EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1508-1517 ISSN 1307-5543 – ejpam.com Published by New York Business Global Approximation of Generalized Biaxisymmetric Potentials in Lβ-Norm Devendra Kumar1,2 1 Department of Mathematics, Faculty of Sciences Al-Baha University, P.O.Box-7738 Alaqiq, Al-Baha-65799, Saudi Arabia 2 Research and Post Graduate Studies, Department of Mathematics, M. M. H. College, Model Town, Ghaziabad-201001, U.P., India Abstract. Let F be a real valued generalized biaxisymmetric potential (GBASP) in Lβ on SR, the open sphere of radius R about the origin. In this paper we have obtained the necessary and sufficient conditions on the rate of decrease of a sequence of best harmonic polynomial approximates to F such that F is harmonically continues as an entire function GBASP and determine their (p, q)- order and generalized (p, q)-type with respect to proximate order ρ(r). 2020 Mathematics Subject Classifications: 41A15, 30B10. Key Words and Phrases: Entire functions, generalized biaxisymmetric potentials, harmonic polynomial approximation error, Lβ-norm 1 ≤ β <∞, proximate order and Jacobi polynomials . 1. Introduction Let F = F (x, y) be a real-valued regular solution of the generalized biaxisymmetric potential (GBASP) equation ∂2F ∂x2 + 2µ y ∂F ∂y + ∂2F ∂y2 + 2ν x ∂F ∂x = 0, µ, ν > 0, (1.1) which are even in x and y. A polynomial of degree n which is even in x and y is said to be a GBASP polynomial of degree n if it satisfies (1.1). A GBASP F, regular about origin, have local expansions of the form F (x, y) = ∞∑ n=0 anR (µ− 1 2 ,ν− 1 2 ) n (x, y), (1.2) DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4815 Email address: d kumar001@rediffmail.com (D. Kumar) https://www.ejpam.com 1508 © 2023 EJPAM All rights reserved. D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1509 R (µ− 1 2 ,ν− 1 2 ) n (x, y) = (x2 + y2)nP (µ− 1 2 ,ν− 1 2 ) n ( (x 2−y2) (x2+y2) )/P (µ− 1 2 ,ν− 1 2 ) n (1), where x = r cos θ, y = r sin θ and P (µ− 1 2 ,ν− 1 2 ) n (t) are Jacobi polynomials [1, 17]. The series (1.2) can be represented in (r, θ) by F ≡ F (r, θ) = ∞∑ n=0 anr 2nP (µ− 1 2 ,ν− 1 2 ) n (cos 2θ). Let SR = {(x, y) : x2 + y2 < R2}, 0 < R ≤ ∞, be the open sphere of radius R about the origin and SR be the closure of SR. In this paper we consider those GBASP F ∈ Lβ(SR), 1 ≤ β < ∞, that harmonically continue as an entire function GBASP. The characteristic feature follows from the rate of convergence of a sequence of best GBASP polynomial approximates to F in Lβ(SR). The concepts of index-pair (p, q), p ≥ q ≥ 1, (p, q)-order and (p, q)-type were introduced by Juneja et al. [15, 16]. Following the Juneja et al. [15, 16] the (p, q)-order of an entire GBASP function is defined as lim sup r→∞ log[p]M(r, F ) log[q] r = ρ(p, q) ≡ ρ, and, the function having (p, q)-order ρ(b < ρ(p, q) <∞) is said to be of (p, q)-type T if lim sup r→∞ log[p−1]M(r, F ) (log[q−1] r)ρ = T (p, q) ≡ T, where M(r, F ) = maxx2+y2 exp[q−1] 1, is said to be a proximate order of an entire function with index-pair (p, q) if (i) ρ(r) → ρ(p, q) ≡ ρ as r → ∞, b < ρ <∞; (ii) ∧ [q](r)ρ ′(r) → 0 r → ∞, where ρ′(r) denotes the derivative of ρ(r), and ∧ [q](r) = ∏q i=0 log [i] r. The (p, q)-type T ∗ of F with respect to a given proximate order ρ(r) is defined as lim sup r→∞ log[p−1]M(r, F ) (log[q−1] r)ρ(r) = T ∗(p, q) ≡ T ∗. If the quantity T ∗ is different from zero and infinity then ρ(r) is said to be the proximate order of a given GBASP function F with index-pair (p, q). D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1510 P.A. McCoy [14] obtained the results by using integral operator method [2, 4, 6–8], but our method is different from McCoy [14] and the results are the extension of those of McCoy [14]. For the purpose of motivation, it is significant to mention that the Euler-Poisson Darboux equation, arising in gas dynamics, is viewed in terms of equation (1.1) after a transforma- tion and has a variety of physical interpretations. The solution of equation (1.1) which satisfies a suitable radiation condition, corresponding to scattered waves, and their singu- larities are related to the quantum states of the scattered particles. The GBASP play an important role in many aspects of mathematical physics, in particular, in an understanding of compressible flow in the transonic region (see [14]). The limit µ ↓ ν produces the generalized axisymmetric potential equation. Reduction of the GBASP equation to the harmonic function follows from the limit µ ↓ 0 that also reduces the zonal harmonics to the circular harmonics. These functions form complete sets for even harmonic, respectively analytic functions, regular at the origin. The GBASP functions, then, are natural extensions of harmonic or analytic functions. Let Aβ(SR) denote the space of GBASP that is regular and analytic in SR with finite norm ∥ F ∥β,R= [ ∫ ∫ SR |F |pdxdy] 1 β , 1 ≤ β <∞, where ∥ . ∥β,R denotes the Lβ-norm. The best polynomial approximation error for the GBASP is defined by Eβ n(F,R) = inf gR,n∈PR,n {||F − gR,n||β,R}, n = 0, 1, . . . , (1.3) with PR,n = PR,n(z) = Pn( z R); where Pn denotes the set of all GBASP polynomials of degree no higher than n. For β = ∞, the above norm is sup norm. For each n there is an extremal GBASP polynomial g∗R,n ∈ PR,n for which ∥ F − g∗R,n ∥β,R= Eβ n(F,R). For GBASP functions there is a large literature concerning the growth and approxi- mation of this topic. Kasana and Kumar [10] studied the growth and approximation of solutions (not necessarily entire) of certain elliptic partial differential equations. They obtained the characterization of q-type and lower q-type (q ≥ 2) of a GBASP having