International Journal of Analysis and Applications
ISSN 2291-8639
Volume 2, Number 1 (2013), 26-37
http://www.etamaths.com

SLOW GROWTH AND OPTIMAL APPROXIMATION OF

PSEUDOANALYTIC FUNCTIONS ON THE DISK

DEVENDRA KUMAR

Abstract. Pseudoanalytic functions (PAF) are constructed as complex com-

bination of real-valued analytic solutions to the Stokes-Betrami System. These
solutions include the generalized biaxisymmetric potentials. McCoy [10] con-

sidered the approximation of pseudoanalytic functions on the disk. Kumar et

al. [9] studied the generalized order and generalized type of PAF in terms of
the Fourier coefficients occurring in its local expansion and optimal approx-

imation errors in Bernstein sense on the disk. The aim of this paper is to

improve the results of McCoy [10] and Kumar et al. [9]. Our results apply
satisfactorily for slow growth.

1. Introduction

Generalized biaxisymmetric potential (GBASP) that are harmonic at the origin
may be expanded, in analogy with the Taylor’s series for analytic functions of a sin-
gle complex variable, in a convergent series of homogeneous harmonic polynomials
on an open set.

Pseudoanalytic functions are constructed as complex combinations of real-valued
function pair that are analytic solutions of Stokes-Beltrami System (SBS); a gener-
alization of the Cauchy-Riemann equations that is linked to the GBASP equation by
eliminating one of the dependent variables from the system. Pseudoanalytic func-
tions provide sufficient basis for the transformation of Bernstein’s ideas through
transform and special function methods. The real part of pseudoanalytic function
i.e., eliminating the harmonic conjugate gives the theory of GBASP. The GBASP
equation frequently found in the summability theory of [2] Jacobi series as

r
∂

∂r

{
rα+β+1ρ(α,β)(θ)r

∂u

∂r

}
+

∂

∂θ

{
rα+β+1ρ(α,β)(θ)

∂u

∂θ

}
= 0

ρ(α,β)(θ) = (sin θ/2)2α+1(cos θ/2)2β+1,α ≥ β > −
1

2

where (r,θ) are the plane polar coordinates. The domain of the potential u is
a simply connected region with smooth boundary in the upper half plane C+ =
C∩Re(z) ≥ 0. The existence of a harmonic conjugate ν of u is implied in the sense

2010 Mathematics Subject Classification. 30E10, 41A20.
Key words and phrases. Approximation errors, generalized order and types, pseudoanalytic

functions and Stokes-Beltrami System.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

26



SLOW GROWTH AND OPTIMAL APPROXIMATION 27

of the generalized Stokes-Beltrami System (SBS);

r
∂ν

∂r
= −rα+β+1ρ(α,β)(θ)

∂u

∂θ
∂ν

∂θ
= rα+β+1ρ(α,β)(θ)r

∂u

∂r

a system that reduces to the Cauchy-Riemann equations of analytic function theory
in the limit α = β = −1

2
. Following along the lines of analytic function theory, a

pseudoanalytic function [1,5,12] (PAF) is defined as the complex combination

F(reiθ) = u(r,θ) + iν(r,θ)

of a real valued analytic function pair formed from the potential u and the principal
branch of its harmonic conjugate ν = ν(r,θ).

The disk DR(R > 1) of maximum radius on which a PAF F exists, is designated
by F ∈ P(DR). If F is an entire PAF, it has no singularities in the finite C+−
plane and writes F ∈ P(C).

Kumar [8] studied the relationship between the pseudoanalytic functions and
Bergman-Gilbert type integral operators for GBASP and polynomial approxima-
tion. Bergman [3] and Gilbert [5] generalize the operation of taking the real part.
They obtained bounded linear operators which transform analytic functions to so-
lution u, where integral operators are developed to provide the transformation
from analytic functions to solutions of corresponding elliptic equation. Bers [4]
and Vekua [11] have also extended function theory so that solutions u of elliptic
equations can be obtained as u = Re(f), where f is a pseudoanalytic function shar-
ing many of the properties associated with classical analytic functions of a single
complex variable. Also McCoy [10] considered the approximation of pseudoanalytic
functions on the disk and obtained some coefficient and Bernstein type growth the-
orems. Kapoor and Nautiyal [6] characterized the order and type of GBASP’s (not
necessarily entire) in terms of rates of decay of approximation errors on both sup
norm and Lδ−norm, 1 ≤ δ < ∞. All these authors have not studied the generalized
growth of pseudoanalytic functions on the disk. Our results and methods are differ-
ent from all those authors mentioned above and apply satisfactorily for slow growth.

