International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 22-29
http://www.etamaths.com

ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE

FUNCTIONS ON ALGEBRAIC VARIETIES

DEVENDRA KUMAR∗

Abstract. Let X be a complete intersection algebraic variety of codimension m > 1 in Cm+n. In
this paper we characterized the classical growth parameters order and type for transcendental entire

functions f ∈⊕(X), the space of holomorphic functions on the complete intersection algebraic variety
X, in terms of the best polynomial approximation error in Lp-norm, 0 < p ≤ ∞, on a L − regular
non-pluripolar compact subset K of Cm+n.

1. Introduction

The growth of transcendental entire functions in one complex variable case is well represented
in the work of B.Ja Levin [11] and Boas [2]. In several complex variables the standard reference is
the work of P.Lelong and L.Gruman [10] and Ronkin’s book [14]. Einstein-Matthews and Kasana [3]
studied the growth parameters (p,q) − order and (p,q) − type introduced by Juneja et al.([6],[7]) of
transcendental entire functions f : Cn → C. Einstein-Matthews and Clement Lutterodt [4] extended
the results studied in [3] to transcendental entire functions f : X → C, defined on a complete inter-
section algebraic variety X in Cm+n of codimension m > 1, and obtained the growth parameters in
terms of the sequence of extremal polynomials occurring in the development of f. It has been noticed
that the growth parameters of f : X → C in terms of approximation errors is not studied so far. The
aim of this paper is to bridge this gap and to study the results obtained in [4] in terms of the best
approximation errors in Lp-norm, 0 < p ≤∞.

A.R. Reddy ([13],[14]) characterized the growth parameters in terms of approximation errors for a
function continuous on [-1,1]. T. Winiarski ([21],[22]) studied the growth of entire functions in terms of
Lagrange polynomial approximation errors with respect to sup norm on a compact subset K (positive
capacity) of C and Cn,n > 1. Kasana and Kumar [8] generalized the results of Winiarski [22] by using
the concept of index-pair (p,q). Adam Janik [5] characterized the generalized order of entire functions
by means of polynomial approximation and interpolation on compact subsets of Cn, using the Siciak
extremal function ([18],[19]). In [5] Adam Janik extended the results of S.M. Shah [17] in the case
n = 1,K = [−1, 1] and Winiarski [22]. Srivastava and Kumar [20] extended and improved the results
of Adam Janik [5]. But our work is different from all these authors.

The text has been divided into four parts. Section 1 consists of an introductory exposition of the
topic and Section 2 contains some definitions and notations. In Section 3, we have given Zeriahi’s
Bernstein-Markov type inequality with two lemmas in which first one is due to Zeriahi extending the
classical Cauchy inequality and second is concerned with a sequence of extremal polynomials. Finally,
in Section 4, we prove two theorems for a transcendental entire function f ∈ ⊕(Cm+n), the space of
holomorphic functions on the complete intersection algebraic variety X and studied the growth pa-
rameters order and type in terms of Lp-approximation error on a L−regular non-pluripolar compact
subset of Cm+n.

2010 Mathematics Subject Classification. 41A20, 30E10.
Key words and phrases. algebraic variety; approximation error; extremal polynomials and plurisubharmonic

functions.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

22



ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 23

2. Definitions and Notations

Following the definition of Einstein and Kasana [3], we have
Let v : Cm+n → R+ := r ∈ R : r > 0 be a real-valued function such that the following properties hold:
(i) v(z + w) ≤ v(z) + v(w) : z,w ∈ Cm+n,
(ii)v(bz) = |b|v(z) : z ∈ Cm+n,b ∈ C,
(iii) v(z) = 0 ⇐⇒ z = 0.

Here v is a norm on Cm+n and it exhausts the complex space Cm+n by a family of sublevel sets
{Ωc}c≥1 which are defined by

Ωc = {z ∈ Cm+n : v(z) ≤ c,c ∈ R}.
Let ϕ : Cm+n → R+. Define Mϕ,v(r) = supv(z)≤r ϕ(z), the maximum modulus of ϕ with respect to

the norm v for each r ∈ R+. We say that the transcendental entire function f : Cm+n → C is of order
ρ, if log |f| is of order ρ, where

(2.1) ρ = lim sup
r→∞

log Mf,v(r)

log r

If ρ < +∞, f is said to have maximal, normal or minimal type if

(2.2) σ = lim sup
r→∞

Mf,v(r)

rρ
,

is infinite, finite or zero.

