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

Authors: Devendra Kumar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The present paper is concerned with the rational approximation of functions holomorphic on a domainG⊂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.


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## Body

## 1. Introduction

LetK 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:
(1.1)En=En(f,K)=inf⁡r∈Rn‖f-r‖K
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
(1.2)limsup⁡n→∞En1/n≤1d,
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:
(1.3)limsup⁡n→∞(E1E2⋯En)1/n2≤1d.The last relation implies Walsh's inequality (1.2) and the following upper estimate for liminfn→∞En1/n:
(1.4)liminf⁡n→∞En1/n≤1d2.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 orderρ(α,β,f) of the rate of growth of entire functions f was introduced by Šeremeta [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 domainC∖(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}.Letα and β be continuous positive functions on [a,∞) satisfying the following properties:(i)
limx→∞α(x)=+∞, and limx→∞β(x)=+∞;(ii)
limx→∞(β(x+o(x))/β(x))=1;(iii)
α-1(log(1/℧β(x)))/α-1(log(1/℧′β(x)))=o(x) as x→0 for all ℧′>℧>0.Letf 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 functionsα(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 thatf 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,
(1.5)limsup⁡n→∞exp⁡α(n)[β(log⁡+(((E0E1⋯En)1/n(n+1)d)))]ρ≤T(α,β,f),
where log+x=max(0,logx) for x≥0.Proof.
Let us assume thatT(α,β,f)<∞. Fix arbitrary numbers T′′>T′>T(α,β,f). For n=1,2,…, we set
(1.6)δn=min⁡(14,β-1[T′′exp⁡(-α(n))]1/ρ).
We haveδn→0 as n→∞. Using (1.5) for all sufficiently large values of n,n≥n1, we set
(1.7)log⁡‖f‖γ2,n≤α-1{log⁡(T′[β(δn)]-ρ)}=α-1{ρlog⁡(1(T′1/ρβ(δn)))}.
From (1.6), we have
(1.8)n=α-1{log⁡(T′′[β(δn)]-ρ)}.
In (1.7), γ2,n defined as subsets of the extended complex plane C:
(1.9)γk,n={z:w(z)=εk,n},Dn={z:w(z)>ε0,n},
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:
(1.10)(E0E1⋯En)1/n(n+1)d≤(Cnm(n+1)!n8n)1/n(n+1)×exp⁡(α-1(ρlog⁡(1/T′1/ρβ(δn)))α-1(ρlog⁡(1/T′′1/ρβ(δn)))+δnC(K,M)).
Now, using property (iii), we get(1.11)log⁡+((E0E1⋯En)1/n(n+1)d)≤δnC(K,M)+o(δn).
It gives(1.12)limsup⁡n→∞exp⁡α(n)[β(log⁡+(E0E1⋯En)1/n(n+1)d)]ρ≤T′′.
On lettingT′′→T(α,β,f), the proof is complete.In the consequence of Theorem1.1, we have the following.Corollary 1.2.
With the assumption of Theorem1.1, the following inequalities are valid:
(1.13)limsup⁡n→∞exp⁡(α(n))[β(log⁡+En1/nd)]ρ≤T(α,β,f),(1.14)limsup⁡n→∞exp⁡(α(n))[β(log⁡+En1/nd2)]ρ≤T(α,β,f).Proof.
Using the factEn≤En-1≤⋯≤E0, we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that
(1.15)liminf⁡n→∞exp⁡α(n)[β(log⁡+(En1/nd2))]ρ>T′>T(α,β,f).
Then, for sufficiently large values ofn, we get
(1.16)β[log⁡+(En1/nd2)]>[T′exp⁡α(n)]1/ρ
or
(1.17)log⁡+End2n≥nβ-1[T′exp⁡(α(n))]1/ρ.
Since the functionsα and β are increasing, (1.17) gives
(1.18)log⁡+((E0E1⋯En)1/n(n+1)d)≥(∑k=0nkβ-1{[T′/exp⁡α(n)]1/ρ+c})n(n+1)≥β-1{[T′exp⁡α(n)]1/ρ+cn(n+1)},
where c is a constant. Using (ii), we get
(1.19)[β(log⁡+(E0E1⋯En)1/n(n+1)d)]ρ≥[T′exp⁡α(n)]
or
(1.20)liminf⁡n→∞exp⁡α(n)[β(log⁡+(E0E1⋯En)1/n(n+1))]ρ≥T′>T(α,β,f)
which contradicts (1.5). Thus, (1.14) is valid.