Let p and q are two positive functions defined on (0,∞), strictly increasing and
infinitely differentiable such that

lim
x→∞

p(x) = lim
x→∞

q(x) = ∞,

lim
x→∞

p(cx)

p(x)
= 1,

lim
x→∞

q((1 + w(x))x)

q(x)
= 1, lim

x→0
w(x) = 0,

p(x/q−1(cp(x))) = (1 + o(x))p(x), for x →∞,

lim
x→∞

∣∣∣∣d(q−1(cp(x)))d(log x)
∣∣∣∣ ≤ b,

b is a non zero positive constant and d(u) means the differential of u.



28 KUMAR

Kumar et al. [9] defined the (p,q)−order and (p,q)−type (or generalized order
and generalized type) of pseudoanalytic functions F ∈ P(DR) with radial limits as:

ρ0(p,q) = lim sup
r→R

p(log MR(F))

q(R/(R−r))
,

σ0(p,q) = lim sup
r→R

p(log MR(F))

[q(R/(R−r))]ρ0(p,q)
, 1 < r < R,

where

Mr(F) = max{|F(reiθ)| : reiθ ∈ DR},r < R.
In [9] Kumar et al. studied the generalized order ((p,q)−order) and generalized

type ((p,q)−type) of a PAF in terms of the Fourier coefficients occurring in its local
expansion and optimal approximation errors in Bernstein sense on the disk. They
obtained these results by using the following condition

lim
x→∞

∣∣∣∣d(q−1(cp(x)))d(log x)
∣∣∣∣ < b,

this condition does not hold for p = q. Therefore the results fail to exist for p = q.

In this paper we shall improve the results of Kumar et al. [9] by using the con-
cept of generalized order of slow growth introduced by Kapoor and Nautiyal [7]
with the help of general function as:

Let L denote the class of functions h satisfying the following conditions:

(i) h is defined on (0,∞), strictly increasing to infinity differentiable such that
lim
x→∞

h(x) = ∞.

(ii) limx→∞
h((1+w(x)).x)

h(x)
= 1, for every function w, such that limx→0 w(x) = 0.

Let ∆ denote the class of function h satisfying condition (i) and

(iii) limx→∞
h(cx)
h(x)

= 1, for every c > 0.

Let Ω be the class of functions h satisfying (i) and (iv) and Ω be the
class of functions satisfying (i) and (v) where

(iv) there exists a γ ∈ Ω and x0,K1 and K2 such that, for all x > x0.

0 < K1 ≤ lim
x→∞

d(h(x))

d(γ(log x))
≤ K2 < ∞

(v) limx→∞
d(h(x))
d(log x)

= K3, 0 < K3 < ∞.
Now we define the (p,p)−order and (p,p)−type of F ∈ P(DR) (or generalized

growth) by

ρ(p,p) = lim sup
r→R

p(log Mr(F))

p(R/(R−r))
,

σ(p,p) = lim sup
r→R

p(log Mr(F))

[p(R/(R−r))]ρ(p,p)
.

Kapoor and Nautiyal [7] showed that classes Ω and Ω are contained in ∆. Fur-
ther, Ω ∪ Ω = φ.



SLOW GROWTH AND OPTIMAL APPROXIMATION 29

2. Generalized Order and Generalized Type with Fourier
Coefficients of Pseudoanalytic Functions

The purpose of this section is to establish the relationship of the generalized
growth (p−growth) of pseudoanalytic functions in a disk with Fourier coefficients
occurring in its local expansion.

In a neighborhood of the origin, the pseudoanalytic function PAF F ∈ P(DR)
has the local expansion

F(reiθ) =

∞∑
n=0

anwnFn(re
iθ),reiθ ∈ DR,

Fn(re
iθ) = un(r,θ) + iνn(r,θ),n = 0, 1, 2, . . . and an real-valued. Write

(2.1) lim sup
n→∞

p(n)

p
[

n
log(n2α+1|an|Rn)

] = µ(p,p).
First we prove

Lemma 2.1. Let p(x) ∈ Ω and µ > 0. For every r > 1, the maximum of the
function

x → w(x,r) = x log(r/R) +
x

p−1(p(x)/µ)

is reached for x = xr solution of the equation

(2.2) x = p−1
{
µp

[
1 −d log(p−1(p(x)/µ))/d(log x)

log(R/r)

]}
.

