On the Rational Approximation of

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Volume 2012, Article ID 908104,10pages doi:10.1155/2012/908104

Research Article

On the Rational Approximation of

Analytic Functions Having Generalized Types of Rate of Growth

Devendra Kumar

Department of Mathematics, Research and Post Graduate Studies, M.M.H. College, Model Town, Ghaziabad 201001, UP, India

Correspondence should be addressed to Devendra Kumar,d kumar001@rediﬀmail.com Received 21 March 2012; Accepted 2 July 2012

Academic Editor: Narendra Govil

Copyrightq2012 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.

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

1. Introduction

LetK be a compact subset of the extended complex planeCand letEn be the error in the best uniform approximation of a functionf holomorphic onKonKin the classRn of all rational functions of ordern:

EnEn

f, K inf

r∈Rn

f−r

K 1.1

for each nonnegative integern, where · Kis the supremum norm onK.

In view of Walsh’s inequality1, iffis holomorphic onC\M, whereMis a compact set inCandM∩Kφ, then

lim sup

n→ ∞ E_n^1/n≤ 1

d, 1.2

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wheredexp1/CK, MandCK, Mis the capacity of the condenserK, M,see2–4, for the definition and properties of the capacity.

The theory of Hankel operators permits one5–7to estimate the order of decrease of the productE1E2· · ·En:

lim sup

n→ ∞ E1E2· · ·En^1/n² ≤ 1

d. 1.3

The last relation implies Walsh’s inequality1.2and the following upper estimate for lim inf_{n→ ∞}En^1/n:

lim inf

n→ ∞ E_n^1/n≤ 1

d². 1.4

The present paper is concerned to results that make the inequalities1.2,1.3and 1.4more precise for analytic functions having generalized types of the rate of growth of the maximum modulus in the domain of analyticity off.

The generalized order ρα, β, f of the rate of growth of entire functions f was introduced by ˇSeremeta 8, who obtained a characterization of ρα, β, f in terms of the coeﬃcients of the power series off. In8, the relationship between the generalized order of entire functionsf 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 papers9–12. The degree of rational approximation of entire functions of a finite generalized order is investigated in6.

Now let us consider the Dirichlet problem in the domainC\K∪Mwith boundary function equal to 1 on ∂Mand to 0 on∂K. Here, K andM be disjoint compact sets with connected complements in the extended complex planeCsuch that their boundaries consist of finitely many closed analytic Jordan curves. Since the domainC\K∪Mis regular with respect to the Dirichlet problem, this problem is solvable. Letwzbe the solution which is extended by continuity toC:wz 1 forz∈Mandwz 0 forz∈K. For 0< ε <1, let γε {z:wz ε}.

Let α and β be continuous positive functions on a,∞ satisfying the following properties:

ilimx→ ∞αx ∞, and limx→ ∞βx ∞;

iilim_x_{→ ∞}βx ox/βx 1;

iiiα⁻¹log1/βx/α⁻¹log1/βx oxasx → 0 for all>>0.

Letf be holomorphic onG C\M. We define the generalized orderρα, β, fand generalized typeTα, β, foffin the domainGby the formulae:

aρα, β, f lim sup_ε_→1αlogfγε/βlog1/1−ε, bTα, β, f lim sup_ε→₁αfγε/1/1−ε^ρα,β,f, wherefγ_εmax_z∈γε|fz|.

It is easy to see that for the functionsαx log_px, p ≥ 2, andβx xproperties i–iiiwill hold. The following theorem gives the characterization of the rate of decay of productE0E1· · ·Enfor functionsfhaving fast rates of growth of the maximum modulus. So to avoid some trivial cases, we will assume that lim_ε_→1fγε∞.

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Theorem 1.1. Suppose thatf is holomorphic onG, αand βsatisfy conditions (i)–(iii), andf has generalized orderρα, β, f>0 and generalized typeTα, β, fin the domainG. Then,

lim sup

n→ ∞ expαn β

log

E0E1· · ·En^{1/nn 1}d_ρ

≤T α, β, f

, 1.5

where log xmax0,logxforx≥0.

Proof. Let us assume that Tα, β, f < ∞. Fix arbitrary numbers T > T > Tα, β, f. For n1,2, . . ., we set

δnmin 1 4, β⁻¹

Texp−αn_1/ρ

. 1.6

We haveδn → 0 asn → ∞. Using1.5for all suﬃciently large values ofn, n≥n1, we set

logf

γ2,n≤α⁻¹ log

T

βδn_−ρ α⁻¹

ρlog

1 T^1/ρβδn

.