Let K be a compact subset of Cm+n, which is nonpluripolar on each irreducible component of a
complete intersection variety X. The Siciak extremal function VK associated to K has been studied
extensibly by Siciak [18] and Sadullaev ([15],[16]) and is defined as:

VK = sup{u(z) : u ∈ i(X); u(ζ) ≤ 0,ζ ∈ K,z ∈ X}
where the subcone i(X) is given by

i(X) = {u(z) : u ∈ PSH(X); u(z) ≤ log(‖ z ‖ +1) + Cu,z ∈ X}
here Cu is a constant depending only on the cone of plurisubharmonic function (PSH)u and ‖ . ‖ is
the Euclidean norm on Cm+n.
The upper semi-continuous regularization of VK is defined on X by

V ∗K(z) = lim sup
ζ→z

VK(ζ),ζ ∈ K,z ∈ X,

V ∗K(z) is PSH(X) and satisfies

V ∗K(z) ≤ log(‖ z ‖ +1) + O(1),as ‖ z ‖→ +∞.
If VK is continuous on Cm+n, then VK = V ∗K ∈ i. It is given in [18] that if, for all z ∈ K,VK is
continuous, then VK is continuous on X. In this case we say that K is L− regular in X. We define
the sublevel sets of the extremal function VK by setting

Ωα = {z ∈ X : VK(z) ≤ α},α > 1,α ∈ R,
and sublevel sets of the upper semi-continuous regularization V ∗K of VK by

Ωr = {z ∈ Cm+n : expV ∗K(z) ≤ r},r > 1.
It has been observe that the sequence of sublevel sets {Ωr}r>1 exhausts the complex space Cm+n. For
f : Cm+n → C a transcendental entire function, set

MK,f (r) = sup
z∈Ωr

|f(z)|,r > 1.

It can be easily shown that log+ MK,f (r) and log
+ MK,v(r) give the same order given by

(2.3) ρ ≡ ρ(f) = lim sup
r→∞

log log+ MK,f (r)

log r
.



24 DEVENDRA KUMAR

If 0 < ρ < +∞, the type of f : Cm+n → C is defined by

(2.4) σ ≡ σ(f) = lim sup
r→∞

log+ MK,f (r)

rρ
.

Zeriahi [23] constructed an orthogonal polynomial basis {Ak}k≥1 for the space ⊕(X). The basis is
orthogonal in the Hilbert space L2(X,µ), essentially by means of the Hilbert-Schmidt process, here µ
be the extremal capacity measure on K given by µ = (ddcVK. Further details on this positive Borel
measure µ supported on K can be obtained from the paper of E. Bedford and B.A. Taylor [1].

Let Pd(Cm+n) denote the C-vector space of polynomials πd : Cm+n → C of degree ≤ d for d ≥ 1.
Let L2P (K,µ) denote the closed subspace of the Hilbert space L

2(K,µ) generated by the restriction to
K of polynomials πd ∈ Pd(Cm+n) of degree (πd) ≤ d, for d ≥ 1. Then every function f ∈ L2P (K,µ)
has a power series expansion of the form

(2.5) f = Σk≥1fkAk,

with

fk =
1

∆2k(K)

∫
K

f.Akdµ, ∆k(K) = (

∫
K

|Ak|2dµ)
1
2 ,k ≥ 1,

here . is the dot product of vectors.
Let Lp(K,µ),p ≥ 1 denote the class of all functions such that

‖ f ‖Lp(K,µ)= (
∫
K

|f|pdµ)
1
p < ∞,

then we define the best polynomial approximation error in Lp-norm, p ≥ 1, by
(2.6) E

p
d(K,f) = inf{‖ f −πd ‖Lp(K,µ),πd ∈ Pd(C

m+n)}.
If the extremal function VK associated with K is continuous for every z ∈ K, then VK is continuous on
X and L− regular, so instead of defining sublevel sets for the upper semi-continuous regularization,
we define the same for VK by setting

Ωr = {z ∈ X : VK(z) < log r,r ∈ R,r > 1}.
Then we have

VK(z) ≥
1

sk
log(

|Ak|
ak(K)

),

where

ak(K) = max
z∈K
|Ak(z)|, |Ak|Ωr ≤ ak(K)r

sk,sk = degree(Ak).

