Approximation of Entire Functions of Slow Growth

In the present paper, we study the polynomial approximation of entire functions in Banach spaces (B(p, q, κ) space, Hardy space and Bergman space). The coefficient characterizations of generalized type of entire functions of slow growth have been obtained in terms of the approximation errors.


Introduction and Notation
The growth and the value distribution of Taylor entire function were studied for a long time.Many researches have done in-depth research and many important results, which have been obtained in [1−8] .For example, S. M. Shah [1] and S. K. Bajpai [2] gave some different characterizations on the coefficients and the maximum modulus, the maximum term, and the rank of the maximum term for the Taylor entire function of fast growth ρ = ∞.On the other hand, G. P. Kapoor, A. Nautiyal [3−4] and R. Ganti, G. S. Srivastava [5] continued this work and defined generalized order and generalized type for the Taylor entire function of slow growth ρ = 0 in a new way, as follows: Let Ω denote the class of functions h(x) satisfying the following conditions (H, i) and (H, ii).
In this paper, we defined generalized order and generalized type is essentially due to R. Ganti and G. S. Srivastava [5] , but we extend the range of α(x), and then extend some know results of entire function of slow growth. Let be an entire function.Set M (r) ≡ M (r, f ) = max Then M (r), u(r) and v(r) are called respectively the maximum modulus, the maximum term, and the rank of the maximum term of f (z) for |z| = r.Elements in the range set of v(r) are called principle indices.
Normally, the growth of f (z) is measured in terms of order ρ and type τ defined as under: Definition 1. [5] The order and the type of entire function can be defined as: lim sup for 0 < ρ < ∞ various researches have give different characterizations for entire function of fast growth.In this paper we defined the generalized order and generalized type in a new way with the help of general function as following: Let Λ denote the class of functions h(x) satisfying the conditions (H, i) and (H, iii).
(H, iii) where ln [0] x = x, ln [1] x = ln x, ln [p] x = ln [p−1] ln x, so we can easy testify that for every c > 0, that is, h(x) is slowly increasing.
From (H, ii) and (H, iii), we know that α(x) ∈ Ω is a special case of α(x) ∈ Λ, when p = 1.Definition 2. Let α(x) ∈ Λ, then the generalized order and generalized type also can be defined in this way as under: Recently, R. Ganti and G. S. Srivastava [5] also considered the approximation of entire functions in Banach spaces.Thus, let f (z) be an analytic function in the unit disc U = {z ∈ C : |z| < 1} and we set |f (re iθ )| q dθ} 1/q , q > 0.
let H q denote the Hardy space of function f (z) satisfying the condition and let H ′ q denote the Bergman pace of function f (z) satisfying the condition , then H q and H ′ q are Banach spaces for q ≥ 1.Following [5], we say that a function f (z) which is analytic in U belongs to the spaces B(p, q, k) if It is known [7] that B(p, q, k) is Banach spaces for p > 0 and q, k ≥ 1, otherwise it is a Frechet space.Further [8] , Definition 3. Let X denote one of the Banach spaces defined above, the approximation errors of entire function f (z) can be defined as: where P n consists of algebraic polynomial of degree at most n in complex variable z.
For α(x) ∈ Λ, p = 1, R. Ganti and G. S. Srivastava [5] have obtained the characterization of generalized type of f (z) in terms of the error E n (f, X) defined above.But, For α(x) ∈ Λ, p > 1, p ∈ N + haven't been proved so far.In the present paper, we have made an attempt to finish it.
Our main conclusions are as follows: Theorem 1.Let α(x) ∈ Λ, then some necessary and sufficient conditions of the entire function f (z) with generalized order ρ is Theorem 3. Let α(x) ∈ Λ, then some necessary and sufficient conditions of the entire function f (z) ∈ B(p, q, k) to be of generalized type τ having finite generalized order ρ, 1 < ρ < ∞ is Assuming that the condition of Theorem 3 are satisfied and ξ(α) is a positive number, some necessary and sufficient conditions for a function f (z) ∈ H q to be an entire function of generalized type ξ(α) have finite generalized order ρ is that

Preliminary lemmas
In this section, we will give some lemmas which play an important role in the proof of theorems.

