INTRODUCTION
The partial differential equation:
is called the Generalized Axially Symmetric Helmholtz Equation (GASHE) and the solutions of Eq. 1 are called GASHE functions. A GASHE function u, regular about the origin, has the following BesselGegenbauer series expansion:
where, x = rcosθ, y = rsinθ, J_{v+n} are Bessel function
of first kind and C^{v}_{n} are Gegenbauer polynomials. A GASHE
function u is said to be entire if the series (2) converges absolutely and uniformly
on the compact subsets of the whole (x,y)places it is known (Gilbert,
1969) for an entire GASHE function u that:
The growth of entire function f(z) is measured by order ρ and type T defined as under:
where:
be the maximum modules.
In function theory, the growth parameters may be completed from the Taylor's
coefficients or Chebyshev polynomial approximations. Function theoretic methods
extended these properties to harmonic functions in several variables (Gilbert
and Colton, 1963; Gilbert, 1969; McCoy,
1979). (McCoy, 1992) studied the rapid growth of
entire function solution of Helmholtz equation in terms of order ρ and
type T using the concept of index. He obtained some bounded on the order and
type of entire function solution of Helmholtz equation that reflect their antecedents
in the theory of analytic functions of a single complex variable. Recently,
(Kumar and Arora, 2010) studied some results generalized
axisymmetric potentials. In this paper we have studied the slow growth of entire
GASHE function u by using the concept of generalized order (Kapoor
and Nautiyal, 1981) in Banach spaces (B(p,q,m)) spaces, Hardy space and
Bergman space).
Seremeta (1970) defined the generalized order and generalized
type with the help of general functions as follows.
Let
denote the class of functions h satisfying the following conditions:
(i) 
h(x) is defined on [a,α) and is positive, strictly increasing
differentiable and tends to ∞ as x→∞ 
(ii) 

for every function n(x) such that n(x)α
as x→∞. 

Let Δ denote the class of functions h satisfying
condition (i) and: 
(iii) 

for every c>0 that is, h(x) is slowly
increasing. 

For entire function f(z) and functions α(x)εΔ,
β(x)εL*, (Seremeta, 1970), proved that: 
Further, for α(x) εL^{0}, β^{1} (x)ε L^{0}, γ (x) εL^{0}:
where, 0<ρ<∞ is a fixed number.
It has been noticed that above relations were obtained under certain conditions
which do not hold if α = β. To define this scale, (Kapoor
and Nautiyal, 1981) defined generalized order ρ(α,f) of slow growth
with the help of general functions as follows.
Let Ω be the class of functions h(x) satisfying (i) and (iv) there exists a δ(x) ε Ω and x_{0}, K_{1} and K_{2} such that:
(iv) 


Let
be the class of functions h(x) satisfying (i) and (v): 
(v) 