Following the Siciak [18] we observe that if K is L−regular then

lim sup
d→∞

(E
p
d(K,f))

1
d =

1

R
< 1

if and only if f has an analytic continuous to

{z ∈ Cm+n; VK(z) < log(
1

R
)}.

3. Auxiliary Results

In this section we shall state some preliminary results which will be used in the sequel.

First we state Zeriahi’s Bernstein-Markov type inequality [23]:
BM:For all � > 0, there exists a constant C� > 0 such that

(3.1) sup
z∈K
|f(z)| ≤ C�(1 + �)deg(f)(

∫
K

|f|2dµ)
1
2

for every holomorphic function f with polynomial growth on the complete intersection algebraic vari-
ety X and K is a nonpluripolar compact subset of X.
Now we state the following lemmas of Zeriahi extending the classical Cauchy inequality.



ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 25

Lemma 3.1. Let f = Σk≥0fkAk be a holomorphic function on X. Then for every θ > 1, there
exist an integer Nθ and a constant Cθ > 0 such that

(3.2) |fk|rsk∆k(K) ≤ Cθ
(r + 1)Nθ

(r − 1)2n−1
|fk|Ωrθ,

for every r > 1,k ≥ 1, where Cθ and Nθ are independent of r,k and f.
Lemma 3.2. If K is an L−regular, then the sequence of extremal polynomials {Ak}k≥1 satisfies

(3.3) lim
k→∞

(
|Ak(z)|
νk

)
1
sk = exp(VK(z)),νk =‖ Ak ‖L2(K,µ),

for every z ∈ Cm+n and

(3.4) lim
k→∞

(
|Ak(z)|
νk

)
1
sk = 1.

4. Main Results

In this section we shall prove our main theorems. Moreover, we shall characterize the classical
growth parameters order and types of transcendental entire function in terms of Lp-approximation
error defined by (2.6).

Theorem 4.1.If f : X → C is a transcendental entire function on X with a series expansion (2.5)
with respect to the orthogonal polynomial basis {Ak}k≥1, then f ∈ Lp(K,µ), 1 < p ≤ ∞ is of finite
order if and only if

(4.1) ρ = lim sup
k→∞

sk log sk
− log(Epsk(K,f))

< +∞,

and ρ = ρ1, where E
p
sk

(K,f) is defined by (2.6).
Proof. First we have to prove that ρ ≤ ρ1. If ρ1 = ∞, then nothing to be prove. Assume that ρ1 < ∞
and let � > 0. For a sufficiently large k, from (4.1) we have

0 ≤
sk log sk

− log(Epsk(K,f))
≤ ρ1 + �

or

(4.2) Epsk(K,f) ≤ (sk)
−sk
ρ1+� .

Since adding a polynomial will not change the order of a function. Thus, for r ≥ 1 and a0(K) = 0, we
can assume that following inequality holds for every k ≥ 0,

(4.3) MK,f (r) ≤ Σk≥1|fk|ak(K)rsk.

Now we will proceed the proof in two steps (p ≥ 2) and (1 < p < 2). Let f = Σk≥0fk.Ak be an element
of Lp(K,µ).

Step 1. If f ∈ Lp(K,µ) with p ≥ 2, then f = Σ∞k=0fk.Ak with convergence in L
2(K,µ),

fk =
1

ν2k

∫
K

f.Akdµ,k ≥ 1,νk ≡ ∆k(K),

or

=
1

ν2k

∫
K

(f −Psk−1).Akdµ.

It gives

|fk| ≤
1

ν2k

∫
K

|(f −Psk−1)|.|Ak|dµ,



26 DEVENDRA KUMAR

now using Bernstein−Walsh inequality and Hölder′s inequality we have for any � > 0

(4.4) |fk|νk ≤ C�(1 + �)skE
p
sk−1(K,f),k ≥ 0.

Step 2. If 1 ≤ p < 2, let p′ such that 1
p

+ 1
p′

= 1 then p′ ≥ 2. By Hölder′s inequality we get

|fk|ν2k ≤‖ f −Psk−1 ‖Lp(K,µ)‖ Ak ‖Lp′(K,µ) .