Proof of Theorems
The proof of theorem 1 Let M (σ) = M (r), σ = ln r, λ n = n, then the Dirichlet series can be convert to Taylor entire functions, so the proof of theorem 1 can get from theorem 1 of paper [9].The proof of theorem 2 Case I: When p = 1, R. Ganti and G. S. Srivastava [5] has been proved that using (4), we have we suppose 0 < τ < ∞, then for every ε > 0, ∃r > r 0 , we have It follow (4) that, when r is large enough, we obtain α(ln r − 1) = ( 1 , ).Now proceeding to limits, we can get the above inequality obviously holds when τ = ∞.
Case II: Conversely, let We suppose B < ∞, then for every ε > 0 and for all n ≥ n 0 , we have That is to say ∀ε > 0, ∃n 0 > 0, when n > n 0 , we have In addition, when r is large enough, there exists we have and the above inequality, it following that In view of Lemma 4, So from Case I and Case II, the proof is completed.The proof of theorem 3 Case I: When p = 1, R. Ganti and G. S. Srivastava [5] has been proved that It follows (4) that, Case II: When p = 2, 3 • • • , we prove this part in two steps.Fristly, we consider the space B(p, q, k), q = 2, 0 < p < 2, and k ≥ 1.Let f (z) ∈ B(p, q, k) be of generalized type τ with generalized order ρ.Then from theorem 2, we have For a given ε > 0, and all n > m = m(ε), we have where τ = τ + ε.
a j z j be the n th partial sum of the Taylor series of the function f (z).Following [6] , we can get where B(a, b)(a, b > 0) denotes the bata function.By using (10), we get where ] 1/ρ } , since α(x) is increasing and j > n + 1, we obtain Owing to ψ(α) < 1, from (11) and (12), we have For n > m, (13) yields Now proceeding to limits, we have For reverse inequality, by [6] , we have, Then for sufficiently large n, we can get by applying limits and in view of (9), we obtain It follow from ( 15) and (17) that, In the second steps, we consider the space B(p, q, k) for 0 < p < q, q = 2, and q, k ≥ 1. Gvaradze [7] show that, for p ≥ p 1 , q ≤ q 1 , and k ≤ k 1 , if at least one of the inequality is strict, then the strict inclusion B(p, q, k) ⊂ B(p 1 , q 1 , k 1 ) holds and the following relation is true: For any function f (z) ∈ B(p, q, k), the last relation yields For the general case B(p, q, k), q = 2, we prove the necessity of condition (10).Let f (z) ∈ B(p, q, k) be an entire transcendental function having finite generalized order ρ(α; f ) whose generalized type is defined by (9).Using the relation (10), for n > m we estimate the value of the best polynomial approximation as follows .
For n > m, from the above inequality, we have (1−ψ(α)) Since ψ(α) < 1, and α ∈ Λ, proceeding to limits and using (14), we have For the reverse inequality, let 0 < p < q < 2 and k, q ≥ 1.By (19), where p 1 = P, q 1 = 2, and k 1 = k, and the condition (10) is already proved for the space B(p, 2, k), we obtain then for n is large enough, we have By applying limits and from (9), we obtain Now we assume that 2 ≤ p < q.Set q 1 = q, k 1 = k, and 0 < p 1 < 2 in the inequality (19), where p 1 is an arbitrary fixed number.Substituting p 1 for p in (20), we can get From the above inequality and applying the same analogy as in the previous case 0 < p ≤ 2 < q, for sufficiently large n, we obtain by applying limits and in view of (9), we obtain From the relations (15), (17) and the above inequality, we obtain the required relation (18).So from Case I and Case II, the proof is completed.
Using estimate (23) we prove the above inequality in the case q = ∞.
Conversely, Let the generalized type τ of an an entire transcendental function f (z) having finite generalized order ρ as follows and using the relation (10), we have On account of (12) E n (H q ; f ) = f − g n (f ) Hq Remark: An analogy of Theorem 4 for Bergman Spaces follows from (5) and from Theorem 3 for q = ∞.