(Kapoor and Nautiyal,
1981) showed that classes Ω and
are contained in Δ. Further, Ω ∩ =
φ and they defined the generalized order ρ(α,f) for entire
functions f(z) of slow growth as: 
where, α(x) either belongs to Ω or to .
Vakarchuk and Zhir (2002) considered the approximation
of entire functions in Banach spaces. Thus, let f(z) be analytic in the unit
disc U_{1} = {zεC:z<1} and we get:
Let H_{q} denote the Hardy space of functions f(z) satisfying the condition:
and let H_{q}, denote the Bergman space of functions f(z) satisfying the condition:
For q = ∞, let f_{H∞}, = f_{H∞}
= sup{f(z), zεU_{1}}. Then H_{q} and H_{q}, are
Banach spaces for q≥1. (Vakarchuk and Zhir, 2002),
we say that a function f(z) which is analytic in U_{1} belongs to the
space B(p,q,m) if:
0<p<q≤4, 0<m<∞ and:
It is known Gvaradge (1994) that B(p,q,m) is a Banach
space for p>0 and q,m≥1, otherwise it is a Frechet space. Further Vakarchuk
(1994):
Let X denote one of the Banach spaces defined above and let:
where, P_{n} consists of algebraic polynomials of degree at most n in complex variable z.
Vakarchuk and Zhir (2002) studied the generalized order
of f(z) in terms of the errors E_{n}(f,x) defined above. It has been
noticed that these results do not hold good when α = β = γ, i.e.,
for entire functions of slow growth.
It is significant to mention here that characterization of coefficient and Chebyshev approximation error of entire function GASHE, u in certain Banach spaces by generalized order of slow growth have not been studied so far. In this paper, we have made an attempt to bridge this gap. Moreover, we have obtained some bounds on generalized order of entire function GASHE u in certain Banach spaces (B(p,q,m)) space, Hardy space and Bergman spaces) in terms of coefficients and Chebyshev approximation errors.
It is important to write here that the function α(x) = log_{p}(x),
p≥1 and α(x) = exp ((logx)^{δ}), 0<δ<1, satisfy
the condition αεΔ. For α(x) = logx, our results gives the
logarithmic order in place of generalized order. So if a function f has a finite
logarithmic order of finite generalized order with α(x) = log_{p}x,
p≥1, then the order ρ of f is equal to zero.
NOTATIONS
• 
We shall write θ(ξ) of θ_{1}(ξ) 
• 
F[x;c] = α^{1} [cα(x)], c is a positive constant 
• 
E[f[x;c]] is an integral part of the function F 
MAIN RESULTS
Now we shall prove our main results.
Theorem 1: Let α (x) ∈ ,
then the entire GASHE function u(x,y) is of generalized order ρ(u), 1≤ρ(u)≤∞,
if and only if:
where:
and r_{*}>1. 
Proof: Suppose α (x) ∈
and ρ(u)<∞. Then for every ε>0, there exists r(ε)
such that:
or:
Now using the orthogonality property of Gegenbauer polynomials (Gilbert,
1969) and the uniform convergence of series (2), we have:
Further, from the well known series expansion of J_{v+n}(kr), we have:
and so for n≥[(kr)^{2}], where [x] denotes the integral part of x, we have:
From Eq. 11 and 12 and the using the
relation:
and:
for n≥[(kr)^{2}], we now get:
where:
Since:
as n→∞. We can choose constants K_{*}<∞ and r_{*}>1 such that:
Thus, for n≥[(kr)^{2}], Eq. 14 yields:
Using Eq. 10, we obtain:
The larger factor is minimized at:
This leads to:
Since α (x) ∈
as n→∞, we have:
Conversely, let:
Suppose L(u)<∞. Then for given ε>0, there exists n_{0},≥n_{0}(ε) such that:
where,
The inequality:
is satisfied with some n = n(r). Then:
From Eq. 18, we have:
We can take .
Let us consider the function:
We have:
As x→∞, in view of the assumption of theorem, for finite:
is bounded. So there is an A>0 such that for x≥x_{1}, we have:
We can take A<log2. It may be seen that inequalities Eq.
18 and 19 hold for
Let n_{0}, = max (n_{0}(ε), E[x_{1}]+1). For r>r_{1}
(n_{0}), we have:
From Eq. 19 and 20 it gives that:
This leads to the fact that if for r>r_{1}(n_{0}), we let x*(r) designate the point where:
then
where:
We have:
For r>r_{1}(n_{0}),
Since, α (x) ∈ ⊆ Δ
now proceeding to limits we obtain:
Combining Eq. 16 and 21 the proof is
immediate.
Theorem 2: Let α (x) ∈
then the entire GASHE function u(x,y) is of generalized order ρ(u) if ρ(u)≤θ(L(f*)).
where:
Proof: Using Eq. 2 with 13, we
get:
where, K is a constant and:
It follows from Eq. 3 that f*(z) is an entire function. Since:
it follows from Eq. 22 that:
Now applying (Kapoor and Nautiyal, 1981) to the function
f*(z) we get the required results.
Theorem 3: Let α (x) ∈
and u(x,y) be a GASHE function in the disc z≤r_{0}. Then the generalized
order of u(x,y) satisfy:
(i) 