Proof. The proof follows on the lines of [9, Lemma 2.1] by simple calculation
replacing q by p.

Lemma 2.2. Let F(reiθ) =
∑∞
n=0 anwnFn(re

iθ),F ∈ P(DR). For every 1 < r < R
and p(x) ∈ Ω, we put

M(r,F) = sup
n
{‖anwn‖rn,r > 0},

‖F‖ =

{
‖F‖δ =

[∫ ∫
D1
|F|δrdrdφ

]
, 1 ≤ δ < ∞

‖F‖∞ = M1(F), δ = ∞
,

and

ρ0(p,p) = lim sup
r→R

p(log M(r,F))

p(R/(R−r))
then

ρ(p,p) ≤ ρ0(p,p).
Proof. Let

f(z) =

∞∑
n=0

anz
n

M(r,f) ≤
∞∑
n=0

|an|rn ≤
∞∑
n=0

|an|wnrn,



30 KUMAR

substituting r = rξR1−ξ(r/R)1−ξ in above inequality we get

M(r,f) ≤
∞∑
n=0

|an|wn(rξR1−ξ)n(r/R)(1−ξ)
n

, (r/R) < 1

or

M(r,f) ≤
∞∑
n=0

sup(|an|wn(rξR1−ξ)n(r/R)(1−ξ)
n

)

or

M(r,f) ≤ M(r
′
,F)

∞∑
n=0

(r/R)(1−ξ)
n

,r
′

= (rξR1−ξ),

≤ M(r
′
,F)

1

1 − (r/R)1−ξ
or

log M(r,f) ≤ log M(r
′
,F) − log(1 − (r/R)1−ξ).

If the function r → M(r
′
,F) is bounded, then ρ(p,p) = ρ0(p,p) = 0. So we can

assume that M(r
′
,F) →∞ as r → R. Then, for every r sufficiently close to R

p(log M(r,f))

p(R/(R−r))
≤
p(log M(r

′
,F) − log(1 − (r/R)1−ξ))

p(R/(R−rξR1−ξ))
.
p(R/(R−rξR1−ξ))

p(R/(R−r))
.

Since

p(R/(R−rξR1−ξ))
p(R/(R−r))

→ 1 as r → R,

we obtain by passing to limits on both sides

ρ(p,p) ≤ ρ0(p,p).

Hence the proof is complete.

Theorem 2.1. Let Let p(x) ∈ Ω and F(reiθ) =
∑∞
n=0 anwnFn(re

iθ,F ∈ P(DR),reiθ ∈
DR such that µ(p,p) defined by (2.1) is finite. Then F is the restriction of a pseudo
analytic function in P(DR)(R > 1) and its (p,p)−order ρ(p,p) = µ(p,p).

Proof. It can be seen [9] that for every 1 < r < R the series
∑∞
n=0 anwnFn(re

iθ)
is convergent in DR. Now first we show that ρ(p,p) ≤ µ(p,p). By (2.1) we have for
every ε > 0, there exists n(ε) such that for every n > n(ε),

(2.3) p(n) ≤ (µ(p,p) + ε)p
(

n

log(|an|n2α+1Rn)

)
,

since

(2.4) log(|an|n2α+1rn) = n log(r/R) + log(|an|n2α+1Rn)

using (2.3) in (2.4) we get

(2.5) log(|an|n2α+1rn) ≤ n


log(r/R) + 1

p−1
(
p(n)
µ

)

 .



SLOW GROWTH AND OPTIMAL APPROXIMATION 31

Choose

n = xr = p
−1


µp


1 −d

(
p−1

(
p(x)
µ

))
/d(log x)

log(R/r)




 .

Using the properties of the function p

(
d log(p−1( p(x)µ ))

d(log x)
= 0(1)

)
, and the function

t → log t, (log(1 + x) = (1 + o(1)).x,x → 0), we have

n = xr = (1 + o(1))p
−1 [µp(R/(R−r))] ,

and replacing in (2.5), we have

log(|an|n2α+1rn) ≤ (1 + o(1))p−1(µp(R/(R−r)))
(

log(r/R) +
1

R/(R−r)

)
since R

R−r > 1, it gives

log(|an|n2α+1rn) ≤ C0p−1(µp(R/(R−r)).

By the properties of function p and proceeding the limit supremum as r suffi-
ciently close to R we get

lim sup
r→R

p(log M(r,F))

p(R/(R−r))
≤ µ = µ + ε,

or

ρ(p,p) ≤ µ(p,p).

