EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 2, 2019, 486-498 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global (p, q)-Growth of Entire Harmonic Functions in Rn in Terms of Approximation Errors Devendra Kumar1,2, Rifaqat Ali3,∗ 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, College of Science, King Khalid University, P.O.Box:9004, Postal Code:61413. Abha, Saudi Arabia, K.S.A. Abstract. The relationship between the generalized growth parameters of an entire harmonic function in space Rn, n ≥ 3, with the rate of its best harmonic polynomial approximation error and ratios of these errors of functions harmonic in the ball of radius R has been studied. 2010 Mathematics Subject Classifications: 30E10, 41A15 Key Words and Phrases: Entire harmonic function, approximation errors, n-dimensional spaces, (p, q)-order and (p, q)-type, harmonic polynomials and spherical harmonics 1. Introduction The approximation of entire functions on compact sets was studied by Srivastava and Kumar [14,15] and obtained generalized growth parameters in terms of approximation and interpolation error. Similar studies have been done for harmonic functions. The harmonic functions play an important role not only in theoretical mathematics but also in Physics and mechanics to describe different stationary processes. Therefore, it is significant to mention here that the study of generalized growth parameters of a harmonic function in an n-dimensional spaces has relevance. Harmonic functions can be expanded into series in spherical harmonics in space Rn, n ≥ 3 and in the adjoined Legendre polynomials in space R3. The growth characteristics of harmonic functions in terms of the coefficients of their expansion into series as well as not related the expansion coefficients, in particular, in terms of the norm of their gradient at the origin were obtained. Also, the growth of harmonic function in terms of approximation errors by harmonic polynomials in Rn, n ≥ 3 was considered by various authors (see,[3,6-13]). The aim of the present work is to investigate ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3373 Email addresses: d kumar001@rediffmail.com (D.Kumar), (rifaqat.ali1@gmail.com ; rrafat@kku.edu.sa (R. Ali)) http://www.ejpam.com 486 c© 2019 EJPAM All rights reserved. D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 487 conditions under which a harmonic function in the ball of n-dimensional space continues to the entire harmonic function, and to derive formulae for the generalized growth parameters (p, q)-order, lower(p, q)-order, (p, q)-type and lower(p, q)-type of harmonic function in space in terms of harmonic polynomial approximation errors. Here p and q are integers such that p ≥ q ≥ 1. Let u be an entire harmonic function in Rn and has a Fourier-Laplace series expansion [16] u(rx) = ∞∑ k=0 Y (k)(x;u)rk, (1.1) where x ∈ Sn = {x ∈ Rn : |x| = 1} a unit sphere in Rn centered at the origin Y (k)(x;u) =a (k) 1 Y (k) 1 (x) + a (k) 2 Y (k) 2 (x) + · · ·+ a(k)γk Y (k) γk (x), a (k) j = (u, Y (k) j ) = Γ(n/2) 2(π) n 2 ∫ Sn u(x)Y (k) j (x)dS, j = 1, γk, γk = (2k + n− 2)(k + n− 3)! k!