EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 258-268
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Coefficients Characterization of Entire Harmonic
Functions in Terms of Norm of Gradients at Origin in

Rn, n ≥ 3

Devendra Kumar1,2, Rajeev Kumar Vishnoi3,∗

1 Department of Mathematics, Faculty of Sciences Al-Baha University, P.O.Box-1988, Alaqiq,
Al-Baha-65431, Saudi Arabia, K.S.A.
2 Research and Post Graduate Studies, Department of Mathematics, M. M. H. College,
Model Town, Ghaziabad-201001, U.P., India
3 Department of Mathematics, Vardhaman College Bijnor-246701, U.P., India

Abstract. Coefficient characterizations of generalized order, lower order and generalized type of
entire harmonic function having the spherical harmonic expansion throughout a neighborhood of
the origin in Rn have been obtained in terms of norm of gradients at origin.

2020 Mathematics Subject Classifications: 31B05, 42A16

Key Words and Phrases: Norm of gradient, entire harmonic functions, generalized orders,
generalized type and growth of zero orders.

1. Introduction

In the study of entire functions of one complex variable, the main issues are the rela-
tionship between the growth of such functions and behavior of Taylor coefficients. Several
authors such as Srivastava and Kumar [20], Kumar [11,14], Harfaoui [8] and others inves-
tigated growth parameters of entire functions in terms of Taylor’s series coefficients and
polynomial approximation errors in different norms. Similar studies have been done for
harmonic functions by Kumar [12,13], Kumar and Kasana [15] and Armitage [1] as they
have series expansion in terms of spherical harmonics in Rn. Some times it is useful to
study the growth of harmonic functions in terms of norm of their gradient at the origin in
n-dimensional space. Such results are equivalent to characterization in terms of spherical
harmonic coefficients. Results of one kind can not obtained directly from the other, and
thus require separate study. Also, the problem to investigate the growth characteristics
of harmonic functions in terms that are not related to series expansion coefficients. The

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3636

Email addresses: d kumar001@rediffmail.com (Devendra Kumar),
rajeevvishnoi100@gmail.com (Rajeev Kumar Vishnoi)

http://www.ejpam.com 258 c© 2020 EJPAM All rights reserved.



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 259

derivatives of a harmonic functions at the origin are equal to complicated linear combina-
tions of the spherical harmonic coefficients. The relevance of this study is due to the fact
that the harmonic functions play very important role in theoretical mathematical research,
physics and mechanics to express stationary processes.

Therefore, the aim of this paper is to characterize the generalized growth parameters
(generalized order, lower order and generalized type) in the sense of Sheremeta [18] of
entire harmonic functions in terms of norm of gradient at origin .

It is significant to mention here that time dependent problems in R3 leads to the study
of entire harmonic functions in R4.

A function H(x), x ∈ Rn which has continuous partial derivatives of second order and
satisfies Laplace,s differential equation

n∑
i=1

∂2H

∂x2
i

= 0

is said to be harmonic in n-dimensional space Rn.

The function H has the spherical harmonic expansion throughout a neighborhood of
the origin in Rn as

H(x) =
∞∑
k=0

Hk(x), (1.1)

where Hk(x) is a harmonic homogeneous polynomial of degree k in x1, x2, . . . xn having
real coefficients [3, pp. 47]. These polynomials are known as spherical harmonics.

Let Sn = {x ∈ Rn : |x| = 1} be a unit sphere in Rn. The series (1.1) also can be
expressed as

H(x) =

∞∑
k=0

dk∑
j=1

ajkQ
j
k(
x

r
)rk, |x| = r, (1.2)

where {Qjk}
dk
j=1 be an orthonormal basis for Hk with respect to the scalar product

< f, g >=
1

wn

∫
Sn
f(x)g(x)dσ1,

while

wn =
2πn/2

Γ(n/2)

denotes the area of Sn and dσ1 is the element of surface area on Sn.



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 260

Also (See [21, pp.145])

dk =
(n+ 2k − 2)(n+ k − 3)!

k!(n− 2)!

is the dimension of vector space Hk and

ajk =
1

wn

∫
Sn
H(x)Qjk(x)dσ1.

If the series (1.2) converges uniformly on the sphere |x| = (
∑n

i=1 x
2
i )

1
2 = δ, then we have

ajk =
1

δ2k+n−1wn

∫
|x|=δ

H(x)Qjk(x)dσ,

where dσ = δn−1dσ1 is the element of surface area on the sphere |x| = δ.

For each n-tuple a = (a1, a2, . . . , an) of non-negative integers, we define |a| = a1 +a2 +

· · ·+ an, a! = a1!a2! . . . an! and Da = ∂|a|
∂
a1
x1
∂
a2
x2
...∂anxn

.

