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

Abstract

The present paper is concerned with the rational approximation of functions holomorphic on a domain , 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 be a compact subset of the extended complex plane and let be the error in the best uniform approximation of a function (holomorphic on ) on in the class of all rational functions of order : for each nonnegative integer , where is the supremum norm on .

In view of Walsh's inequality [1], if is holomorphic on , where is a compact set in and , then where and is the capacity of the condenser , (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 :

The last relation implies Walsh's inequality (1.2) and the following upper estimate for :

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 .

The generalized order of the rate of growth of entire functions was introduced by Šeremeta [8], who obtained a characterization of in terms of the coefficients of the power series of . In [8], the relationship between the generalized order of entire functions and the degree of polynomial approximation of 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 with boundary function equal to 1 on and to 0 on . Here, and be disjoint compact sets with connected complements in the extended complex plane such that their boundaries consist of finitely many closed analytic Jordan curves. Since the domain is regular with respect to the Dirichlet problem, this problem is solvable. Let be the solution which is extended by continuity to for and for . For , let .

Let and be continuous positive functions on satisfying the following properties:(i), and ;(ii);(iii) as for all .

Let be holomorphic on . We define the generalized order and generalized type of in the domain by the formulae:(a), (b), where .

It is easy to see that for the functions , and properties (i)–(iii) will hold. The following theorem gives the characterization of the rate of decay of product for functions having fast rates of growth of the maximum modulus. So to avoid some trivial cases, we will assume that .

Theorem 1.1. Suppose that is holomorphic on and satisfy conditions (i)–(iii), and has generalized order and generalized type in the domain . Then, where for .

Proof. Let us assume that . Fix arbitrary numbers . For , we set
We have as . Using (1.5) for all sufficiently large values of , we set
From (1.6), we have
In (1.7), defined as subsets of the extended complex plane : where , and . It is given [13] that , and , consist of finitely many closed analytic curves whose lengths are bounded from above by a positive quantity not depending on . It is assumed that and are positively oriented with respect to and , respectively.
In view of (1.8) for , we may use the inequality 3.1 of [13] in the form:
Now, using property (iii), we get
It gives
On letting , 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:

Proof. Using the fact , we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that
Then, for sufficiently large values of , we get or
Since the functions and are increasing, (1.17) gives where is a constant. Using (ii), we get or which contradicts (1.5). Thus, (1.14) is valid.

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

For a function analytic in a domain , the type of in can be defined by (b) for and :

For and , the property (iii) fails to hold. However, we have the following: and we may repeat the arguments involving (1.10), we get

Taking , and in (2.2), for sufficiently large values of we have or

We summarize the above facts in the following.

Theorem 2.1. Let have an order and generalized type in the domain . Then,

By the inequality , one gets the following.

Corollary 2.2. With the assumption of Theorem 2.1

Theorem 2.1 also gives us the following corollary.

Corollary 2.3. With the assumption of Theorem 2.1,

Proof. Let
Then, from the relation we obtain which contradicts the inequality (2.6).

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

A continuous positive function on belongs to the class , if this function satisfies the following.

is strictly increasing on , for any .

Let . We define order and type of in by the formulae:

The following results are concerned with the degree of rational approximation of functions having type . The functions , and , satisfy the condition . For , the parameter is called the logarithmic type of in [14].

Theorem 2.4. Let , analytic in , be of order , and type . Then,

Proof. The inequality (2.16) holds for obviously. Now, let and as . Fix . Then, for sufficiently close to , from (2.15), we have
Define . Using [13, Equation (3.1)] with (2.17), for all sufficiently large values of , we have
Since is strictly increasing, for , we get
In view of (2.13), (2.19) gives
In order to complete the proof, it remains to let tend to .

Now, we have the following corollaries.

Corollary 2.5. With assumption of Theorem 2.4,

The proof is immediate in view of .

For , let

Corollary 2.6. Let a function , analytic in , be of order , and type where is continuously differentiable on and for all the function as or is increasing and
Then,

Proof. We may assume that . Let
For sufficiently large values of ,
Since is increasing, we get
We see that
Thus,
From this and (2.26), we get which contradicts (2.16). Hence the proof is complete.

Remark 2.7. The function , and , 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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Copyright

Copyright © 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.