# p

Authors: Mohammed Harfaoui; Loubna Lakhmaili; Abdellah Mourassil
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

Letfz=∑k=0+∞anzn be a nonconstant entire function and Mf,r=maxz=rfz.It is well known that the functionr⟼logMf,r is an indefinitely increasing convex function of logr. To estimate the growth of f precisely, Boas (see [1]) has introduced the concept of order, defined by the number ρ0≤ρ≤+∞:(1)ρ=lim supr⟶+∞log logMf,rlogr.It is known that the order and type of an entire functionfz=∑n=0+∞anzn are given, respectively, by(2)ρf=lim supn⟶∞nlnnln1/an⋅σf=1ρelim supn⟶∞nanρ/n.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 orderρ,0<ρ<+∞, is said to be of type σ,0≤σ≤+∞, if(3)σ=lim supr⟶+∞logMf,rrρ.Iff 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 p≥q≥1, they have defined the number(4)ρp,q=lim supr⟶+∞logpMf,rlogqr,b≤ρp,q≤+∞, where b=0 if p>q and b=1 if p=q, where log0x=x, and logpx=loglogp−1x, for p≥1.The functionf is said to be of index-pair p,q if ρp−1,q−1 is a nonzero finite number. The number ρp,q is called the p,q-order of f.Bajpai et al. have also defined the concept of thep,q-type σp,q, for b<ρp,q<+∞, by(5)σp,q=lim supr⟶+∞logp−1Mf,rlogq−1rρp,q.In their works, the authors established the relationship ofp,q-growth of f with respect to the coefficients an in the Maclaurin series of f.We also have many results in terms of polynomial approximation in the classical case. LetK 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(6)EnK,f=f−TnK,where the norm .K is the maximum on K and Tn 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(7)limn⟶+∞n1/ρEnK,fn,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:(8)limn⟶+∞EnK,fn=ρeσ2−ρ,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=capK as followsIfK is a compact subset of the complex plane ℂ, of positive logarithmic capacity, then(9)limn⟶+∞nEnK,fρ/n=cρeρσ,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 capD0,1=1.The authors considered, respectively, the Taylor development off with respect to the sequence znn and the development of f with respect to the sequence ωnn defined by(10)ωnz=∏j=1j=nz−ηnj,n=1,2,…,where ηn=ηn0,ηn1,…,ηnn 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 sequenceEkK,fkk 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 Lp-norm for a compact of ℂn.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 (p,q-order and p,q-type), studied by K. Reczek (see [10]) in terms of coefficient of the development pnn 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 ofℂ (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 classℛ0 (see [11]), where ℛ0 is the class of functions f such that on some compact circular disk Δ0 (depending on f) we have(11)limn⟶+∞infrnsupz∈Δ0f−rnz1/n=0,where rn ranges over the rational functions of type n with poles off Δ0.


## 2. Auxiliary Results: The Newton-Padé Approximants

First, we recall some definitions and notations which will be used later.Definition 1.
IfΔ is a compact subset of ℂ, we define its logarithmic capacity (transfinite diameter) by(12)capΔ=limn⟶+∞infrnPnΔ1/n,where Pn ranges over all polynomials of degree n with leading coefficient 1 and PnΔ=supΔPnz.
LetΔ be a compact subset of the complex plane ℂ such that capΔ>0, and f is a complex function defined and bounded on Δ. For n∈ℕ, put (error of best rational approximation)(13)ϵnΔ,f=infrnf−rnΔ.
We will denote byR, the class of functions f, such that on some compact circular disk Δ (depending on f) we have(14)limn⟶+∞ϵnΔ,fn=0,where rn ranges over the rational functions of type n (rn=Pn/Qn) with poles off Δ.Remark 1.
If we letrn 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]):(15)NForp∈ℤ, putlogpx=loglogp−1x;log0x=x;Λp=∏k=1plogkx,exppx=expexpp−1x;exp0x=x,Epx=∏k=0pexpkx.Lemma 1 .
(see [2]).
With the above notations we have the following results:(16)E−px=xΛp−1x,Λ−px=xEp−1x,ddxexppx=Epxx=1Λ−p−1x,ddxlogpx=E−pxx=1Λp−1x,Ep−1x=x,ifp=0,logp−1logx−log2x+olog3x,ifp=1,2,…,limx⟶+∞expEp−2x=e,ifp=2,1,ifp≥3,limx⟶+∞expp−1Ep−2−1x1/x=e,ifp=2,1,ifp≥3.
For more details of these results, see [2].
Letznn=1∞ be a sequence of complex numbers. Suppose that f is a function holomorphic in a neighbourhood of the set zn:1≤n<∞. Denote by Rn,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 fn,m satisfy the following conditions:(1)
fn,m∈Rn,m(2)
The functionf−fn,m/ωn+m+1 is holomorphic at each point zi for 1≤i≤n+m+1
For each couplen,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 znn=1∞. In the sequel, we will consider the sequences of Newton-Padé approximants fn,m with m fixed and with n tending to infinity. It will be useful to simplify the notations. Denote(17)fn=fn,m=pnqn,where(18)pnz=∑i=0npnizi,(19)qnz=z−zn,1,…,z−zn,mn,where zn,1,…,zn,mn are the poles of the approximant fn. Then, the polynomials pn and qn have no common divisors of degree higher than zero. Assume that(20)zn,1≤⋯≤zn,mn.