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

For a functionf analytic in a domain G, the type of f in G can be defined by (b) for α(x)=logx and β(x)=x:
(2.1)T=limsup⁡ε→1log⁡‖f‖γ(ε)(1/1-ε)ρ.Forα(x)=logx and β(x)=x, the property (iii) fails to hold. However, we have the following:
(2.2)α-1(clog⁡(1/β(x)))α-1((c+1)log⁡(1/β(x)))=x,
and we may repeat the arguments involving (1.10), we get
(2.3)(E0E1⋯En)1/n(n+1)d≤(cn(n+1)!n8n)1/n(n+1)×exp⁡(α-1(log⁡[1/T′1/ρβ(δn)]ρ)α-1(log⁡[1/T′′1/ρβ(δn)]ρ)+δnC(K,M)).TakingT′′=T′+1, and x=T′′1/ρδn in (2.2), for sufficiently large values of n we have
(2.4)n((T′+1)1/ρlog⁡+(E0E1⋯En)1/n(n+1)d)ρ≤T′
or
(2.5)limsup⁡n→∞n[log⁡+(E0E1⋯En)1/n(n+1)d]ρ≤TT+1.We summarize the above facts in the following.Theorem 2.1.
Letf have an order ρ>0 and generalized type T in the domain G. Then,
(2.6)limsup⁡n→∞n[log⁡+(E0E1⋯En)1/n(n+1)d]ρ≤TT+1.By the inequalityEn≤En-1≤⋯≤E0, one gets the following.Corollary 2.2.
With the assumption of Theorem2.1(2.7)limsup⁡n→∞n[log⁡+(En1/nd)]ρ≤TT+1.Theorem2.1 also gives us the following corollary.Corollary 2.3.
With the assumption of Theorem2.1,
(2.8)liminf⁡n→∞n[log⁡+(En1/nd2)]ρ≤TT+1.Proof.
Let(2.9)liminf⁡n→∞n[log⁡+(En1/nd2)]ρ>T1>TT+1.
Then, from the relation(2.10)lim⁡n→∞∑k=0nk1-1/T1n2-1/T1=12-1/T1,
we obtain
(2.11)liminf⁡n→∞n[log⁡+((E1E2⋯En)1/n(n+1)d)]ρ≥T1>TT+1,
which contradicts the inequality (2.6).Now, we defineα-type of f to classify functions having slow rates of growth.A continuous positive functionh on [a,+∞) belongs to the class Λ, if this function satisfies the following.h is strictly increasing on [a,+∞),
(2.12)lim⁡x→∞h(x)=+∞,(2.13)lim⁡x→+∞h(cx)h(x)=1,
for any c>0.Letα∈Λ. We define α-order and α-type of f in G by the formulae:
(2.14)ρ(α,f)=limsup⁡ε→1α(log⁡‖f‖γ(ε))α(log⁡(1/(1-ε))),(2.15)T(α,f)=limsup⁡ε→1α(log⁡‖f‖γ(ε))[α(1/(1-ε))]ρ.The following results are concerned with the degree of rational approximation of functions havingα-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.
Letf, analytic in G, be of α-order ρ(α,f)≥1, and α-type T(α,f),α∈Λ. Then,
(2.16)limsup⁡n→∞α[((E0E1⋯En)1/n(n+1)dn)][α(n)]ρ(α,f)≤T(α,f).Proof.
The inequality (2.16) holds for T(α,f)=∞ obviously. Now, let T(α,f)<∞ and ∥f∥γ(ε)→∞ as ε→1. Fix T′>T(α,f). Then, for ε sufficiently close to 1, from (2.15), we have
(2.17)‖f‖γ(ε)≤α-1[T′[α(11-ε)]ρ],ρ(α,f)≡ρ.
Defineδn=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
(2.18)E0E1⋯Endn(n+1)≤(n+1)!cnn8nexp⁡(n+1)(log⁡(α-1(T′[α(n)]ρ))+1C(K,M)).
Sinceα is strictly increasing, for n≥n0, we get
(2.19)(E0E1⋯En)1/n+1dn≤c1α-1[T′[α(n)]ρ].
In view of (2.13), (2.19) gives
(2.20)limsup⁡n→∞α[(E0E1⋯En)1/n+1dn][α(n)]ρ≤T′.
In order to complete the proof, it remains to letT′ tend to T(α,f).Now, we have the following corollaries.Corollary 2.5.
With assumption of Theorem2.4,
(2.21)limsup⁡n→∞α(Endn)[α(n)]ρ(α,f)≤T(α,f).The proof is immediate in view ofEn≤En-1≤⋯E0.Forc>0, let
(2.22)F[x,c,ρ]=log⁡(α-1(c[α(x)]ρ)).Corollary 2.6.
Let a functionf, analytic in G, be of α-order ρ(α,f)≥1, and α-type T(α,f) where α∈Λ is continuously differentiable on [a,+∞) and for all 1<c<∞ the function x(F(x,c,ρ))′=O(1) as x→∞ or is increasing and
(2.23)lim⁡x→∞x(F(x,c,ρ))′F(x,c,ρ)=0.
Then,(2.24)liminf⁡n→∞α(End2n)[α(n)]ρ(α,f)≤T(α,f).Proof.
We may assume thatT(α,f)<∞. Let
(2.25)liminf⁡n→∞α(End2n)[α(n)]ρ>T′>T(α,f).
For sufficiently large values ofn,
(2.26)α((E0E1⋯En)1/n+1dn)[α(n)]ρ≥α(exp⁡[(1/(n+1))(∑k=1nF[k,T′,ρ]+c)])[α(n)]ρ.
SinceF[x,T′,ρ] is increasing, we get
(2.27)∑k=1n-1‍F[k,T′,ρ]≤∫1n‍F[x,T′,ρ]dx≤∑k=2n‍F[k,T′,ρ],∫1n‍F[x,T′,ρ]dx=nF[n,T′,ρ]-F[1,T′,ρ]-∫1n‍x(F[x,T′,ρ])′dx.
We see that(2.28)1nF[n,T′,ρ]∫1n‍x(F[x,T′,ρ])′dx⟶0asn⟶∞.
Thus,(2.29)(1/(n+1))∑k=1nF[k,T′,ρ]F[n,T′,ρ]⟶1asn⟶∞.
From this and (2.26), we get
(2.30)liminf⁡n→∞α((E0E1⋯En)1/n+1dn)[α(n)]ρ≥α(exp⁡F[n,T′,ρ])[α(n)]ρ≥T′>T(α,F)
which contradicts (2.16). Hence the proof is complete.Remark 2.7.
The functionα(x)=logpx,p≥1, and α(x)=exp(logx)ρ,0<δ<1, satisfy the assumptions of Corollary 2.6.



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