(ii) 

where:
and:
Proof: Let GASHE u(x,y) be analytic in the disc U = {zεC: z≤r_{0}} and set:
(Vakarchuk and Zhir, 2002), we say that a function
which is analytic in U belongs to the space B(p,q,m) if:
0<p<q≤∞, 0<m<∞ and:
We have for PεP_{n}, that:
In view of Eq. 22 we get:
or:
where:
From Theorem 1, for any given ε>0 and all n>n_{0} = n_{0} (ε), we have:
We shall prove the result in two steps. First we consider the space B(p,q,m), q = 2, 0<p<2 and m≥1. Let:
be the nth partial sum of the Taylor series of the function f*(z). (Vakarchuk
and Zhir, 2002) and using (Reddy, 1972) extension
of Bernstein theorem for given ε>0 there is an n_{0}(ε)>0
such that:
Using Eq. 23 we get:
for all n≥n_{0}(ε), where B(a,b)(a,b>0) denotes the beta function. By using Eq. 24, we obtain:
where:
Set:
Since, φ(α)<1, by virtue of Eq. 25 and 26
we get:
For n>n_{0}, Eq. 27 gives:
But:
Hence:
It gives:
Applying the limits, we get:
or:
Using the orthogonality property of Gegenbauer polynomials (Gilbert,
1969) and the uniform convergence of the series (2), for any gεπ_{n1},
we get:
From Eq. 1215 with (Vakarchuk
and Zhir, 2002) in above we get:
Then for sufficiently large n, we have:
Applying limits and using (Kapoor and Nautiyal, 1981)
for, we get:
or:
Now we consider the spaces B(p,q,m) for 0<p<q, q ≠ 2 and q, m≥1.
(Gvaradge, 1994) showed that, for p≥p_{1},
q≤q_{1} and m≤m_{1} if at least one of the inequalities
is strict, then the strict inclusion B(p,q,m)⊂B(p_{1},q_{1},m_{1})
holds the following relation is true:
for any u∈B(p,q,m), the last relation gives:
Let u∈B(p,q,m) be an entire transcends function solution of Helmholtz Eq. 1 having finite generalized order ρ(u). Consider the function:
Now:
Using Eq. 23 we get:
For n>n_{0}, from Eq. 33 we get:
Since φ(α)<1 and α∈,
applying the limits and using Eq. 28, we get:
For (ii) inequality, let 0<p<q<2 and m,q≥1. By Eq. 32, where p = p_{1}, q = 2 and m_{1} = m_{2} and the condition Eq. 24 is already proved for the space B(p,2,m), we get:
Now let 0<p≤2<q. Since, we have:
Therefore,
Then for sufficiently large n, we have:
By proceeding to limits and from Kapoor and Nautiyal (1981),
we obtain:
Now we assume that 2≤p<q. Set q_{1} = q, m_{1} = m and
0<p_{1}<2 in the inequality Eq. 36, where p_{1}
is an arbitrary fixed number, Substituting p_{1} for p in Eq.
36, we get:
Using Eq. 38 and following the same analogy as in the previous
case 0<p≤2<q, for sufficiently large n, we have:
Proceeding to limits and using (Kapoor and Nautiyal, 1981),
we get:
Combining Eq. 31, 35, 37
and 39, we obtain the result (ii). This completes the proof
of Theorem 3.
Theorem 4: Let u(x,y)∈H_{q} be a GASHE function on the
disc
and α(x)∈ .
Then the generalized order of u(x,y) satisfy:
(i) 

(ii) 

where:
and:
Proof: We can obtain the relation:
In view of Eq. 26, we get:
Thus gives:
Applying the limits, we get:
This completes the proof of (i). (ii) In view of Eq. 8, we see that:
where, ζ_{q} is a constant independent of n and u.
Using Eq. 36 with 40 we get:
or:
For the Hardy space H∞, we have:
Using Eq. 41, the inequality (40) is true for q = ∞.
Hence the proof is completed.