Using Lemma 2.2 we obtain

(2.6) ρ(p,p) ≤ µ(p,p).

Now we show that ρ(p,p) ≥ µ(p,p). By the definition of ρ(p,p), for every ε > 0,
there exists rε ∈]1,R[ such that for every r ≥ rε we have

(2.7) log Mr(F) ≤ p−1[(ρ(p,p) + ε)p(R/(R−r))].

Since for r ∈]1,R[,

(2.8) log(|an|n2α+1Rn) = −n log(r/R) + log(|an|n2α+1rn).

Thus

p(n)

p
(

C1n
log(|an|n2α+1Rn)

) ≤ ρ(p,p) + ε.
Now proceeding to limits supremum as n →∞, we get

(2.9) µ(p,p) ≤ ρ(p,p).

Combining (2.6) and (2.9) we complete the proof.

Let F =
∑∞
n=0 anwnFn(re

iθ) be pseudo analytic function of (p,p)−order ρ =
ρ(p,p) and write

T(p,p) = lim sup
n→∞

p(n/ρ){
p
(

ρ
(ρ−1)

n
log(|an|n2α+1Rn)

)}ρ−1 .



32 KUMAR

Now we prove

Lemma 2.3. Let p(x) ∈ Ω and F =
∑∞
n=0 anwnFn(re

iθ). For every r ∈]1,R[,

σ1(p,p) = lim sup
r→R

p(log M(r,F))

(p(R/(R−r)))ρ
,

then

σ(p,p) ≤ σ1(p,p).

Proof. The proof can be obtain by using the same reasoning as in the proof of
Lemma 2.2 as

p(log M(r,f))

[p(R/(R−r))]ρ
≤
p(log M(RξR1−ξ,F) − log(1 − (r/R)1−ξ))

p(R/(R−rξR1−ξ))
.
p(R/(R−rξR1−ξ))

p(R/(R−r))
.

Proceeding the limit, we get

σ(p,p) ≤ σ1(p,p).

In view of (2.7) and [10, eq. 2.8], (2.8) gives that

log(|an|n2α+1Rn) ≤−n log(r/R) + log(n + 2)n2α+1A) + p−1[(ρ(p,p) + ε)p(R/(R−r))].

or

log(|an|n2α+1Rn)
n

≤ ϕ(r,n)

where

ϕ(r,n) = log(R/r) +
1

n
log((n + 2)n2α+1A) +

1

n
p−1[(ρ(p,p) + ε).p(R/(R−r))]

and A = ‖ρ(α,β)‖δ′, 1δ +
1
δ′

= 1.

For r sufficiently close to R and for sufficiently large n,ϕ(r,n) is equivalent to
log(R/r) for n →∞ and log(R/r) is equivalent to R−r

r
= R

r
− 1 for r → R. Then

ϕ(r,n) = (1 + o(1)) log(R/r) as n →∞,

and

log(R/r) = (1 + o(1))
(R−r)

r
(r → R).

Therefore for r sufficiently close to R and n sufficiently large

log(|an|n2α+1Rn)
n

≤ (1 + o(1))(R/r − 1).

Substituting

r =
R[

1 + p−1
(

p(n)
(ρ(p,p)+ε)

)],
and applying the properties of function p, we obtain

log(|an|n2α+1Rn) ≤ C1
n[

p−1
(

p(n)
(ρ(p,p)+ε)

)],



SLOW GROWTH AND OPTIMAL APPROXIMATION 33

Theorem 2.2. Let p(x) ∈ Ω and F =
∑∞
n=0 anwnFn(re

iθ) be a pseudoanalytic
function on the closed unit disk. If F is of finite generalized (p,p)−order ρ(p,p),
and

(2.10) T(p,p) = lim sup
n→∞

p(n/ρ){
p
(

ρ
ρ−1.

n
log(|an|n2α+1Rn)

)}ρ−1 < ∞.
Then F is the restriction of a pseudoanalytic function in P(DR)(R > 1) and its

(p,p)−type σ(p,p) = T(p,p).

Proof. The function F is the restriction of a pseudoanalytic function in P(DR) by
the definition of T(p,p) and arguments used in Theorem 2.1. Put T = T(p,p),ρ =
ρ(p,p); σ = σ(p,p).