In view of [2] for each ξ ∈ C∞(Rn) and non-negative integer k, we define the norm of
the kth gradients of H at origin by

|∇kH(0)| = (
k!

2k

∑
|a|=k

[DaH(0)]2

a!
)
1
2 . (1.3)

It has been proved [5] that the series (1.1) converges absolutely and uniformly on
compact subsets of the open ball |x| < R, where

R−1 =
√

2 lim sup
k→∞

(
|∇kH(0)|

k!
)
1
k . (1.4)

Definition (1.3) and equality (1.4) immediately show that the series (1.1) converges
absolutely and uniformly on compact subsets of open ball |x| < R, where

R−1 = lim sup
k→∞

(
|∇kH(0)|

k!
)
1
k , (1.5)

and such convergence can not obtain within any larger ball centered at origin.

Fryant and Shankar [4] proved the following lemma.

Lemma A. Let H(x) =
∑∞

k=0Hk(x) is uniformly convergent in a neighborhood of the
origin in Rn. Then for all r < R,

M2(r,H) ≤M(r,H) ≤ N(r,H),



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 261

where M(r,H) = max|x|=r |H(x)|,

M2(r,H) = [Γ(n/2)
∞∑
k=0

|∇kH(0)|2

k!Γ(k + n/2)
r2k]

1
2 ,

and

N(r,H) =
√

Γ(n/2)
∞∑
k=1

√
dk

|∇kH(0)|√
k!Γ(k + n/2)

rk.

Here the upper bound of M(r,H) holds for all r ≥ 0, and the lower bound of M(r,H)
obtains for all r such that the spherical harmonic series H is uniformly convergent on the
sphere |x| = r.

We define the order ρ of H as

ρ = lim sup
r→∞

log logM(r,H)

log r
, 0 ≤ ρ ≤ ∞,

and when 0 < ρ <∞, the type T is defined as

T = lim sup
r→∞

logM(r,H)

rρ
, 0 ≤ T ≤ ∞.

Fugard [6] characterized the order and type of an entire harmonic function in terms of
the mth gradient defined above. Also, Kumar and Singh [16] investigated these results for
non entire case. Srivastava [19] improved Fugard’s results and obtained generalized order
and generalized type. In this paper, we extend the results of Srivastava [19].

2. Generalized Growth

Let ξ : [a,∞) → R for some a ≥ 0, such that ξ(x) is positive, strictly increasing and
differentiable and tends to ∞ as x → ∞. Then ξ is said to belong to the class L0 if for
every real valued function φ(x) such that φ(x)→ 0 as x→∞, ξ satisfies

lim
x→∞

ξ[(1 + φ(x))x]

ξ(x)
= 1,

and belongs to the class Λ if for all c, 0 < c <∞, we have the stronger condition

lim
x→∞

ξ(cx)

ξ(x)
= 1.

Let α, β ∈ L0,Λ, following the analogy with [18], we define generalized and lower
generalized order of the entire harmonic function H ∈ Rn by



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 262

ρ(α, β,H) = lim sup
r→∞

α(logM(r,H))

β(r)
, λ(α, β,H) = lim inf

r→∞

α(logM(r,H))

β(r)
.

Now we prove

Theorem 2.1. Let H be a harmonic function in a neighborhood of the origin in Rn,
satisfying one of the following conditions:
(i). For α, β ∈ Λ, F (t, c) = β−1(cα(t)), 0 < c <∞,

lim
t→∞

d(logF (t, c))

d(log t)
= O(1).

(ii). For α, β ∈ L0,

lim
t→∞

d(logF (t, c))

d(log t)
= p, 0 < p <∞,

then the generalized order ρ(α, β,H) of entire harmonic function H is determined by

ρ(α, β,H) = lim sup
k→∞

α(pk)

β(epR[ |∇kH(0)|
k! ]

−1
k )
.

Proof. Consider the entire functions of single complex variable z:

f1(z) =
√

Γ(n/2)
∞∑
k=1

|∇kH(0)|√
k!Γ(k + n/2)

(
z

R
)k,

and

f2(z) =
√

Γ(n/2)
∞∑
k=1

√
dk

|∇kH(0)|√
k!Γ(k + n/2)

(
z

R
)k.

Since

Γ(k + n/2)

Γ(k + 1)
= k

n
2
−1,

we have
√
dk
√

Γ(n/2)|∇kH(0)|√
k!Γ(k + n/2)

'
√
dk
√

Γ(n/2)|∇kH(0)|
k!k(n−2)/4

,

as

(dk)
1
2 = [

(n+ 2k − 2)(n+ k − 3)!

k!(n− 2)!
]
1
2 = [

(n+ k − 3)!

(k − 1)!
]
1
2 = k

(n−2)
2 .