(n− 2)! . Here dS is the element of the surface area on the sphere Sn, (u, Y (k) j ) is the scalor product in L2(Sn) and Y (k) is a spherical harmonic of degree k, k ∈ Z+ = {0, 1, 2, . . . , } on the unit sphere Sn(n ≥ 2) [16]. Let Bn R = {y ∈ Rn : |y| ≤ R} be the ball of radius R in space Rn, n ≥ 3 centered at the origin, and Bn R be the closure of Bn R. We denote HR, the class of harmonic functions in Bn R and continuous on Bn R, 0 < R <∞. Let πk be the set of harmonic polynomials of degree≤ k. The approximation error of function u ∈ HR by harmonic polynomials P ∈ πk be defined as E (k) R (u) = inf P∈πk {max y∈BnR |u(y)− P (y)|}. (1.2) For u ∈ HR continue to the entire harmonic function of n-dimensional space Rn, n ≥ 3, it is known [17 ,p.45] that lim k→∞ (E (k) R (u)) 1 k = 0. (1.3) The concept of order ρ(F ) and lower order λ(F ) of an entire function F (z) = ∑∞ n=0 anz n was introduced by R.P. Boas [1] as ρ(F ) = lim r→∞ sup log logM(r, F ) log r , λ(F ) = lim r→∞ inf log logM(r, F ) log r . The concept of type T (F ) and lower type t(F ) has been introduced when the entire functions have same nonzero finite order. An entire function of order ρ, 0 < ρ <∞, is said to be of type T (F ) and lower type t(F ) if T (F ) = lim r→∞ sup logM(r, F ) rρ(F ) , t(F ) = lim r→∞ inf logM(r, F ) rρ(F ) , D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 488 where 0 < ρ(F ) <∞,M(r, F ) = max0≤θ≤2π |F (r, θ)|. For the class of order ρ(F ) = 0 and ρ(F ) = ∞, the type can not be defined. To refine the above concept of order and type, Juneja et. al., [4,5] introduced the concept of (p, q)- orders and (p, q)-types. Therefore, we define the (p, q)-order and lower(p, q)-order as ρ(p, q, F ) = lim sup r→∞ log[p]M(r, F ) log[q] r , λ(p, q, F ) = lim inf r→∞ log[p]M(r, F ) log[q] r , (1.4) b ≤ ρ(p, q, F ) ≤ ∞, b = 0 if p > q and b = 1 if p = q. The (p, q)-type and lower(p, q)-type are defined as T (p, q, F ) = lim sup r→∞ log[p−1]M(r, F ) log[q−1] rρ(p,q,F ) , t(F ) = lim inf r→∞ log[p−1]M(r, F ) log[q−1] rρ(p,q,F ) , (1.5) where log[0](x) = x and log[m](x) = log[m−1] log(x) for m ≥ 1. Notations: P (α) = P (α, p, q) = {α if p > q, {1 + α if p = q = 2, {max(1, α) if 3 ≤ p = q <∞, {∞ if p = q =∞, } and M(α) = M(α, p, q) = { 1 eα if (p, q) = (2, 1), {(α− 1)(α−1) αα if (p, q) = (2, 2), {1 if p ≥ 3, } From [4] we define the relations between(p, q)-order, lower(p, q)-order, the coefficients of F (z) and ratios of these successive coefficients as following: Theorem A. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ), then ρ(p, q, F ) = P (L(p, q, F )) where L(p, q, F ) = lim sup n→∞ log[p−1] n log[q] |an|− 1 n . Theorem B. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ), then ρ(p, q, F ) = P (L∗(p, q, F )) D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 489 where L∗(p, q, F ) = lim sup n→∞ log[p−1] n log[q] | anan+1 | . Theorem C. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ) and (| anan+1 |) a nondecreasing function of n for n > n0 then λ(p, q, F ) = P (l(p, q, F )) where l(p, q, F ) = lim inf n→∞ log[p−1] n log[q] |an|− 1 n . Theorem D. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ) and (| anan+1 |) a nondecreasing function of n for n > n0 then λ(p, q, F ) = P (l∗(p, q, F )) where l∗(p, q, F ) = lim inf n→∞ log[p−1] n log[q] | anan+1 | . From [5] we define the relation between (p, q)-type, lower(p, q)-type and the coefficients of F (z) as: Theorem E. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ) and (p, q)-type T (p, q, F ) if and only if T = MV , where V (p, q, F ) = lim sup n→∞ log[p−2] n (log[q−1] |an|− 1 n )ρ−A . with A = 1 if (p, q) = (2, 2) and A = 0 if (p, q) 6= (2, 2). Theorem F. Let F (z) = ∑∞ n=0 anz n be an entire function of (p, q)-order ρ(p, q, F ), lower(p, q)-type t(p, q, F ) and (| anan+1 |) a nondecreasing function of n for n > n0 then t = Mv, where v(p, q, F ) = lim inf n→∞ log[p−2] n (log[q−1] |an|− 1 n )ρ−A . 2. Auxiliary Results In this section we will prove some auxiliary results which will be used in the sequel. Consider the two functions f and g of complex variable z: f(z) = ∞∑ k=0 √ (2ν)!√ 2(2ν + 1)!(k + 2ν)2ν E (k) R (u)( z R )k (2.1) D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 490 and g(z) = ∞∑ k=1 4 (2ν)! (k + 2ν)2νE (k) R (u)( z R )k. (2.2) In view of [17, pp. 47] we see that if u is an entire function then f and g are also entire functions of the complex variable z. Using Lemma 3 with inequality (8) of [17], we get m(r, f) ≤M(r, u) ≤ |Y (0)(ξ, u)|+M(r, g) (2.3) where m(r, f) is the maximum term of power series of function f(z) on the circle{z : |z| = r}, and M(r, g) = max|z|=r |g(z)|. Lemma 2.1. Let f and g be defined by (2.1) and (2.2). Then the (p, q)-orders and (p, q)-types of f and g respectively are equal. Proof. First we consider the case (p, q) = (2, 1), 1 ρ(2, 1, f) = lim inf k→∞ − 1 k log( √ (2ν)!√ 2(2ν+1)!(k+2ν)2ν R−kE (k) R (u)) log k , = lim inf k→∞ logRk(E (k) R (u))−1 + log √ 2(2ν + 1)!(k + 2ν)2ν − log √ (2ν)! k log k , = lim inf k→∞ logRk(E (k) R (u))−1 k log k + 1, then 1 ρ(2,1,f) ≥ 1 and ρ(2, 1, f) ≤ 1. This implies necessarily we have ρ(2, 1, f) = 0 to define ρ(p, q, f). Now ρ(2, 1, f) = 0⇒ lim inf k→∞ 1 ρ(2, 1, f) = +∞ ⇒ lim inf k→∞ logRk(E (k) R (u))−1 k log k = +∞ ⇒ lim inf k→∞ k log k logRk(E (k) R (u))−1 = 0 ⇒ lim inf k→∞ log k! logRk(E (k) R (u))−1 = 0 ⇒ (E (k) R (u)R−1) 1 k → 0. D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 491 For p ≥ q > 1, we have 1 L(p, q, f) = lim inf k→∞ log[q−1](− 1 k log( √ (2ν)!√ 2(2ν+1)!(k+2ν)2ν R−kE (k) R (u))) log[p−1] k = lim inf k→∞ log[q−1]( log[R−k(E (k) R (u))]−1 k + log √ 2(2ν+1)!(k+2ν)2ν k )− log √ (2ν)! k log[p−1] k = lim inf k→∞ 1 log[p−1] k log[q−1]( logR−k(E (k) R (u))−1 k )× (1 + log √ 2(2ν + 1)!(k + 2ν)2ν log[R−k(E (k) R (u))]−1 − log √ (2ν)! log[R−k(E (k) R (u))]−1 ) = lim inf k→∞ log[q−1](− 1 k log(R−kE (k) R (u))) log[p−1] k + log(1 + o(1)) log[p−1] k = lim inf k→∞ log[q−1](− 1 k log(R−kE (k) R (u))) log[p−1] k , and 1 L(p, q, g) = lim inf k→∞ log[q−1](− 1 k log( 4 (2ν)!(k + 2ν)2νE (k−1) R (u)R−k)) log[p−1] k = lim inf k→∞ log[q−1]( log[R−k(E (k) R (u))]−1 k + log(2ν)! k − 1 k log 4(k + 2ν)2ν) log[p−1] k = lim inf k→∞ log[q−1](− 1 k log(R−kE (k) R (u))) log[p−1] k + log(1 + o(1)) log[p−1] k = lim inf k→∞ log[q−1](− 1 k log(R−kE (k) R (u))) log[p−1] k , Using Theorem A, we see that the function f and g have same (p, q)-order, it leads to the fact that ρ(p, q, f) = ρ(p, q, g) = ρ. Now we consider the (p, q)-type for q = 2 as D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 492 1 v(p, q, f) = lim inf k→∞ (− 1 k log( √ (2ν)!√ 2(2ν+1)!(k+2ν)2ν R−kE (k) R (u)))ρ−1 log[p−2] k = lim inf k→∞ ( log[R−k(E (k) R (u))]−1 k + log √ 2(2ν+1)!(k+2ν)2ν k − log √ (2ν)! k )ρ−1 log[p−2] k = lim inf k→∞ 1 log[p−2] k ( logR−k(E (k) R (u))−1 k )ρ−1× (1 + log √ 2(2ν + 1)!