1.7

From1.6, we have

nα⁻¹ log

T

βδn_−ρ

. 1.8

In1.7,γ2,ndefined as subsets of the extended complex planeC:

γk,n{z:wz εk,n},

Dn{z:wz> ε0,n}, 1.9

whereε0,n k/2n, ε1,n k/n, ε2,n1−δn, k 0,1,2, andn 1,2, . . .. It is given13that γ0,n, γ1,n, andγ2,n, n 1,2, . . ., consist of finitely many closed analytic curves whose lengths are bounded from above by a positive quantity not depending onn. It is assumed thatγ0,n

andγ2,nare positively oriented with respect toDnand{z:wz> ε2,n}, respectively.

In view of1.8forn≥maxn0, n1, we may use the inequality 3.1 of13in the form:

E0E1· · ·En^{1/nn 1}d≤

Cⁿmn 1!n⁸ⁿ_{1/nn 1}

×exp

⎛

⎜⎝α⁻¹ ρlog

1/T^1/ρβδn α⁻¹

ρlog

1/T^1/ρβδn δn

CK, M

⎞

⎟⎠.

1.10

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Now, using propertyiii, we get

log

E0E1· · ·En^{1/nn 1}d

≤ δn

CK, M oδn. 1.11

It gives

lim sup

n→ ∞ expαn β

log E0E1· · ·En^{1/nn 1}d_ρ

≤T. 1.12

On lettingT → Tα, β, f, the proof is complete.

In the consequence ofTheorem 1.1, we have the following.

Corollary 1.2. With the assumption ofTheorem 1.1, the following inequalities are valid:

lim sup

n→ ∞ expαn β

log E^1/n_n d_ρ

≤T α, β, f

, 1.13

lim sup

n→ ∞ expαn β

log E^1/n_n d²_ρ

≤T α, β, f

. 1.14

Proof. Using the factEn≤En−1 ≤ · · · ≤E0, we obtain1.13immediately from1.5. To prove 1.14, let us suppose that

lim inf

n→ ∞ expαn β

log

E^1/n_n d²_ρ

> T> T α, β, f

. 1.15

Then, for suﬃciently large values ofn, we get

β log

E^1/n_n d²

>

T expαn

1/ρ

1.16

or

log End²ⁿ≥nβ⁻¹ T

expαn 1/ρ

. 1.17

Since the functionsαandβare increasing,1.17gives

log

E0E1· · ·En^{1/nn 1}d

≥ _n

k0kβ⁻¹

T/expαn1/ρ

c

nn 1

≥β⁻¹

T expαn

1/ρ

c

nn 1

,

1.18

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wherecis a constant. Usingii, we get

β

log E0E1· · ·En^{1/nn 1}d_ρ

≥ T

expαn

1.19

or

lim inf

n→ ∞ expαn β

log E0E1· · ·En^{1/nn 1}_ρ

≥T> T α, β, f

1.20

which contradicts1.5. Thus,1.14is valid.

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

For a functionf analytic in a domainG, the type offinGcan be defined bybforαx logxandβx x:

Tlim sup

ε→1

logf

γε

1/1−ε^ρ. 2.1 Forαx logxandβx x, the propertyiiifails to hold. However, we have the following:

α⁻¹ clog

1/βx

α⁻¹

c 1log

1/βx x, 2.2

and we may repeat the arguments involving1.10, we get E0E1· · ·En^{1/nn 1}d≤

cⁿn 1!n⁸ⁿ_{1/nn 1}

×exp

⎛

⎜⎝α⁻¹ log

1/T^1/ρβδn_ρ α⁻¹

log

1/T^1/ρβδn_ρ δn

CK, M

⎞

⎟⎠.

2.3

TakingTT 1, andxT^1/ρδnin2.2, for suﬃciently large values ofnwe have n

T 1_1/ρ

log E0E1· · ·En^{1/nn 1}d

ρ≤T 2.4

or

lim sup

n→ ∞ n

log E0E1· · ·En^{1/nn 1}dρ

≤ T

T 1. 2.5

We summarize the above facts in the following.

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Theorem 2.1. Letfhave an orderρ >0 and generalized typeTin the domainG. Then,

lim sup

n→ ∞ n

log E0E1· · ·En^{1/nn 1}dρ

≤ T

T 1. 2.6

By the inequalityEn≤En−1≤ · · · ≤E0, one gets the following.

Corollary 2.2. With the assumption ofTheorem 2.1 lim sup

n→ ∞ n log

E^1/n_n d_ρ

≤ T

T 1. 2.7

Theorem 2.1also gives us the following corollary.

Corollary 2.3. With the assumption ofTheorem 2.1, lim inf

n→ ∞ n log

E^1/n_n d²ρ

≤ T

T 1. 2.8

Proof. Let

lim inf

n→ ∞ n log

E^1/n_n d²_ρ

> T1> T

T 1. 2.9

Then, from the relation

nlim→ ∞

_n

k0k^1−1/T¹

n^2−1/T¹ 1

2−1/T1, 2.10

we obtain

lim inf

n→ ∞ n log

E1E2· · ·En^{1/nn 1}d_ρ

≥T1> T

T 1, 2.11

which contradicts the inequality2.6.