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 263

Therefore,

√
dk
√

Γ(n/2)|∇kH(0)|√
k!Γ(k + n/2)

'
√

Γ(n/2)|∇kH(0)|k(n−2)/4

k!
,

or

lim
k→∞

[

√
dk
√

Γ(n/2)|∇kH(0)|√
k!Γ(k + n/2)

]−
1
k ' [

|∇kH(0)|
k!

]−
1
k .

Hence f1(z) and f2(z) defined above are entire functions in view of (1.5).

Using Lemma A, we obtain

µ(r, f1) ≤M(r,H) ≤M(r, f2), (2.1)

where µ(r, f1) is the maximum term of the power series expansion of function f1(z) on the
circle |z| = r and M(r, f2) = max|z|=r |f2(z)|.

We see that

|f1(z)|2 =
∞∑
k=0

{
√

Γ(n/2)|∇kH(0)|√
k!Γ(k + n/2)

}2(
r

R
)2k

+
∑
k 6=m
{
√

Γ(n/2)|∇kH(0)||∇m|H(0)|√
k!m!Γ(k + n/2)Γ(m+ n/2)

}( z
R

)k(
z

R
)m.

(2.2)

Using (2.2) with the estimate [6, p.290] of M2(r,H), we get

M(r, f1 ≥M2(r,H) ≥ B(
r

R
)k
|∇kH(0)|

k!
, (2.3)

where B is a finite constant. Since µ(r, f1) is the maximum term of f1(z) then by a result
of Valiron [22, p.34], we obtain

logM(r, f1) ' logµ(r, f1) as r →∞. (2.4)

Now taking into account the definition of ρ(α, β,H) with (2.1) and (2.4) we get

ρ(α, β, f1) ≤ ρ(α, β,H) ≤ ρ(α, β, f2).

Applying the coefficient formula of generalized order of an entire function of one com-
plex variable [18] and bearing in mind that β ∈ Λ or L0, we obtain

ρ(α, β, f1) = ρ(α, β, f2 = lim sup
k→∞

α(pk)

β(epR[ |∇kH(0)|
k! ]−

1
k )
. (2.5)



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 264

Remark 2.1. If α(x) = β(x) = log x, we get the classical order ρ(H) in terms of norm
of gradient at the origin studied by Fugard [6, Thm.2.1],

ρ(H) = lim sup
k→∞

log k

[ |∇kH(0)|
k! ]−

1
k

, ρ(H) = ρ.

Remark 2.2. If α(x) = x, β(x) = xρ, p = 1
ρ , then (2.5) gives the formula for the

classical type T (H) obtained by Fugard [6, Thm.2.6],

R(T (H)ρe)
1
ρ = lim sup

k→∞
k

1
ρ (
|∇kH(0)|

k!
)
1
k .

Remark 2.3. If α(x) = x, β(x) = xρ(x), where ρ(x) is the proximate order of the entire
function H, then the formula for the generalized type T ∗(H) with respect to proximate
order ρ(x) is given by

R(T ∗(H)ρe)
1
ρ = lim sup

k→∞
θ(k)(

|∇kH(0)|
k!

)
1
k ,

where x = θ(k)⇔ k = xρ(x).

Theorem 2.2. Let H be a harmonic function in a neighborhood of the origin in
Rn, n ≥ 3, for which

λ(α, β,H) ≥ lim inf
k→∞

α(pk)

β(epR[ |∇kH(0)|
k! ]−

1
k )
. (2.6)

If the function

µ(k) = { |∇kH(0)|
|∇k+1H(0)|

}
√

(k + 1)(k + n/2)

be a nondecreasing function of k for all large values of k and one of the (i),(ii) conditions
of Theorem 2.1 is satisfied, then inequality in (2.6) converts in equality.

Proof. As f1(z) is defined above is an entire function and

logM(r, f1) ' logM(r,H) as r →∞.

Hence f1(z) is also of generalized lower order λ(α, β, f1). Since under the assumption
√

Γ(n/2)|∇kH(0)|√
k!Γ(k+n/2)√

Γ(n/2)|∇k+1H(0)|√
(k+1)!Γ(k+1+n/2)

' |∇kH(0)|
|∇k+1H(0)|

√
(k + 1)(k + n/2)

is non-decreasing function of k. Now applying [17, Thm.2] with (2.1), for the function
f1(z) and f2(z) we get the required result.



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 265

3. Growth of Entire Harmonic Functions of Zero Order

To study the growth of entire functions of zero order, Kapoor and Nautiyal [10] defined
a new class of functions as follows:
The class of functions ξ(x) denoted by Ω which satisfies:
(i). ξ(x) is positive, defined on [a,∞), differentiable, strictly increasing and tends to ∞
as x→∞.
(ii). ξ(x) such that

lim
x→∞

d(ξ(x))

d(log x)
= K, 0 < K <∞.