But ‖ Ak ‖Lp′(K,µ)≤ C ‖ Ak ‖K= Cak(K), now by Bernstein−Markov inequality we have

|fk|ν2k ≤ CC�(1 + �)
sk ‖ f −Psk−1 ‖Lp(K,µ),

it gives

(4.5) |fk|ν2k ≤ C
′
�(1 + �)

skEpsk(K,f).

From (4.4) and (4.5), we get for p ≥ 1

(4.6) |fk|νk ≤ A�(1 + �)skEpsk(K,f),

where A� is a constant depends only on �.

Now using Zeriahi′sBernstein−Markov type inequality in (4.3) and (4.6), we obtain

MK,f (r) ≤ Σk≥0|fk|C�(1 + �)skνkrsk ≤ Σk≥0A�C�(1 + �)2skEpsk(K,f)r
sk,

using inequality (4.2) in above, we get

MK,f (r) ≤ C′�Σk≥0(1 + �)
2sk(sk)

−sk
ρ1+� rsk = Σ1 + Σ2,

where

Σ1 = Σ1≤k≤(2r(1+�)2)(ρ1+�) (1 + �)
2sk(sk)

−sk
(ρ1+�) rsk

and

Σ1 = Σk≥(2r(1+�)2)(ρ1+�) (1 + �)
2sk(sk)

−sk
(ρ1+�) rsk.

In Σ2, we have (r(1 + �)
2k)

−1
(ρ1+�) ≤ 1

2
, so that Σ2 ≤ 1, and

Σ1 ≤ (F(r,�))(ρ1+�)Σk≥1(sk)
−sk

(ρ1+�)

where F(r,�) = (r(1 + �)2)(2r(1+�)
2)

or

Σ1 ≤ K1 exp((2r(1 + �)2)(ρ1+�) log(r(1 + �)2)) ≤ K2 exp(r(ρ1+�)),

for some constants K1 > 0,K2 > 0. Hence it follows from definition of order given by (2.3) that
ρ ≤ ρ1 + �, since � > 0 is arbitrary, it gives

(4.7) ρ ≤ ρ1.

In order to prove the reverse inequality i.e., ρ1 ≤ ρ we consider the polynomial of degree sk as

Psk(z) = Σ
k
j=0fjAj,

then

(4.8) E
p
sk−1(K,f) ≤ Σ

∞
sj=sk

|fj| ‖ Aj ‖Lp(K,µ)≤ C0Σ∞sj=sk+1|fj| ‖ Aj ‖K,k ≥ 0,p ≥ 1.

In the consequences of Lemmas 3.1 and 3.2, we obtain the following inequality

(4.9) |fk|ak(K) ≤
MK,f (r)

rsk
,r > 0.

Using (4.9)in (4.8), we get

(4.10) E
p
sk−1(K,f) ≤ C0Σ

∞
sj=sk+1

MK,f (r)r
−sj = C0MK,f (r)

(r∗/r)(sk+1)

1 − (r∗/r)



ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 27

for all sufficiently large sk and all r > r
∗,r∗ > 1. Here C0 is some fixed number.

For all sufficiently large sk and r > 2r
∗, (4.10) gives

(4.11) E
p
sk−1(K,f) ≤ γMK,f (r)(r

∗/r)sk,

where γ is a constant independent of sk and r.

If ρ1 = 0, then nothing to be prove. Let us assume that 0 < ρ1 < ∞. If ρ1 < ∞, define ρ∗ = ρ1 −�,
for small � > 0, so that ρ1 > 0. Let ρ

∗ > 0 be arbitrary if ρ1 = +∞. Then for infinitely many indices
k ≥ 1, from (4.1) we have

sk log sk ≥ ρ∗ log(Epsk(K,f))
−1

or

(4.12) log Epsk(K,f) ≥
−sk log sk

ρ∗
.

Using (4.12) in (4.11) we get

(4.13) log MK,f (r) ≥
−sk log sk

ρ∗
+ sk log(

r

r∗
) − log γ.

The minimum value of right hand side of (4.13) is obtained at
rsk
r∗

= (esk)
1
ρ∗ and substituting the

value of (
rsk
r∗

) in (4.13) we obtain the following inequality

log MK,f (r) ≥
sk
ρ∗
− log γ

or

log log MK,f (rsk)

log rsk − log r∗
≥ ρ∗(

log sk − log ρ∗

log sk + 1
).

