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

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.


Introduction
Let K be a compact subset of the extended complex plane C and let E n be the error in the best uniform approximation of a function f holomorphic on K on K in the class R n of all rational functions of order n: 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 lim sup 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 The last relation implies Walsh's inequality 1.2 and the following upper estimate for lim inf n → ∞ E 1/n n : 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 order ρ 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 domain C \ 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}.Let α and β be continuous positive functions on a, ∞ satisfying the following properties: Let f 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: where It is easy to see that for the functions α x log p x, p ≥ 2, and β x x properties i -iii will hold.The following theorem gives the characterization of the rate of decay of product E 0 E 1 • • • E n for functions f having fast rates of growth of the maximum modulus.So to avoid some trivial cases, we will assume that lim ε → 1 f γ ε ∞.

Theorem 1.1. Suppose that f 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
where log x max 0, log x for x ≥ 0.
Proof.Let us assume that T α, β, f < ∞.Fix arbitrary numbers T > T > T α, β, f .For n 1, 2, . .., we set We have δ n → 0 as n → ∞.Using 1.5 for all sufficiently large values of n, n ≥ n 1 , we set 1.7 From 1.6 , we have In 1.7 , γ 2,n defined as subsets of the extended complex plane C: In view of 1.8 for n ≥ max n 0 , n 1 , we may use the inequality 3.1 of 13 in the form:

1.10
International Journal of Mathematics and Mathematical Sciences Now, using property iii , we get On letting T → T α, β, f , the proof is complete.
In the consequence of Theorem 1.1, we have the following.
Corollary 1.2.With the assumption of Theorem 1.1, the following inequalities are valid: Then, for sufficiently large values of n, we get 1.17 Since the functions α and β are increasing, 1.17 gives where c is a constant.Using ii , we get which contradicts 1.5 .Thus, 1.14 is valid.

Rational Approximation of Analytic Functions Having Slow Rates of Growth
For a function f analytic in a domain G, the type of f in G can be defined by b for α x log x and β x x: For α x log x and β x x, the property iii fails to hold.However, we have the following: and we may repeat the arguments involving 1.10 , we get

2.3
Taking T T 1, and x T 1/ρ δ n in 2.2 , for sufficiently large values of n we have We summarize the above facts in the following.

International Journal of Mathematics and Mathematical Sciences
Theorem 2.1.Let f have an order ρ > 0 and generalized type T in the domain G.Then, By the inequality Then, from the relation which contradicts the inequality 2.6 .Now, we define α-type of f to classify functions having slow rates of growth.
A continuous positive function h on a, ∞ belongs to the class Λ, if this function satisfies the following.
h is strictly increasing on a, ∞ , lim for any c > 0.
Let α ∈ Λ.We define α-order and α-type of f in G by the formulae:

2.15
The following results are concerned with the degree of rational approximation of functions having α-type T α, f .The functions α x log p x, p ≥ 1, and α x exp log x δ , 0 < δ < 1, satisfy the condition α Λ.For α x log x, the parameter T α, f is called the logarithmic type of f in G 14 .

2.19
In view of 2.13 , 2.19 gives lim sup

2.20
In order to complete the proof, it remains to let T tend to T α, f .Now, we have the following corollaries.

International Journal of Mathematics and Mathematical Sciences
Corollary 2.5.With assumption of Theorem 2.4,

2.22
Corollary 2.6.Let a function f, 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 Proof.We may assume that T α, f < ∞.Let lim inf n → ∞ α E n d 2n α n ρ > T > T α, f .

2.25
For sufficiently large values of n,

2.26
Since F x, T , ρ is increasing, we get x F x, T , ρ dx.