The generalized order ρ(α, α, f), generalized lower order λ(α, α, f) and generalized type
of the entire function f(z) were defined as:

ρ(α, α,H) = lim sup
r→∞

α(logM(r, f))

α(log r)
, λ(α, α, f) = lim inf

r→∞

α(logM(r, f))

α(log r)
,

T (α, α, f) = lim sup
r→∞

α(logM(r, f))

[α(log r)]ρ
,

where α(x) ∈ Ω and 1 ≤ λ(α, α, f) ≤ ρ(α, α, f) ≤ ∞.

The coefficient characterizations of entire function f(z) =
∑∞

n=0 akz
k were also ob-

tained as follows:

ρ(α, α, f) = 1 + lim sup
k→∞

α(k)

α(log |ak|−
1
k )
. (3.1)

If | akak+1
| be a non-decreasing function of k, then

λ(α, α, f) = 1 + lim inf
k→∞

α(k)

α(log |ak|−
1
k )
. (3.2)

Also, for α(x) ∈ Ω, Ganti and Srivastava [7] obtained

T (α, α, f) = lim sup
k→∞

α(kρ )

{α( ρ
ρ−1 log |ak|−

1
k )}ρ−1

,

provided dF (k;T,ρ)
d(log x) = O(1) as x→∞ for all T, 0 < T <∞.

Ning Juhong and Chen Qing [9] improved above results by introducing a new class Ω∗

(the extension of Ω).



D. Kumar, R.K. Vishnoi / Eur. J. Pure Appl. Math, 13 (2) (2020), 258-268 266

The class of functions ξ(x) ∈ Ω∗ satisfies (i) and (iii)

(iii).limx→∞
d(ξ(x))

d(log[q] x)
= K, 0 < K <∞, q ≥ 1, q ∈ N+,

where log[q] x = log[q−1] log x, log[0] = x.

The class ξ(x) also satisfies L0 and Λ.
It is clear that α(x) ∈ Ω is a particular case of α(x) ∈ Ω∗. for q = 1.

Ning Juhong and Chen Qing [9] obtained the following coefficient characterization:
Let α(x) ∈ Ω∗, then some necessary and sufficient conditions of the entire function

f(z) having generalized order ρ is

lim sup
r→∞

α(logM(r, f))

α(log r)
− 1 = lim sup

k→∞

α(k)

α(log |ak|−
1
k )

for q = 1, (3.3)

lim sup
k→∞

α(k)

α(log |ak|−
1
k )
≤ lim sup

r→∞

α(logM(r, f))

α(log r)

≤ lim sup
k→∞

α(k)

α(log |ak|−
1
k )

+ 1, for q = 2, 3, . . . , .

(3.4)

For α(x) ∈ Ω∗, the entire function f(z) of generalized order ρ, 1 < ρ < ∞ having the
generalized type T if and only if

lim sup
r→∞

α(logM(r, f))

[α(log r)]ρ
= lim sup

k→∞

α(kρ )

{α(log |ak|−
1
k )}ρ−1

for q = 1, (3.5)

lim sup
r→∞

α(logM(r, f))

[α(log r)]ρ
= lim sup

k→∞

α(kρ )

{α(log |ak|−
1
k )}ρ

for q = 2, 3, . . . . (3.6)

Now we prove

Theorem 3.1. Let α(x) ∈ Ω∗, then necessary and sufficient conditions for H to be
continued to the entire harmonic function in space Rn, n ≥ 3 having generalized order
ρ1(α, α,H) is

lim sup
k→∞

α(k)

α(log[ |∇kH(0)|
k! ]−

1
k )
≤ lim sup

r→∞

α(logM(r,H))

α(log r)

≤ lim sup
k→∞

α(k)

α(log[ |∇kH(0)|
k! ]−

1
k )

+ 1 for q = 2, 3, . . . .

(3.7)

Proof. Applying the method of proving Theorem 2.1 and taking (3.4) into account
with properties of α(x), we obtain the required result (3.7).



REFERENCES 267

Theorem 3.2. Let α(x) ∈ Ω∗, then the function H can be continued to the entire har-
monic function in space Rn, n ≥ 3, having generalized order ρ1(α, α,H), 1 < ρ1(α, α,H) <
∞, is of generalized type T1(α, α,H) if, and only if

lim sup
r→∞

α(logM(r,H))

[α(log r)]ρ1
= lim sup

k→∞

α(k)

[α(log[ |∇kH(0)|
k! ]−

1
k )]ρ1

for q = 2, 3, . . . .

Proof. The result follows on using (3.6) for the entire function f1(z).

Remark 3.1. Theorems 3.1 and 3.2 have been proved by Srivastava [19] for q = 1.

Acknowledgements

The authors are thankful to the editor for his useful comments, and the referees for
their valuable suggestions which improved the paper.

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