Proceeding the limits and taking the definition (2.3) into account, we get

(4.14) ρ = lim sup
r→∞

log log MK,f (r)

log r
≥ lim sup

r→∞

log log MK,f (rsk)

log rsk
≥ ρ∗.

Since ρ∗ is arbitrary real number, smaller than ρ, it gives that ρ ≥ ρ1. Now in view of (4.7) the result
is immediate. Hence the proof is completed.
Theorem 4.2 If f : X → C is a transcendental entire function on X with a series expansion (2.5)
with respect to the orthogonal polynomial basis {Ak}k≥1, then f ∈ Lp(K,µ) with a finite order
ρ(0 < ρ < ∞) has finite type σ(0 < σ < ∞) if and only if

eρσ = lim sup
k→∞

sk(E
p
sk

(K,f))
ρ
sk < +∞,

and σ1 = σ, where E
p
sk

(K,f) is given by (2.6).
Proof. Let δ1 = eρσ1. For given � > 0 and δ1 > 0, we have for sufficiently large k

(4.15) sk(E
p
sk

(K,f))
ρ
sk ≤ δ1 + �

or

sk log sk
− log(Epsk(K,f))

≤
ρ

1 − log(δ1+�
sk

)
.

Now it follows from Theorem 4.1 that the order of f is at most ρ.
Now let us consider that 0 < δ1 < ∞ and we have to show that σ ≤ δ1eρ = σ1. From (4.15) we get

(4.16) Epsk(K,f) ≤ (
δ1 + �

sk
)
sk
ρ .

Consider

(4.17) |f(z)| ≤ Σk≥1|fk||Ak|Ωr ≤ Σk≥1|fk|ak(K)r
sk.

Further we will proceed the proof by considering two cases:



28 DEVENDRA KUMAR

Case 1. Let p ≥ 2, then we have f = Σk≥0fkAk because f ∈ L2(K,µ),Lp(K,µ) ⊂ L2(K,µ) and
{Ak}k is a basis of L2(K,µ). Consider the series Σk≥0fkAk in Cm+n and it can be easily seen that
this series converges uniformly on every compact subsets of Cm+n to an entire function. Using the
Bernstein−Markov inequality (BM) in (4.17) we get

|f(z)| ≤ C�Σk≥1|fk|(1 + �)skνkrsk,

it gives from (4.6) that

|f(z)| ≤ C�Σk≥1|fk|(1 + �)2skEpsk(K,f)r
sk.

Now in view of (4.16) we have

|f(z)| ≤ C′�Σk≥1((1 + �)
2ρ(

δ1 + �

sk
)rρ)

sk
ρ = C′�Σ1 + Σ2.

Let us assume the function

φ(s) = ((r(1 + �)2)ρ(
δ1 + �

s
))
s
ρ ,s > 0.

This function attains its maximum value at

s = (
δ1 + �

e
)(r(1 + �)2)ρ

and the value is equal to exp((δ1+�
eρ

)(r(1 + �)2)ρ). Hence for any constant K1 > 0,

C′�Σ1 = C
′
�Σ1≤k≤2(δ1+�)(1+�)2ρrρ((

δ1 + �

sk
)(1 + �)2ρrρ)

sk
ρ

≤ 2(δ1 + �)(1 + �)2ρrρ exp((
δ1 + �

eρ
)(r(1 + �)2)ρ)

≤ K1 exp((
δ1 + �

eρ
)(r(1 + �)2)ρ),

and

C′�Σ2 = C
′
�Σk>2(δ1+�)(1+�)2ρrρ((

δ1 + �

sk
)(1 + �)2ρrρ)

sk
ρ

≤ C′�Σk≥1
1

2k
= K2 < ∞.

Thus from above discussion we get σ ≤ δ1
eρ

.

Case 2. For the case 1 ≤ p < 2 and f ∈ Lp(K,µ) by (BM) inequality and Hölder′s inequality we
get again the inequality (4.6). Now proceeding on the lines of proof of Case 1, the result is immediate.