If T < ∞ by (2.10), for every ε > 0, there exists n0 ≤ n such that

p(n/ρ) ≤ (T + ε)
{
p

(
ρ

ρ− 1
n

log(|an|n2α+1Rn)

)}ρ−1
or

(2.11) log(|an|n2α+1Rn) ≤
ρ

(ρ− 1)
n

p−1
((

1
T
p(n/ρ)

)1/(ρ−1)),T = T + ε.
Since

log(|an|n2α+1rn) ≤ n log(r/R) + log(|an|n2α+1Rn).

Using (2.11), we get

log(|an|n2α+1rn) ≤ n log(r/R) +
ρ

ρ− 1
n

p−1
((

1
T
p(n/ρ)

)1/(ρ−1)).
For every r ∈]1,R[, and r sufficiently close to R, we put

φ(x,r) = x log(r/R) +
ρ

ρ− 1
x

p−1
((

p(x/ρ)

T

)1/(ρ−1)).
then

∂φ(x,r)

∂x
= log(r/R) +

ρ

ρ− 1
d

dx




x

p−1
((

p(x/ρ)

T

)1/(ρ−1))

 .

Then the maximum of the function x → φ(x,r) is reached for x = xr where xr
is the unique solution of the equation

∂φ

∂x
(x,r) = 0.

If we put S = S(x,T, 1
ρ−1 ) = p

−1
((

1
T
p(x/ρ)

)1/(ρ−1))
, then

φ(x,r) = x log(r/R) +
x

S



34 KUMAR

we have

∂φ

∂x
(x,r) = 0 ⇔ log(r/R) +

ρ

ρ− 1

(
S −xdS

dx

S

)
= 0,

or

log(R/r) =
ρ

ρ− 1


1 − d(log S)d(log x)

S


 ,

as dS
dx

= dS
d(log x)

d(log x)
dx

= 1
x

dS
d(log x)

.

Since

log(R/r) = log

(
r −R
R

+ 1

)
∼
(
r −R
R

)(
as

r −R
R

→ 0
)

and ∣∣∣∣∣d
[
log
(
p−1((p(x/ρ))1/(ρ−1))

)]
d log x

∣∣∣∣∣ ≤ b,
where b is a constant positive. Then by the properties of function p ∈ Ω, we have

S = (1 + o(1))
ρ

ρ− 1

(
R

R−r

)
,

thus

p−1

((
p(x/ρ)

T

)1/ρ−1)
= (1 + o(1))

ρ

ρ− 1

(
R

R−r

)
,

therefore

xr = (1 + o(1))ρp
−1(T(p(R/(R−r)))1/(ρ−1)).

Substituting in the relation (2.11), we have

log(|an|n2α+1rn) ≤ sup φ(x,r) = φ(xr,r).
Replacing x by xr in this last relation we get

log(|an|n2α+1rn) ≤
(1 + o(1))ρ−1

ρ
p−1(T(p(R/(R−r)))ρ−1)
R/(R−r)

.

Since R
R−r > 1 and

ρ−1
ρ

< 1, then

log(|an|n2α+1rn) ≤ Cp−1(T(p(R/(R−r)))ρ−1).
Then

log(M(r,f)) ≤ Cp−1(T(p(R/(R−r)))ρ−1).
Thus

p(log M(r,f))

p(R/(R−r))
≤ T.

Proceeding to the limit supremum as r → R, we get
σ(p,p) ≤ T(p,p).

The result is obviously holds for T = ∞.



SLOW GROWTH AND OPTIMAL APPROXIMATION 35

To complete the proof it remains to show that σ(p,p) ≥ T(p,p). Put σ =
σ(p,p) + ε,ρ = ρ(p,p).

Suppose that σ < ∞. By definition of σ(p,p), we have for every ε > 0, there
exist r0 ∈]1,R[, such that for every r > r0(R > r > n0 > 1)
(2.12) log Mr(F) ≤ p−1[σ(p(R/(R−r)))ρ].

Since for every r ∈]1,R[
log(|an|n2α+1Rn) = −n log(r/R) + log(|an|n2α+1rn)

then in view of (2.12) and [10,eq. 2.8], we get

log(|an|n2α+1Rn) ≤−n log(r/R) + log((n + 2)n2α+1A) + p−1[σ(p(R/(R−r)))ρ].
Let

log(|an|n2α+1Rn)
n

≤ w(r,n)

where

w(r,n) = − log(r/R) +
1

n
log((n + 2)n2α+1A) +

1

n
p−1[σ(p(R/(R−r)))ρ].

For r sufficiently close to R we have

lim
n→∞

w(r,n) = − log(r/R) = log(R/r).

