2.18)
Since 𝛼 is strictly increasing, for 𝑛≥𝑛0, we get 𝐸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛≤𝑐1𝛼−1î€ºğ‘‡î…ž[]𝛼(𝑛)𝜌.(2.19)
In view of (2.13), (2.19) gives limsupğ‘›â†’âˆžğ›¼î‚ƒî€·ğ¸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)ğœŒâ‰¤ğ‘‡î…ž.(2.20)
In order to complete the proof, it remains to let ğ‘‡î…ž tend to 𝑇(𝛼,𝑓).

Now, we have the following corollaries.

Corollary 2.5. With assumption of Theorem 2.4, limsupğ‘›â†’âˆžğ›¼î€·ğ¸ğ‘›ğ‘‘ğ‘›î€¸[]𝛼(𝑛)𝜌(𝛼,𝑓)≤𝑇(𝛼,𝑓).(2.21)

The proof is immediate in view of 𝐸𝑛≤𝐸𝑛−1≤⋯𝐸0.

For 𝑐>0, let 𝐹[]𝛼𝑥,𝑐,𝜌=log−1[](𝑐𝛼(𝑥)𝜌).(2.22)

Corollary 2.6. Let a function 𝑓, analytic in 𝐺, be of 𝛼-order 𝜌(𝛼,𝑓)≥1, and 𝛼-type 𝑇(𝛼,𝑓) where 𝛼∈Λ is continuously differentiable on [ğ‘Ž,+∞) and for all 1<𝑐<∞ the function 𝑥(𝐹(𝑥,𝑐,𝜌))=𝑂(1) as ğ‘¥â†’âˆž or is increasing and limğ‘¥â†’âˆžğ‘¥(𝐹(𝑥,𝑐,𝜌))î…žğ¹(𝑥,𝑐,𝜌)=0.(2.23)
Then, liminfğ‘›â†’âˆžğ›¼î€·ğ¸ğ‘›ğ‘‘2𝑛[]𝛼(𝑛)𝜌(𝛼,𝑓)≤𝑇(𝛼,𝑓).(2.24)

Proof. We may assume that 𝑇(𝛼,𝑓)<∞. Let liminfğ‘›â†’âˆžğ›¼î€·ğ¸ğ‘›ğ‘‘2𝑛[]𝛼(𝑛)𝜌>ğ‘‡î…ž>𝑇(𝛼,𝑓).(2.25)
For sufficiently large values of 𝑛, 𝛼𝐸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌≥𝛼∑exp(1/(𝑛+1))𝑛𝑘=1𝐹𝑘,ğ‘‡î…žî€»,𝜌+𝑐[]𝛼(𝑛)𝜌.(2.26)
Since 𝐹[𝑥,ğ‘‡î…ž,𝜌] is increasing, we get 𝑛−1𝑘=1𝐹𝑘,ğ‘‡î…žî€»â‰¤î€œ,𝜌𝑛1𝐹𝑥,ğ‘‡î…žî€»,𝜌𝑑𝑥≤𝑛𝑘=2𝐹𝑘,ğ‘‡î…žî€»,,𝜌𝑛1𝐹𝑥,ğ‘‡î…žî€»î€º,𝜌𝑑𝑥=𝑛𝐹𝑛,ğ‘‡î…žî€»î€º,𝜌−𝐹1,ğ‘‡î…žî€»âˆ’î€œ,𝜌𝑛1𝑥𝐹𝑥,ğ‘‡î…ž,ğœŒî€»î€¸î…žğ‘‘ğ‘¥.(2.27)
We see that 1𝑛𝐹𝑛,ğ‘‡î…žî€»î€œ,𝜌𝑛1𝑥𝐹𝑥,ğ‘‡î…ž,ğœŒî€»î€¸î…žğ‘‘ğ‘¥âŸ¶0asğ‘›âŸ¶âˆž.(2.28)
Thus, ∑(1/(𝑛+1))𝑛𝑘=1𝐹𝑘,ğ‘‡î…žî€»,𝜌𝐹𝑛,ğ‘‡î…žî€»,𝜌⟶1asğ‘›âŸ¶âˆž.(2.29)
From this and (2.26), we get liminfğ‘›â†’âˆžğ›¼î‚€î€·ğ¸0𝐸1⋯𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌≥𝛼exp𝐹𝑛,ğ‘‡î…ž,𝜌[]𝛼(𝑛)ğœŒâ‰¥ğ‘‡î…ž>𝑇(𝛼,𝐹)(2.30) which contradicts (2.16). Hence the proof is complete.

Remark 2.7. The function 𝛼(𝑥)=log𝑝𝑥,𝑝≥1, and 𝛼(𝑥)=exp(log𝑥)𝜌,0<𝛿<1, satisfy the assumptions of Corollary 2.6.


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