fast rates of growth in terms of ratio of approximation errors in Lβ- norm. In [12], Kumar obtained some results for GBASP and the polynomial approximation of pseudo analytic functions, while in [13] Kumar obtained the characterization of growth parameters in terms of axially symmetric harmonic polynomial and Lagrange polynomials approxima- tion errors in n-dimensions. In the present paper, using a different technique, we derive formulae for the (p, q)-order and generalized (p, q)-type with respect to a proximate order, of entire GBASP functions in terms of GBASP polynomials approximation errors in Lβ- norm. Our results extend and improve the results obtained by McCoy [14]. D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1511 2. Lemmas and Results To prove our main results the following lemmas are required. Lemma 2.1. Let F ∈ Aβ(SR), then for all n ∈ N the following inequality holds: |an|R2n+2 0 ≤ (πR2 0) 1 η (2n+ 2)((2n+ µ+ ν)C(n, µ, ν))Γ(n+ α+ 1) Γ(α+ 1)Γ(n+ 1) Eβ n−1(F,R0) where C(n, µ, ν) = Γ(n+ 1)Γ(n+ µ+ ν) Γ(n+ µ+ 1 2)Γ(n+ ν + 1 2) , α = max(µ− 1 2 , ν − 1 2) and 1 η + 1 β = 1. Proof. From the orthogonality property of Jacobi polynomials and uniform conver- gence of the series (1.2) on SR, we have anτ 2n =2(2n+ µ+ ν)C(n, µ, ν) ∫ π 2 0 (F (τ, θ)− g∗τ,n−1(τ, θ))× × P (µ− 1 2 ,ν− 1 2 ) n (cos 2θ) sin2µ θ cos2ν θdθ, (2.1) where g∗τ,n−1 ∈ Pτ,n−1, 0 < τ < R0. Using [3, p.168] max −1≤t≤1 |P (µ− 1 2 ,ν− 1 2 )(t)| = Γ(n+ α+ 1) Γ(α+ 1)Γ(n+ 1) (2.2) in (2.1), we obtain |an|τ2n = (2n+ µ+ ν)C(n, µ, ν)Γ(n+ α+ 1) 2Γ(α+ 1)Γ(n+ 1) ∫ 2π 0 |(F (τ, θ)− g∗τ,n−1(τ, θ))|dθ, since F and g∗τ,n−1 are even in x and y. Multiplying both sides of the above inequality by τdτ and integrating from 0 to R0, we get |an|R0 2n+2 = 2(n+ 1)(2n+ µ+ ν)C(n, µ, ν)Γ(n+ α+ 1) 2Γ(α+ 1)Γ(n+ 1) × × ∫ ∫ SR0 |(F (x, y)− g∗R0,n−1(x, y))|dxdy. (2.3) For F ∈ Aβ(SR0), there exists g∗R0,n−1 ∈ PR0,n−1 such that 2Eβ n−1(F,R0) ≥ ∥ F − g∗R0,n−1 ∥β,R0 ≥ ( ∫ ∫ SR0 |(F (x, y)− g∗R0,n−1(x, y))|βdxdy) 1 β ≥ 1 (πR2 0) 1 η ∫ ∫ SR0 |(F (x, y)− g∗R0,n−1(x, y))|dxdy. (2.4) D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1512 Now combining (2.3) and (2.4) we get the required result. Let w = ψ(z) be the univalent function mapping the complement of SR on |w| > 1 such that ψ(∞) = ∞ and ψ′(∞) > 0. Set SR = {z : ψ(z) = r, r > 1}. Then Lemma 2.2. Let F ∈ Aβ(SR) be an entire GBASP function of (p, q)-order ρ and generalized (p, q)-type T ∗ with respect to ρ(r). Then lim sup r→∞ log[p]M(r, F ) log[q] r = ρ, lim sup r→∞ log[p−1]M(r, F ) (log[q−1] r)ρ(r) = T ∗ γ , where M(r, F ) = maxz∈SR |F |, γ = R−ρ for q = 1 and γ = 1, otherwise. This lemma is an immediate consequence of [18, Lemma 3.1]. Lemma 2.3. Let F ∈ Aβ(SR), r ′ > 1, be an entire GBASP function. Then, for all sufficiently large values of n, we have Eβ n(F,R) ≤ KM(r, F )(n+ 1)α+ 1 2 ( r′R