## 3. Thep,q-Growth of Meromorphic Functions

In our work we assume thatp>q≥3.LetMmℂ 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.
Letznn=1∞ be a bounded sequence of complex numbers and let f be a function meromorphic in ℂ, holomorphic in a neighbourhood of the set zn:1≤n<∞. Suppose that f has exactly m poles in C, counted with their multiplicities. Then,(1)
For almost every n there exists the approximantfn(2)
The poles offn tend to the poles of f when n tends to infinity(3)
lim supn⟶∞=fz in ℂ, except for the poles of f(4)
f can be extended to a function of the class Mmℂ
This lemma is a slight modification of the staff theorem, so we omit the proof.Theorem 1.
Letznn=1∞ be a bounded sequence of complex numbers. Let ω be a domain containing the set zn:1≤n<∞. Assume that there exists a limit point of the sequence zn in ω. Let f be a function meromorphic in ω and holomorphic at each point of zn for 1≤n<∞. Assume that for almost every n, there exists the n,m-th Newton-Padé approximant fn with respect to the sequence znn=1∞ and that for some positive numbers μ and ν(21)lim supn⟶+∞logp−2nlogq−2−1/nlogpnnμ≤ν,
Then,(1)
The order off is not greater than ν.(2)
Ifρf=ν then the type of f is not greater than μ.(3)
If(22)lim supn⟶+∞logp−2nlogq−2−1/nlogpnnμ=ν,and if(23)lim supn⟶∞zn,m1/n≤1,then ρf=μ and σf=ν.Proof.
Letz∈C/Dθ, suppose that there exists a sequence nl and a neighbourhood U of the point z such that for every l the function fnl has no poles in U. Then, it can be shown in the previous way that limn⟶∞fnz=limn⟶∞fnlz=fz. So, we have shown that limn⟶∞fnz=fz in C/Dθ except for at most m points. We can choose a number R0 such that for every point(24)z∈CDθB0,R0,limn⟶∞fnz=fz.
Assume thatR0 is so great that MfR is an increasing function for R larger than R0. Let R be greater than R0. Then,(25)MfR≤MfRθ≤fn0C0,Rθ+∑n=n0+1∞fn−fn−1C0,Rθ.
According to (25), we have(26)MfR≤A1Rθn+∑n=n0+1∞pnnθ−2mnRθ+sm+n.
LetK be an arbitrary number greater than μ. Then, it follows from (21) that there exists a number n1≥n0 such that(27)pnn≤exp−nexpq−2logp−2nν1/μ,forn≥n1,if R is large enough, then Rθ+s≤θ−m⋅R. It follows from (25) and (26) that(28)MfR≤A2Rn1+θ−m⋅Rm∑n=n1+1∞exp−nexpq−2logp−2nν1/μθ−3m⋅Rn,where A2 depends only on θ.
LetnR be the smallest integer greater than expp−2Klogq−12θ−3m⋅Rμ. Then, nR is greater than n1, if R is large enough, and the sum ∑n≥nRpnnθ−3m⋅Rn is smaller than 1. Consequently,(29)MfR≤A2Rn1+θ−m⋅RmnR⋅maxexp−nexpq−2logp−2nμ1/νθ−3m⋅Rn+1,when R is large enough. From (29), we get(30)MfR≤A2Rn1+θ−m⋅RmnR⋅Rexp−expq−2logR−ϵn,where A2 depends only on θ, μ, and K. Therefore, we can show that using the general formula of a p,q type(31)Rexp−expq−2logR−ϵn≤expp−1Klogq−1Rμ−ϵ,which implies that the order of f is not greater than μ and if ρf=μ, then the type of f does not exceed K, consequently not greater than ν. 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 MmC. Then, we can write f=φ/Q, where φ is an entire function and Q is a polynomial of the form(32)Qz=t−ξ1,…,t−ξk,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 off is smaller than μ or the type of f is smaller than ν. Then, there exist a number K<ν, such that(33)φz≤expp−1Klogq−1zμ,when z is large enough. Using the Cauchy formula we get from (17) and (32),(34)pnn=12πi∫C0,rφzqnzwm+n+1zdz,for r>s. Using (17), (18), and (33), we obtain the estimation(35)pnn≤r⋅2mrm+zn,mmexpp−1Klogq−1rμminwm+n+1z:z=r,when r is large enough. Put r=expq−1logp−2n/K1/ρ. Then, for almost every n, the estimation (35) is true. Hence, we derive(36)pnn≤exp−nexpq−2logp−2K1/μ<exp−nexpq−2logp−2K1/ν,and this contradicts the assumed equality (22). We have proved 3Â°.


