(p, q)-Growth of Meromorphic Functions and the Newton-Padé Approximant

In this paper, we have considered the generalized growth ((p, q)-order and (p, q)-type) in terms of coefficient of the development pnn 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 C (in the supremum norm). We will also show that the order of meromorphic functions puts an upper bound on the degree of convergence.


Introduction
Let f(z) � +∞ k�0 a n z n be a nonconstant entire function and M(f, r) � max |z|�r |f(z)|.
It is well known that the function r ⟼ log(M(f, r)) is an indefinitely increasing convex function of log(r). To estimate the growth of f precisely, Boas (see [1]) has introduced the concept of order, defined by the number ρ(0 ≤ ρ ≤ +∞): It is known that the order and type of an entire function f(z) � +∞ n�0 a n z n are given, respectively, by ρ f � lim sup n⟶∞ n ln n ln 1/ a n · σ f � 1 ρe lim sup n⟶∞ n a n ρ/n . (2) e 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 ρ, 0 < ρ < +∞, is said to be of type σ, 0 ≤ σ ≤ +∞, if σ � lim sup r⟶+∞ log(M(f, r)) r ρ .
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 indexpair of an entire function.
us, for p ≥ q ≥ 1, they have defined the number ρ(p, q) � lim sup r⟶+∞ log [p] (M(f, r)) Bajpai et al. have also defined the concept of the (p, q)-type σ(p, q), for b < ρ(p, q) < +∞, by In their works, the authors established the relationship of (p, q)-growth of f with respect to the coefficients a n 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 C of positive logarithmic capacity, and f be a complex function defined and bounded on K. For k ∈ N, put where the norm ‖.‖ K is the maximum on K and T n is the nth Chebyshev polynomial of the best approximation to f on K.
Bernstein showed (see [3], p. 14), for K � [− 1, 1], that there exists a constant ρ > 0 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. is 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 g of order ρ and type σ for K � [− 1, 1].
In the same way Winiarski (see [6]) generalized this result to a compact K of the complex plane C of positive logarithmic capacity, denoted by c � cap(K) as follows If K is a compact subset of the complex plane C, of positive logarithmic capacity, then if and only if f is the restriction to K of an entire function of order ρ (0 < ρ < +∞) and type σ.
Recall that the capacity of [− 1, 1] is cap([− 1, 1]) � 1/2, and the capacity of a unit disc is cap(D(0, 1)) � 1. e authors considered, respectively, the Taylor development of f with respect to the sequence (z n ) n and the development of f with respect to the sequence (ω n ) n defined by where η (n) � (η n0 , η n1 , . . . , η nn ) 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 ( ) k tends to zero depends on the growth of the entire function (order and type).
Harfaoui (see [7][8][9]) obtained a result of generalized order and type in terms of approximation in L p -norm for a compact of C n .
Recall that in the paper of Winiarski (see [6]), the author used the Cauchy inequality. e aim of this paper is to generalize the growth ((p, q)-order and (p, q)-type), studied by K. Reczek (see [10]) in terms of coefficient of the development p nn 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 C (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 R 0 (see [11]), where R 0 is the class of functions f such that on some compact circular disk Δ 0 (depending on f ) we have where r n ranges over the rational functions of type n with poles off Δ 0 .

Auxiliary Results: The Newton-Padé Approximants
First, we recall some definitions and notations which will be used later.

Definition 1.
If Δ is a compact subset of C, we define its logarithmic capacity (transfinite diameter) by where P n ranges over all polynomials of degree n with leading coefficient 1 and Let Δ be a compact subset of the complex plane C such that cap(Δ) > 0, and f is a complex function defined and bounded on Δ. For n ∈ N, put (error of best rational approximation) We will denote by R, the class of functions f, such that on some compact circular disk Δ (depending on f ) we have where r n ranges over the rational functions of type n (r n � P n /Q n ) with poles off Δ.
Remark 1. If we let r n 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]): International Journal of Mathematics and Mathematical Sciences Lemma 1 (see [2]).
With the above notations we have the following results: , For more details of these results, see [2]. Let (z n ) ∞ n�1 be a sequence of complex numbers. Suppose that f is a function holomorphic in a neighbourhood of the set (z n : 1 ≤ n < ∞). Denote by R n,m , the set of all rational functions, whose numerators and denominators are polynomials of degrees not greater than n and m, respectively. Let the function f n,m satisfy the following conditions: (1) f n,m ∈ R n,m (2) e function f − f n,m /ω n + m + 1 is holomorphic at each point z i for 1 ≤ i ≤ n + m + 1 For each couple (n, m), there exists at most one function satisfying the above conditions. It is called the (n, m)-th Newton-Padé approximant of the function f with respect to the sequence (z n ) ∞ n�1 . In the sequel, we will consider the sequences of Newton-Padé approximants (f n,m ) with m fixed and with n tending to infinity. It will be useful to simplify the notations. Denote where where z n,1 , . . . , z n,m n are the poles of the approximant f n . en, the polynomials p n and q n have no common divisors of degree higher than zero. Assume that z n,1 ≤ · · · ≤ z n,m n . (20)

The (p, q)-Growth of Meromorphic Functions
In our work we assume that p > q ≥ 3. Let M m (C) be the class of meromorphic functions whose number of poles is not greater than m. e main result of this paper is as follows: is lemma is a slight modification of the staff theorem, so we omit the proof.

International Journal of Mathematics and Mathematical Sciences
n�1 be a bounded sequence of complex numbers. Let ω be a domain containing the set z n : 1 ≤ n < ∞. Assume that there exists a limit point of the sequence (z n ) in ω. Let f be a function meromorphic in ω and holomorphic at each point of z n for 1 ≤ n < ∞. Assume that for almost every n, there exists the (n, m)-th Newton-Padé approximant f n with respect to the sequence (z n ) ∞ n�1 and that for some positive numbers μ and ] lim sup en, (1) e order of f is not greater than ].
(2) If ρ(f) � ] then the type of f is not greater than μ.
Proof. Let z ∈ C/D θ , suppose that there exists a sequence (n l ) and a neighbourhood U of the point z such that for every l the function f n l has no poles in U. en, it can be shown in the previous way that lim n⟶∞ f n (z) � lim n⟶∞ f n l (z) � f(z). So, we have shown that lim n⟶∞ f n (z) � f(z) in C/D θ except for at most m points. We can choose a number R 0 such that for every point Assume that R 0 is so great that M f (R) is an increasing function for R larger than R 0 . Let R be greater than R 0 . en, According to (25), we have Let K be an arbitrary number greater than μ. en, it follows from (21) that there exists a number n 1 ≥ n 0 such that if R is large enough, then (R θ + s) ≤ θ − m · R. It follows from (25) and (26) that where A 2 depends only on θ.
Let n R be the smallest integer greater than exp [p− 2] (K(log [q− 1] (2θ − 3m · R)) μ ). en, n R is greater than n 1 , if R is large enough, and the sum n≥n R |p nn |(θ − 3m · R) n is smaller than 1. Consequently, when R is large enough. From (29), we get where A 2 depends only on θ, μ, and K. erefore, we can show that using the general formula of a (p, q) type which implies that the order of f is not greater than μ and if ρ(f) � μ, then the type of f does not exceed K, consequently not greater than ]. is proves 1 and 2. Now, assume that the conditions (22) and (23) are satisfied. en, of course, f can be extended to a function of the class M m (C). en, we can write f � φ/Q, where φ is an entire function and Q is a polynomial of the form where k is the number of poles of f. en, 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. en, there exist a number K < ], such that 4 International Journal of Mathematics and Mathematical Sciences when |z| is large enough. Using the Cauchy formula we get from (17) and (32), for r > s. Using (17), (18), and (33), we obtain the estimation when r is large enough. Put r � exp [q− 1] (log [p− 2] (n)/K) 1/ρ . en, for almost every n, the estimation (35) is true. Hence, we derive and this contradicts the assumed equality (22). We have proved 3Â°.

Best Rational Approximation in
Terms of (p, q)-Growth e 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, (0 < ρ < ∞). en, Remark 2. A function f is entire of order at most ρ, where r n are replaced by polynomials.
where z � re iϕ and a v and b v are the zeros and poles, respectively, of f Q n − P n . Let Q n (z) � (z − z v ). Since P n is the nth Taylor polynomial to fQ n , and hence majorized by a constant times |Q n | near the origin, we have on |ω| � R : |P n (ω) ≤ (1 + | z v | )|(R const) n by the Walsh-Bernstein lemma. With the usual notation of the Nevanlinna theory, by replacing the integrand with const + log + |f| + log(const n R n (1 + |z v |)) − log R 2n+1 and integrating, using the fact that the Poisson kernel had integral 1 and is bounded for r < R,we get log fQ n − P n (z) ≤ log r 2n+1 − log R 2n+1 + const + const m(R) if |z − b v | > ϵ. Now, if f is of order ≤ ρ, we have by definition that T(R) ≤ R 1/α for any α < ρ − 1 for sufficiently large R, and we get log f Q n − P n (z) ≤ log r R 2n+1 + n log const + log R n We take R � exp q− 1 (log p− 2 (n) α ) for so small r, and the two sums will disappear, and then subtract log|Q n | to get International Journal of Mathematics and Mathematical Sciences 5 log| (f − (P n /Q n ))(z)| ≤ log(r 2n+1 2 n /exp q− 1 (log p− 2 (n) α ) n+1 ) + n log const − log δ n , except when |Q n (z)| ≤ δ n . Exponentiating, we get en, − 1 n log ϵ n (Δ, f) ≥ log r (1/n)+2 2 const for n assez grand − 1 n log ϵ n (Δ, f) ≥ exp q− 2 log p− 2 (n) α , log log p− 2 (n) α log q− 1 (− 1/n)log ϵ n (Δ, f) ≤ 1.

□
Theorem 4. Let f be a meromorphic function of finite order ρ(p, q), 0 < ρ(p, q) < ∞, and finite type. en, where c is a constant.
Proof. For the proof we use exactly the same step of eorem 3

Data Availability
No data were used to support this study.