Now, we defineα-type offto classify functions having slow rates of growth.

A continuous positive functionhona, ∞ belongs to the classΛ, if this function satisfies the following.

his strictly increasing ona, ∞,

xlim→ ∞hx ∞, 2.12

xlim→ ∞

hcx

hx 1, 2.13

for anyc >0.

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Letα∈Λ. We defineα-order andα-type offinGby the formulae:

ρ α, f

lim sup

ε→1

α logf

γε

α

log1/1−ε, 2.14

T α, f

lim sup

ε→1

α logf

γε

α1/1−ε^ρ. 2.15 The following results are concerned with the degree of rational approximation of functions having α-type Tα, f. The functions αx log_px, p ≥ 1, and αx explogx^δ, 0 < δ < 1, satisfy the condition αΛ. For αx logx, the parameterTα, f is called the logarithmic type offinG14.

Theorem 2.4. Letf, analytic inG, be ofα-orderρα, f≥1, andα-typeTα, f, α∈Λ. Then,

lim sup

n→ ∞

α

E0E1· · ·En^{1/nn 1}dⁿ αn^ρα,f ≤T

α, f

. 2.16

Proof. The inequality 2.16 holds for Tα, f ∞ obviously. Now, let Tα, f < ∞ and fγε → ∞ asε → 1. Fix T > Tα, f. Then, for ε suﬃciently close to 1, from 2.15, we have

f

γε≤α⁻¹

T

α 1 1−ε

ρ

, ρ

α, f

≡ρ. 2.17

Defineδnmin1/4, 1/n, n1,2, . . .. Using13, Equation3.1with2.17, for all suﬃciently large values ofn, n≥n0, we have

E0E1· · ·End^{nn 1}≤n 1!cⁿn⁸ⁿexpn 1 log α⁻¹

Tαn^ρ 1

CK, M

. 2.18

Sinceαis strictly increasing, forn≥n0, we get E0E1· · ·En^{1/n 1}dⁿ≤c1α⁻¹

Tαn^ρ

. 2.19

In view of2.13,2.19gives

lim sup

n→ ∞

α

E0E1· · ·En^{1/n 1}dⁿ

αn^ρ ≤T. 2.20 In order to complete the proof, it remains to letTtend toTα, f.

Now, we have the following corollaries.

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Corollary 2.5. With assumption ofTheorem 2.4, lim sup

n→ ∞

αEndⁿ αn^ρα,f ≤T

α, f

. 2.21

The proof is immediate in view ofEn≤E_n−1 ≤ · · ·E0. Forc >0, let

F x, c, ρ

log α⁻¹

cαx^ρ

. 2.22

Corollary 2.6. Let a functionf, analytic inG, be ofα-orderρα, f≥ 1, andα-typeTα, fwhere α∈Λis continuously diﬀerentiable ona, ∞and for all 1 < c <∞the functionxFx, c, ρ O1asx → ∞or is increasing and

xlim→ ∞

x F

x, c, ρ F

x, c, ρ 0. 2.23

Then,

lim inf

n→ ∞

α End²ⁿ αn^ρα,f ≤T

α, f

. 2.24

Proof. We may assume thatTα, f<∞. Let lim inf

n→ ∞

α End²ⁿ

αn^ρ > T> T α, f

. 2.25

For suﬃciently large values ofn, α

E0E1· · ·En^{1/n 1}dⁿ αn^ρ ≥ α

exp

1/n 1_n

k1F k, T, ρ

c

αn^ρ . 2.26

SinceFx, T, ρis increasing, we get n−1

k1

F k, T, ρ

≤ _n

1

F x, T, ρ

dx≤ⁿ

k2

F k, T, ρ

, _n

1

F x, T, ρ

dxnF n, T, ρ

−F 1, T, ρ

− _n

1

x F

x, T, ρ dx.

2.27

We see that

1 nF

n, T, ρ _n

1

x F

x, T, ρ

dx−→0 asn−→ ∞. 2.28

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Thus,

1/n 1_n

k1F k, T, ρ F

n, T, ρ −→1 asn−→ ∞. 2.29

From this and2.26, we get

lim inf

n→ ∞

α

E0E1· · ·En^{1/n 1}dⁿ αn^ρ ≥ α

expF

n, T, ρ

αn^ρ ≥T> Tα, F 2.30

which contradicts2.16. Hence the proof is complete.

Remark 2.7. The functionαx log_px, p≥ 1, andαx explogx^ρ, 0 < δ <1, satisfy the assumptions ofCorollary 2.6.

Acknowledgment

The author is extremely thankful to the reviewers for giving fruitful comments to improve the paper.

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