## 4. Best Rational Approximation in Terms ofp,q-Growth

The aim of this section is to give a generalisation of the following theorems (see [11]).Theorem 2.
Letf be a meromorphic function of order at most ρ, (0<ρ<∞). Then,(37)ϵnΔ,fn<1nα,∀α<1ρ.Remark 2.
A functionf is entire of order at most ρ, (0<ρ<∞), if and only if(38)ϵnΔ,fn<1nα,∀α<1ρ,where rn are replaced by polynomials.Theorem 3.
Letf be a meromorphic function of order≤ρp,q, 0≤ρp,q≤∞. Then,(39)lim supn⟶+∞loglogp−2nαlogq−1−1/nlogϵnΔ,f≤1.Proof.
By the Poisson–Jensen formula, we have(40)logfQn−Pnzz2n+1=12π∫02πlogfQn−PnReiθReiθ2n+1⋅R2−r2R2+r2−2rRcosθ−ϕdθ+∑bv<RlogR2−bv¯zRz−bv−∑av<RlogR2−av¯zRz−av,where z=reiϕ and av and bv are the zeros and poles, respectively, of fQn−Pn. Let Qnz=∏​z−zv. Since Pn is the nth Taylor polynomial to fQn, and hence majorized by a constant times Qn near the origin, we have on ω=R:Pnω≤∏​1+zvRconstn by the Walsh–Bernstein lemma. With the usual notation of the Nevanlinna theory,(41)mR=12π∫02πlog+fReiθdθ,NR=log∏bv<RRbv,TR=NR+mR,by replacing the integrand with const+log+f+logconstnRn∏​1+zv−logR2n+1 and integrating, using the fact that the Poisson kernel had integral 1 and is bounded for r<R,we get(42)logfQn−Pnz≤logr2n+1−logR2n+1+const+constmR+logconstnRn∏​1+zv+∑2r<bv<Rlog2Rbv+2+∑bv≤2rlog2Rϵ,if z−bv>ϵ. Now, if f is of order ≤ρ, we have by definition that TR≤R1/α for any α<ρ−1 for sufficiently large R, and we get(43)logfQn−Pnz≤logrR2n+1+nlog const+logRn∏​1+zv+constR1/α+∑bv≤2rlog2R−∑bv≤2rlogϵ.
We takeR=expq−1logp−2nα for so small r, and the two sums will disappear, and then subtract logQn to get logf−Pn/Qnz≤logr2n+12n/expq−1logp−2nαn+1+nlog const−logδn, except when Qnz≤δn.
Exponentiating, we get(44)ϵnΔ,f=f−PnQnz.
Then,(45)−1nlogϵnΔ,f≥logr1/n+22const+expq−2logp−2nα1+1/n,for n assez grand(46)−1nlogϵnΔ,f≥expq−2logp−2nα,loglogp−2nαlogq−1−1/nlogϵnΔ,f≤1.
Hence, the theorem is proved.Theorem 4.
Letf be a meromorphic function of finite orderρp,q, 0<ρp,q<∞, and finite type. Then,(47)lim supn⟶+∞logp−2αnlogq−2−1/nlogϵnΔ,f≤c,where c is a constant.Proof.
For the proof we use exactly the same step of Theorem 3



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