r )2(n+1), (2.5) where K is a constant independent of n and r and r > 2r′R. Proof. Let us consider the GBASP polynomial gn,r = ∞∑ k=0 akr 2kP (µ− 1 2 ,ν− 1 2 ) k (cos 2θ). Then gn,r ∈ Pn,r. Using the definition of approximation error Eβ n(F,R) for all r, 0 < r < R, we get Eβ n(F,R) ≤|F − gn,r|β,R ≤ ∞∑ k=n+1 |ak|R2k|P (µ− 1 2 ,ν− 1 2 ) k (cos 2θ)| ≤ 1 Γ(α+ 1) ∞∑ k=n+1 |ak|R2kΓ(k + α+ 1) Γ(k + 1) . (2.6) For F ∈ Aβ(SR), we have [5] |ak| ≤ M(r, F ) r2k [(2k + µ+ ν)C(k, µ, ν)C(µ, ν)] 1 2 (2.7) D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1513 for every r < R. Combining (2.6) and (2.7) we get Eβ n(F,R) ≤ M(r, F ) Γ(α+ 1) (C(µ, ν)) 1 2 ∞∑ k=n+1 Γ(k + α+ 1) Γ(k + 1) [(2k+ µ+ ν)C(k, µ, ν)] 1 2 ( R r )2k. (2.8) Since Γ(x+a) Γ(x) ∼ xa as x→ ∞, we have Γ(k + α+ 1) Γ(k + 1) [(2k + µ+ ν)C(k, µ, ν)] 1 2 ∼ √ 2kα+ 1 2 as k → ∞. Hence Γ(k + α+ 1) Γ(k + 1) [(2k + µ+ ν)C(k, µ, ν)] 1 2 < 2 √ 2kα+ 1 2 for all k > k0. Thus, for n > k0 and r > 2r′R, using (2.8) with above inequality, we obtain Eβ n(F,R) ≤ M(r, F ) Γ(α+ 1) 2(2C(µ, ν)) 1 2 ∞∑ k=n+1 kα+ 1 2 ( r′R r )2k ≤ M(r, F ) Γ(α+ 1) 2(2C(µ, ν)) 1 2 (n+)α+ 1 2 ( r′R r )2(n+1) ∞∑ k=0 (1 + k k0 + 1 )α+ 1 2 ( r′R r )2k. Hence the proof is completed from the above inequality. Lemma 2.4. Let F ∈ Aβ(SR), R > R∗, be an entire GBASP function. Then h(z) = ∞∑ n=1 [ 2(n+ 1)(2n+ µ+ ν)C(n, µ, ν)(n+ 1)α Γ(n+ 1) ]2Eβ n−1(F,R)( z R∗ )2n (2.9) is entire. Further, ρ(F ) = ρ(h) and for b < ρ(F ) = ρ(h) <∞, T ∗(F ) = γT ∗(h). Proof. Since [ 2(n+ 1)(2n+ µ+ ν)C(n, µ, ν)(n+ 1)α Γ(n+ 1) ] 1 2n ∼ ( √ 2(n+ 1) √ 2nα+ 1 2 ) 1 n → 1 as n→ ∞, it follows from Lemma 2.2 that h(z) is entire and Eβ n(F,R) ≤ KM(r + 1, F )( r′R r + 1 )2n, we have h(z) = ∞∑ n=1 [ 2(n+ 1)(2n+ µ+ ν)C(n, µ, ν)(n+ 1)α Γ(n+ 1) ]2Eβ n−1(F,R)( z R∗ )2n, D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1514 so we get M( r Rr′ , h) ≤Q(r) +KM(r + 1, F ) ∞∑ n=0 [ r R∗(r + 1) ]2n = Q(r) +K R∗ 2(r + 1)2M(r + 1, F ) (r + 1)2R∗ 2 − r2 , r′ > 1, (2.10) where Q(r) is a polynomial for all sufficiently large value of r. On the other hand, using (1.2), (2.2) and Lemma 2.1, we get | ∞∑ n=0 anr 2nP (µ− 1 2 ,ν− 1 2 ) n (cos 2θ)| ≤ |a0|+ 1 Γ(α+ 1) ∞∑ n=1 |ak|R2nΓ(n+ α+ 1) Γ(n+ 1) ≤ |a0|+ +KK0 ∞∑ n=1 [ 2(n+ 1)(2n+ µ+ ν)C(n, µ, ν)(n+ 1)α Γ(n+ 1) ]2× × Eβ n−1(F,R)( r R0 )2n+2, z ∈ SR, R0 < R. or M(r, F ) ≤M( r R0 , |a0|+KK0h(z)). (2.11) Now the proof follows from (2.10) and (2.11). 3. Main Results In this section we will prove our main results. Theorem 3.1. Let the GBASP F ∈ Aβ(S1), β ≥ 1. Then F harmonically continues as an entire function GBASP if and only if lim n→∞ [Eβ n(F,R)] 1 n = 0. (3.1) Proof. Let F ∈ Aβ(S1), then for 0 < R < 1, F ∈ Aβ(SR). First suppose that F is entire. Then it follows from Lemma 2.3 that lim sup n→∞ [Eβ n(F,R)] 1 n ≤ ( r′R r ), r > 2r′R. Thus, for all sufficiently large r, we have lim sup n→∞ [Eβ n(F,R)] 1 n = 0. To prove only if part, suppose that (3.1) holds, then it follows from (2.11) that series on the right hand side of (1.2) converges uniformly on every compact subset of S∞ and GBASP F is entire. D. Kumar / Eur. J. Pure Appl. Math, 16 (3) (2023), 1508-1517 1515 Theorem 3.2. Let the GBASP F ∈ Aβ(SR), r > 2r′R. Then F harmonically contin- ues as an entire function GBASP of finite (p, q)-order ρ if and only if ρ(p, q) = P (L∗(p, q)), where L∗(p, q) = lim sup n→∞ log[p−1] n log[q][Eβ n(F,R)] − 1 n , and P (L∗(p, q)) = {L∗(p, q) if q < p < ∞, 1 + L∗(p, q) if p = q = 2, max(1 + L∗(p, q)) if 3 ≤ p = q, ∞ if p = q = ∞}. Proof. Using Theorem 3.1, we have F ∈ Aβ(SR) is harmonically continues as an entire function GBASP if and only if h(z) is an entire function. Using Lemma 2.4, F and h(z) have same (p, q)-order. The remaining part of the proof can be obtain easily. Theorem 3.3. Let the GBASP F ∈ Aβ(SR), r > 2r′R. Then F harmonically con- tinues as an entire function GBASP of finite (p, q)-order ρ(b < ρ < ∞) and generalized (p, q)-type T ∗ of F with respect to a proximate order ρ(r) if and only if T ∗(p, q) Mγ = lim sup n→∞ [ ϕ(log[p−2] n) log[q−1][Eβ n(F,R)] − 1 n ]ρ−A, where A = 1 if q = 2, A = 0 if q ̸= 2 and M ≡ M(p, q) = { (ρ−1)(ρ−1) ρρ if (p, q) = (2, 2), 1 eρ if (p, q) = (2, 1), 1 otherwise}. The function ϕ(x) be the unique solution of the equation x = (log[q−1] r)ρ(r)−A ⇔ ϕ(x) = log[q−1] r. Proof. Applying Theorem 3 of Nandan et al. [9] to the function h(z) and resulting characterization of T ∗ = γT ∗(h), with Lemma 2.4, taking together completes the proof. Remark 3.1. For (p, q) = (2, 1), Theorem 3.2 gives the Theorem 2 of P.A. McCoy [14] . Remark 3.2. For (p, q) = (2, 1) and x = ϕ(n) is the function inverse to n = xρ(r), Theorem 3.3 gives the Theorem 3 of P.A. McCoy [14] . 4. Conclusions We estimate formulae for the (p, q)-order and generalized (p, q)-type with respect to a proximate order of entire GBASP functions in terms of GBASP polynomial approximation errors in Lβ-norm, which made it possible to obtain the necessary and sufficient conditions under which a GBASP function harmonically continues to entire GBASP. Our results REFERENCES 1516 improve and extends the results of McCoy [14]. The relevance of our study is due to the fact that GBASP play and important role not only in theoretical mathematical research, but are used in gas dynamics in order to describe different stationary processes. Thus, the special interest are global properties characterising solutions to the partial differential equation that are determined from local properties. Acknowledgements The authors are thankful to the editor for his useful comments, and the referees for their valuable suggestions which improved the paper. References [1] R Askey. 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