International Journal of Mathematics and Mathematical Sciences
Volume 2019 (2019), Article ID 5792549, 6 pages
https://doi.org/10.1155/2019/5792549

Research Article

-Growth of Meromorphic Functions and the Newton-Padé Approximant

Mohammed Harfaoui, Loubna Lakhmaili, and Abdellah Mourassil

University Hassan II Mohammedia, Laboratory of Mathematics, Cryptography, Mechanical and Numerical Analysis, F. S. T., BP 146, Mohammedia 20650, Morocco

Correspondence should be addressed to Mohammed Harfaoui; mharfaoui04@yahoo.fr

Received 4 July 2019; Accepted 24 September 2019; Published 3 November 2019

Academic Editor: Vladimir V. Mityushev

Copyright © 2019 Mohammed Harfaoui et al. 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.

In this paper, we have considered the generalized growth (-order and -type) in terms of coefficient of the development given in the (n, n)-th Newton-Padé approximant of meromorphic function. We use these results to study the relationship between the degree of convergence in capacity of interpolating functions and information on the degree of convergence of best rational approximation on a compact of (in the supremum norm). We will also show that the order of meromorphic functions puts an upper bound on the degree of convergence.

1. Introduction

Let be a nonconstant entire function and .

It is well known that the function is an indefinitely increasing convex function of . To estimate the growth of f precisely, Boas (see [1]) has introduced the concept of order, defined by the number ρ:

It is known that the order and type of an entire function are given, respectively, by

The concept of type has been introduced to determine the relative growth of two functions of the same nonzero finite order. An entire function, of order , is said to be of type , if

If f is an entire function of infinite or zero order, the definition of type is not valid and the growth of such function cannot be precisely measured by the above concept. Bajpai et al. (see [2]) have introduced the concept of index-pair of an entire function. Thus, for , they have defined the number

, where if and if , where , and , for .

The function f is said to be of index-pair if is a nonzero finite number. The number is called the -order of f.

Bajpai et al. have also defined the concept of the -type , for , by

In their works, the authors established the relationship of -growth of f with respect to the coefficients in the Maclaurin series of f.

We also have many results in terms of polynomial approximation in the classical case. Let K be a compact subset of the complex plane of positive logarithmic capacity, and f be a complex function defined and bounded on K. For , put

where the norm is the maximum on K and is the nth Chebyshev polynomial of the best approximation to f on K.

Bernstein showed (see [3], p. 14), for , that there exists a constant such that

is finite, if and only if f is the restriction to K of an entire function of order ρ and some finite type.

This result has been generalized by Reddy (see [4, 5]) as follows:

if and only if f is the restriction to K of an entire function of order ρ and type σ for .

In the same way Winiarski (see [6]) generalized this result to a compact K of the complex plane of positive logarithmic capacity, denoted by as follows

If K is a compact subset of the complex plane , of positive logarithmic capacity, then

if and only if f is the restriction to K of an entire function of order ρ () and type σ.

Recall that the capacity of is , and the capacity of a unit disc is .

The authors considered, respectively, the Taylor development of f with respect to the sequence and the development of f with respect to the sequence defined by

where is the nth extremal points system of K (see [6], p. 260).

We remark that the above results suggest that the rate at which the sequence tends to zero depends on the growth of the entire function (order and type).

Harfaoui (see [7–9]) obtained a result of generalized order and type in terms of approximation in -norm for a compact of .

Recall that in the paper of Winiarski (see [6]), the author used the Cauchy inequality.

The aim of this paper is to generalize the growth (-order and -type), studied by K. Reczek (see [10]) in terms of coefficient of the development which will be defined later.

We use these results to study the relationship between the degree of convergence in capacity of interpolating functions and information on the degree of convergence of best rational approximation on a compact of (in the supremum norm). We will also show that the order of meromorphic functions puts an upper bound on the degree of convergence.

A relation between the degree of convergence (in capacity) of Padé approximants and the degree of best rational is derived for functions in Goncar’s class (see [11]), where is the class of functions f such that on some compact circular disk (depending on f) we have

where ranges over the rational functions of type n with poles off .

2. Auxiliary Results: The Newton-Padé Approximants

First, we recall some definitions and notations which will be used later.

Definition 1. If is a compact subset of , we define its logarithmic capacity (transfinite diameter) by

where ranges over all polynomials of degree n with leading coefficient 1 and .
Let be a compact subset of the complex plane such that , and f is a complex function defined and bounded on . For , put (error of best rational approximation)

We will denote by R, the class of functions f, such that on some compact circular disk (depending on f) we have

where ranges over the rational functions of type n () with poles off .