(k + 2ν)2ν log[R−k(E (k) R (u))]−1 − log √ (2ν)! log[R−k(E (k) R (u))]−1 )ρ−1 = lim inf k→∞ (− 1 k log(R−kE (k) R (u)))ρ−1 log[p−2] k + log(1 + o(1))ρ−1 log[p−2] k = lim inf k→∞ (− 1 k log(R−kE (k) R (u)))ρ−1 log[p−2] k . Similarly for g we have 1 v(p, 2, g) = lim inf k→∞ (− 1 k log(R−kE (k) R (u)))ρ log[p−2] k . Now for the case q ≥ 3, we have 1 v(p, q, f) = lim inf k→∞ log[q−2](− 1 k log( √ (2ν)!√ 2(2ν+1)!(k+2ν)2ν R−kE (k) R (u)))ρ log[p−2] k = lim inf k→∞ log[q−2]( log[R−k(E (k) R (u))]−1 k + log √ 2(2ν+1)!(k+2ν)2ν k − log √ (2ν)! k )ρ log[p−2] k = lim inf k→∞ 1 log[p−2] k log[q−2]( logR−k(E (k) R (u))−1 k )ρ× (1 + log √ 2(2ν + 1)!(k + 2ν)2ν log[R−k(E (k) R (u))]−1 − log √ (2ν)! log[R−k(E (k) R (u))]−1 )ρ−1 = lim inf k→∞ log[q−2](− 1 k log(R−kE (k) R (u)))ρ log[p−2] k + log(1 + o(1))ρ log[p−2] k = lim inf k→∞ (− 1 k log(R−kE (k) R (u)))ρ log[p−2] k . In the same manner for the function g, we obtain 1 v(p, q, g) = lim inf k→∞ (− 1 k log(R−kE (k) R (u)))ρ log[p−2] k . D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 493 Lemma 2.2. Let u be an entire harmonic function of an n-dimensional space n ≥ 3 with (p, q)-order ρ(p, q, u), lower (p, q)-order λ(p, q, u), (p, q)-type T (p, q, u) and lower (p, q)-type t(p, q, u). If f and g are entire functions defined as in (2.1) and (2.2), then ρ(p, q, f) = ρ(p, q, u) = ρ(p, q, g), (2.4) λ(p, q, f) ≤ λ(p, q, u) ≤ λ(p, q, g), (2.5) T (p, q, f) = T (p, q, u) = T (p, q, g), (2.6) t(p, q, f) ≤ t(p, q, u) ≤ t(p, q, g). (2.7) Proof. From (2.3) with (1.4) and (1.5) we have ρ(p, q, f) ≤ ρ(p, q, u) ≤ ρ(p, q, g). (2.8) Since ρ(p, q, f) = ρ(p, q, g), now (2.4) easily obtain by using (2.8). From (2.3) we can get (2.5) immediately. We denote the common value of (p, q)-order of f, g and u and using (2.3) we get log[p−1]m(r, f) (log[q−1] r)ρ ≤ log[p−1]m(r, u) (log[q−1] r)ρ ≤ log[p−1]m(r, g) (log[q−1] r)ρ It proves (2.6) and (2.7). Now let us define αk = maxx∈Sn |Y (k)(x;u)|, βk = √ (2ν)!√ 2(2ν+1)!(k+2ν)2ν E (k) R R−k and γk = 4 (2ν)!(k + 2ν)2νE (k−1) R R−k. 3. Main Results Theorem 3.1. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order ρ(p, q, u) and (p, q)-type T (p, q, u). If ( E (k) R (u) E (k−1) R (u) ) is a nondecreasing function of k for k > k0, then ρ(p, q, u) = P (L(p, q, u)) where L(p, q, u) = lim sup k→∞ log[p−1] k log[q](E (k) R (u)R−k)− 1 k (3.1) and T (p, q, u) = Mv∗(p, q, u) where v∗(p, q, u) = lim sup k→∞ log[p−2] k (log[q−1](E (k) R (u)R−k)− 1 k )ρ−A , ρ(p, q, u) ≡ ρ. (3.2) D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 494 Proof. Corresponding to an entire harmonic function u(rx) = ∑∞ k=0 Y (k)(x;u)rk we define the entire function u(ζx) = ∑∞ k=0 max |Y (k)(x;u)|ζk, |x| = 1 [2], now applying Theorem A, we have ρ(p, q, u) = P (L(p, q, u)) where L(p, q, u) = lim sup k→∞ log[p−1] k log[q](αk) − 1 k . We know that ( αk αk+1 ) is a nondecreasing function of k, k > k0 if ( E (k) R (u) E (k−1) R (u) ) is a nondecreas- ing function of k for k > k0. This implies that ( βk βk+1 ) and ( γk γk+1 ) are also nondecreasing function of k, k > k0. Using [17, Lemma 1] we obtain αk αk+1 ≤ (k + 2ν)2νE (k−1) R (u)R (k + 1 + 2ν)2νE (k) R (u) . Let p(x) = ( x+2ν x+2ν+1)2ν , log p(x) = 2ν log(x+ 2ν)− 2ν log(x+ 1 + 2ν), p′(x) p(x) = 2ν x+ 2ν − 2ν x+ 2ν + 1 , taking w(x) = 2ν x+2ν , w(x) − w(x + 1) > 0 for any x > 0. Hence w(x) is a decreasing function and subsequently p′(x) > 0 for x > 0. Hence ( αk αk+1 ) is nondecreasing if ( E (k) R (u) E (k−1) R (u) ) is nondecreasing function of k for k > k0. The result (3.1) is obtain by using the relation (2.4) and the result (3.2) is found from (2.6). Theorem 3.2. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order ρ(p, q, u) and ( E (k) R (u) E (k−1) R (u) ) is a nondecreasing function of k for k > k0, then ρ(p, q, u) = P (L(p, q, u)) (3.3) where L(p, q, u) = lim sup k→∞ log[p−1] k log[q]( E (k−1) R (u)R E (k) R (u) ) Proof. For an entire function u(zx) = ∑∞ k=0 max |Y (k)(x;u)|zk, using Theorem B we have ρ(p, q, u) = P (L(p, q, u)) D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 495 where L(p, q, u) = lim sup k→∞ log[p−1] k log[q]( αk αk+1 ) if ( αk αk+1 ) is a nondecreasing function of k for k > k0. Applying above relation for the entire function f , we obtain ρ(p, q, f) = P (L(p, q, f)), (3.4) L(p, q, f) = lim sup k→∞ log[p−1] k log[q]( βk βk+1 ) = lim sup k→∞ log[p−1] k log[q]( E (k−1) R (u)R E (k) R (u) (k+1+2ν k+2ν )2ν) = lim sup k→∞ log[p−1] k(log[q−1](log( E (k) R (u)R E (k+1) R (u) ) + 2ν log( k + 1 + 2ν k + 2ν )−1)) = lim sup k→∞ log[p−1] k log[q]( E (k) R (u)R E (k+1) R (u) ) . Similarly for the entire function g, we have L(p, q, g) = lim sup k→∞ log[p−1] k log[q]( E (k) R (u)R E (k+!) R (u) ) . ρ(p, q, g) = P (L(p, q, g)) Now (3.3)follows from (2.4). Theorem 3.3. Let u be an entire harmonic function in Rn, n ≥ 3 with (p, q)-order ρ(p, q, u), lower (p, q)-order λ(p, q, u), lower (p, q)-type t(p, q, u) and let ( E (k) R (u) E (k+1) R (u) ) is a nondecreasing function of k for k > k0, then λ(p, q, u) = P (l∗(p, q, u)) (3.5) where l∗(p, q, u) = lim inf k→∞ log[p−1] k log[q](R−kE (k) R (u))− 1 k and t(p, q, u) = Mv(p, q, u) (3.6) where v(p, q, u) = lim inf k→∞ log[p−2] k (log[q−1](R−kE (k) R (u))− 1 k )ρ−A D. Kumar, R. Ali / Eur. J. Pure Appl. Math, 12 (2) (2019), 486-498 496 Proof. Applying Theorem D to the function f and g we can easily obtain λ(p, q, f) = P (l∗(p, q, f)) and λ(p, q, g) = P (l∗(p, q, g)) where l∗(p, q, f) = lim inf k→∞ log[p−1] k log[q](R−kE (k) R (u))− 1 k , l∗(p, q, g) = lim inf k→∞ log[p−1] k log[q](R−kE (k) R (u))− 1 k Now the result (3.6) follows from (2.5). Similarly applying Theorem F to the entire function f and g, we get t(p, q, f) = Mv(p, q, f), t(p, q, g) = Mv(p, q, g) where v(p, q, f) = lim inf k→∞ log[p−2] k (log[q−1](R−kE (k) R (u))− 1 k )ρ−A v(p, q, g) = lim inf k→∞ log[p−2] k (log[q−1](R−kE (k) R (u))− 1 k )ρ−A The result (3.7) follows from (2.7) and above relations. Acknowledgements The authors are gratful to Akram Ali for his useful comments, discussions and constant encouragement, and the referees for their valuable suggestions which improved the paper. They also would like to express their gratitude to King Khalid University for providing administrative and technical support. REFERENCES 497 References [1] R.P. Boas, Entire Functions, Academic Press, New York, N Y, USA, 1954. [2] A.J. Fryant, Growth of entire harmonic functions in R3, J. Math. Anal. Appl. 66(1978), 599-605. [3] T.B. Fugard, Growth of entire harmonic functions in Rn, n ≥ 2, J. Math. Anal. Appl. 74, Issue 1 (1980), 289-291. [4] O.P. Juneja, G.P. 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