In order to prove the reverse inequality, we note that if δ1 > � > 0, then for infinitely many indices
k

(4.18) Epsk(K,f) ≥ (
δ1 − �
sk

)
sk
ρ .

Now using (4.18) in (4.11) we obtain

log MK,f (rsk) ≥
sk
ρ

log(
δ1 − �
sk

) + sk log(rsk/r
∗) − log γ.

The minimum value of right hand side is attains at
rsk
r∗

= esk
(δ1−�)

. Thus we get

MK,f (rsk) ≥ e
sk
ρ = exp((

δ1 − �
eρ

)rρsk) + O(1).

Proceeding to limits and using the definition (2.4) of type of f ∈ Lp(K,µ), we get

σ ≥
δ1
eρ
.



ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 29

This completes the proof of theorem.
Remark 4.3. Theorem 4.1 and 4.2 also holds for (0 < p < 1) (see[9]).

References

[1] E. Bedford and B.A. Taylor, The complex equilibrium measure of a symmetric convex set in Rn, Trans. Amer. Math.
Soc. 294(1986), 705-717.

[2] R.P. Boas, Entire Functions, Academic Press, New York, 1954.

[3] S.M.Einstein-Matthews and H.S. Kasana, Proximate order and type of entire functions of several complex variables,

Israel Journal of Mathematics 92(1995), 273-284.
[4] S.M. Einstein-Matthews and Clement H. Lutterodt, Growth of transcendental entire functions on algebraic varieties,

Israel Journal of Mathematics 109(1999), 253-271.

[5] Adam Janik, On approximation of entire functions and generalized order,Univ. Iagel. Acta. Math. 24(1984), 321-326.
[6] O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p, q)-order and lower (p, q)-order of an entire function, J. Reine

Angew. Math. 282(1976), 53-67.

[7] O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p, q)-type and lower (p, q)-type of an entire function, J. Reine
Angew. Math. 290(1977), 180-190 .

[8] H.S. Kasana and D. Kumar, On approximation and interpolation of entire functions with index-pair (p, q), Publica-
cions Matematique 38 (1994), No. 4, 681-689.

[9] D. Kumar, Generalized growth and best approximation of entire functions in Lp-norm in several complex variables,

Annali dell’ Universitá di Ferrara VII, 57 (2011), 353-372.
[10] P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Grundlehren der Mathematischen Wis-

senschaften 282, Springer-Verlag, Berlin, 1986.

[11] B. Ja. Levin, Distributions of Zeros of Entire Functions, Translations of Mathematics Monographs 55, Amer. Math.
Soc., Providence, R.I., 1994.

[12] A.R. Reddy, Approximation of an entire function, J. Approx. Theory 3(1970), 128-137.

[13] A.R. Reddy, Best polynomial approximation to certain entire functions, J. Approx. Theory 5(1972), 97-112.
[14] L.I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Amer. Math. Soc., Providence,

R.I., 1974.

[15] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Mathematical Surveys
36(1981), 61-119.

[16] A. Sadullaev, An estimate for polynomials on analytic sets, Mathematics of the USSR-Izvestiya 20(1980), 493-502.
[17] S.M. Shah, Polynomial approximation of an entire function and generalized orders, J. Approx. Theory 19(1977),

315-324.

[18] J. Siciak, Extremal plurisubharmonic functions and capacities in Cn, Lectures in Mathematics 14, Sofia University,
Tokyo, 1982.

[19] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex

variables, Trans. Amer. Math. Soc. 105(1962), 322-357.
[20] G.S. Srivastava and Susheel Kumar, On approximation and generalized type of entire functions of several complex

variables, European J. Pure Appl. Math. 2, no. 4(2009), 520-531.

[21] T.N. Winiarski, Application of approximation and interpolation methods to the examination of entire functions of
n complex variables, Ann. Polon. Math. 28(1973), 97-121.

[22] T.N. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23(1970), 259-273.

[23] A. Zeriahi, Meilleure approximation polynomiale et croissante des fonctions entieres sur certaines varievalgebriques
affines, Annales Inst. Fourier (Grenoble) 37(1987), 79-104.

Department of Mathematics, Al-Baha University, P.O.Box-1988, Alaqiq, Al-Baha-65431, Saudi Arabia,
K.S.A.

∗Corresponding author: d kumar001@rediffmail.com