Remark 1. If we let range over the polynomials of degree n instead of over the rational functions, we get the class of entire functions.
We need the following notations and lemma which will be used in the sequel (see [2]):

Lemma 1 . (see [2]).
With the above notations we have the following results:

For more details of these results, see [2].
Let be a sequence of complex numbers. Suppose that f is a function holomorphic in a neighbourhood of the set . Denote by , the set of all rational functions, whose numerators and denominators are polynomials of degrees not greater than n and m, respectively. Let the function satisfy the following conditions:

(1)(2)The function is holomorphic at each point for

For each couple , there exists at most one function satisfying the above conditions. It is called the -th Newton-Padé approximant of the function f with respect to the sequence . In the sequel, we will consider the sequences of Newton-Padé approximants with m fixed and with n tending to infinity. It will be useful to simplify the notations. Denote

where

where are the poles of the approximant . Then, the polynomials and have no common divisors of degree higher than zero. Assume that

3. The -Growth of Meromorphic Functions

In our work we assume that .

Let be the class of meromorphic functions whose number of poles is not greater than m. The main result of this paper is as follows:

Lemma 2. Let be a bounded sequence of complex numbers and let f be a function meromorphic in , holomorphic in a neighbourhood of the set . Suppose that f has exactly m poles in C, counted with their multiplicities. Then,

(1)For almost every n there exists the approximant (2)The poles of tend to the poles of f when n tends to infinity(3) in , except for the poles of f(4)f can be extended to a function of the class

This lemma is a slight modification of the staff theorem, so we omit the proof.

Theorem 1. Let be a bounded sequence of complex numbers. Let ω be a domain containing the set . Assume that there exists a limit point of the sequence in ω. Let f be a function meromorphic in ω and holomorphic at each point of for . Assume that for almost every n, there exists the -th Newton-Padé approximant with respect to the sequence and that for some positive numbers and

Then,

(1)The order of f is not greater than .(2)If then the type of f is not greater than μ.(3)If

and if

then and .

Proof. Let , suppose that there exists a sequence and a neighbourhood U of the point z such that for every l the function has no poles in U. Then, it can be shown in the previous way that . So, we have shown that in except for at most m points. We can choose a number such that for every point

Assume that is so great that is an increasing function for R larger than . Let R be greater than . Then,

According to (25), we have

Let K be an arbitrary number greater than μ. Then, it follows from (21) that there exists a number such that

if R is large enough, then . It follows from (25) and (26) that

where depends only on θ.
Let be the smallest integer greater than . Then, is greater than , if R is large enough, and the sum is smaller than 1. Consequently,

when R is large enough. From (29), we get

where depends only on θ, μ, and K. Therefore, we can show that using the general formula of a type

which implies that the order of f is not greater than μ and if , then the type of f does not exceed K, consequently not greater than . This proves 1 and 2.
Now, assume that the conditions (22) and (23) are satisfied. Then, of course, f can be extended to a function of the class . Then, we can write , where φ is an entire function and Q is a polynomial of the form

where k is the number of poles of f. Then, of course, the order of φ is equal to the order of f and the type of φ is equal to the type of f.
Assume that either the order of f is smaller than μ or the type of f is smaller than ν. Then, there exist a number , such that

when is large enough. Using the Cauchy formula we get from (17) and (32),

for . Using (17), (18), and (33), we obtain the estimation

when r is large enough. Put . Then, for almost every n, the estimation (35) is true. Hence, we derive

and this contradicts the assumed equality (22). We have proved 3Â°.

4. Best Rational Approximation in Terms of -Growth

The aim of this section is to give a generalisation of the following theorems (see [11]).

Theorem 2. Let f be a meromorphic function of order at most ρ, (). Then,

Remark 2. A function f is entire of order at most ρ, (), if and only if

where are replaced by polynomials.

Theorem 3. Let f be a meromorphic function of order, . Then,

Proof. By the Poisson–Jensen formula, we have

where and and are the zeros and poles, respectively, of . Let . Since is the nth Taylor polynomial to , and hence majorized by a constant times near the origin, we have on by the Walsh–Bernstein lemma. With the usual notation of the Nevanlinna theory,

by replacing the integrand with and integrating, using the fact that the Poisson kernel had integral 1 and is bounded for ,we get

if . Now, if f is of order , we have by definition that for any for sufficiently large R, and we get

We take for so small r, and the two sums will disappear, and then subtract to get , except when .
Exponentiating, we get

Then,

for n assez grand

Hence, the theorem is proved.

Theorem 4. Let f be a meromorphic function of finite order, , and finite type. Then,

where c is a constant.

Proof. For the proof we use exactly the same step of Theorem 3

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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