Then for n sufficiently large and r sufficiently close to R, we have

w(r,n) = (1 + 0(1)) log(R/r),n →∞,
then

(2.13)
1

n
log(|an|n2α+1Rn) ≤ (1 + o(1)) log(R/r).

Assume

(2.14) r =
R(ρ− 1)p−1

(
1
σ
p(n/ρ)

)1/ρ−1
ρ + (ρ− 1)p−1

(
p(n/ρ)
σ

)1/(ρ−1) .
Now using (2.13) and the properties of the function p ∈ Ω and t → log t, for r

sufficiently close to R, we get

log(|an|n2α+1Rn)
n

≤ (1 + o(1))((R/r) − 1).

From (2.14) we have

R

r
− 1 =

ρ + (ρ− 1)p−1
(
1
σ
p(n/ρ)

)1/(ρ−1)
(ρ− 1)p−1

(
p(n/ρ)
σ

)1/(ρ−1) − 1
=

ρ

(ρ− 1)p−1
(
p(n/ρ)
σ

)1/(ρ−1) .
Then for r sufficiently close to R and n sufficiently large we obtain

log(|an|n2α+1Rn)
n

≤
ρ

(ρ− 1)p−1
(
p(n/ρ)
σ

)1/(ρ−1) ,



36 KUMAR

or

(ρ− 1)

ρp−1
(
p(n/ρ)
σ

)1/(ρ−1) ≤ nlog(|an|n2α+1Rn)
or (

1

σ
p(n/ρ)

)1/(ρ−1)
≤ p

(
ρ

(ρ− 1)
n

log(|an|n2α+1Rn)

)
.

Therefore

p(n/ρ)

σ
≤
{
p

(
ρ

(ρ− 1)
n

log(|an|n2α+1Rn)

)}ρ−1
or

p(n/ρ){
p
(

ρ
(ρ−1)

n
log(|an|n2α+1Rn)

)}ρ−1 ≤ σ = σ + ε.
Proceeding to the limit supremum as n →∞ we get

σ(p,p) ≥ T(p,p).
The result is obviously holds for σ(p,p) = ∞.

3. Generalized Growth and Optimal Polynomial Approximation of
Pseudoanalytic Functions

The purpose of this section is to give the relationship between the generalized
order (ρ(p,p)) and generalized type (T(p,p)) of a pseudoanalytic function PAF and
optimal rate of convergence to 0 in the norm defined in Lemma 2.2.

The approximating pseudoanalytic polynomials of (fixed) degree n are taken
from the sets

πn =

[
P : P(reiθ) =

n∑
k=0

ckwkFk(re
iθ),ck real

]
,n = 0, 1, 2, . . . .

The optimal approximates minimize the error ‖F −P‖ for P ∈ πn in Bernstein
sense as

(3.1) En(F) = inf {‖F −P‖ : P ∈ πn}n = 0, 1, 2, . . . .
Lemma 3.1. Let p(x) ∈ Ω and En(F) is defined as (3.1) then

(3.2) lim sup
n→∞

p(n)

p
[

n
log(|an|n2α+1Rn)

] = lim sup
n→∞

p(n)

p
[

n
log(En(F)n2α+1Rn)

],
and
(3.3)

lim sup
n→∞

p(n/ρ){
p
(

ρ
ρ−1

n
log(|an|n2α+1Rn)

)}ρ−1 = lim sup
n→∞

p(n/ρ){
p
(

ρ
ρ−1

n
log(En(F)n2α+1Rn)

)}ρ−1 .
Proof. The proof can be obtain by following on the basis of proofs of Theorem 2.1
and 2.2 and using the identity [10, eq. 12]

lim sup
n→∞

|an|1/n = lim sup
n→∞

[En(F)]
1/n.



SLOW GROWTH AND OPTIMAL APPROXIMATION 37

In the consequence of (3.2) and (3.3) we can prove the following theorem.

Theorem 3.1. Let p(x) ∈ Ω and F be a pseudoanalytic function on P(DR)(R > 1).
Then

(i) Then the generalized (p,p)−order of F is

ρ(p,p) = lim sup
n→∞

p(n)

p
[

n
log(En(F)n2α+1Rn)

]
(ii) The generalized (p,p)−type of F is T(p,p), if and only if,

T(p,p) = lim sup
n→∞

p(n/ρ){
p
(

ρ
ρ−1

n
log(En(F)n2α+1Rn)

)}ρ−1
when 0 < ρ(p,p) < ∞.

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Department of Mathematics, M.M.H. College,Model Town,Ghaziabad-201001, U.P.,
India