APPROXIMATION AND GENERALIZED LOWER ORDER OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES. Susheel Kumar and G. S. Srivastava (Received June, 2013)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 45 (2015), APPROXIMATION AND GENERALIZED LOWER ORDER OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES Susheel Kumar and G. S. Srivastava (Received June, 2013) Abstract. In the present paper, we study the growth properties of entire functions of several complex variables. The characterizations of generalized lower der of entire functions of several complex variables have been obtained in terms of their Tayl s series coefficients. Also we have obtained the characterization of generalized lower der of entire functions of several complex variables in terms of approximation and interpolation errs. 1. Introduction We denote the complex N space by C N. Thus, z C N means that z = (z 1, z 2,..., z N ), where z 1, z 2,..., z N are complex numbers. A function g(z), z C N is said to be analytic at a point ξ C N if it can be expanded in some neighbhood of ξ as an absolutely convergent power series. If we assume ξ = (0, 0,..., 0), then g(z) has representation g(z) = k =0 a k1,k 2,...,k N z k1 1 zk zk N N = a k z k, (1.1) where k = (k 1, k 2,..., k N ) N N 0 and n = k = k 1 + k k N. F r > 0, the maximum modulus S(r, g) of entire function g(z) is given by (see [3]) n=0 S(r, g) = sup{ g(z) : z z z N 2 = r 2 }. F r > 0, the maximum term µ(r) of entire function g(z) is defined as (see [4]) and [5]) µ(r) = µ(r, g) = max { a k r n }. n 0 Also the index k with maximal length n f which maximum term is achieved is called the central index and is denoted by ν(r) = ν(r, g) = k. F generalization of the classical characterizations of growth of entire functions, Seremeta [7] introduced the concept of the generalized der and generalized type using the general growth functions as follows: Let L 0 denote the class of functions h(x) satisfying the following conditions: (i) h(x) is defined on [a, ) and is positive, strictly increasing, differentiable and tends to as x, h[{1+1/ψ(x)}x] h(x) = 1, f every function ψ(x) such that ψ(x) as x. (ii) lim x Let Λ denote the class of functions h(x) satisfying conditions (i) and 2010 Mathematics Subject Classification 30B10, 30D20, 32K05. Key wds and phrases: Entire function, Maximum term, Central index, Generalized lower der, Approximation errs, Interpolation errs.

2 12 SUSHEEL KUMAR and G. S. SRIVASTAVA h(cx) (iii) lim x h(x) = 1, f every c>0, that is h(x) is slowly increasing. If g(z) is an entire function and functions α(x) Λ, (x) L 0, then the generalized der ρ(α,, g) of g(z) is defined as (see [2]) α [log S(r, g)] ρ(α,, g) = lim sup. r (log r) F an entire function g(z) and functions α(x) Λ, (x) L 0, we define the generalized lower der λ(α,, g) of g(z) as α [log S(r, g)] λ(α,, g) = lim inf. (1.2) r (log r) Following Bose and Sharma ([1], p ) we can easily show that the generalized lower der λ(α,, g) of g(z) can be expressed in terms of central index as α { ν(r) } λ(α,, g) = lim inf r (log r). (1.3) Let K be a compact set in C N and let. K denote the supremum nm on K. The function [ ] Φ K (z) = sup p(z) 1/n : p polynomial, deg p n, p K 1, where n = 1, 2,... and z C N, is called the Siciak extremal function of the compact set K (see [2] and [3]). Given a function f defined and bounded on K, f n = 1, 2,..., we put E 1 n(f, K) = f t n K ; E 2 n(f, K) = f l n K ; E 3 n+1(f, K) = l n+1 l n K ; where t n denotes the n th Chebyshev polynomial of the best approximation to f on K and l n denotes the n th Lagrange interpolation f f with nodes at extremal points of K (see [2] and [3]). Kumar and Srivastava ([6], Thm. 2.1) have obtained coefficient characterizations of lower der of entire functions of several complex variables in terms of their Tayl s series coefficients. In the present paper we have obtained the characterizations of generalized lower der of entire functions of several complex variables in terms of their Tayl s series coefficients. Also we have obtained the characterizations of generalized der of entire functions of several complex variables in terms of approximation and interpolation errs. 2. Main Results Now we prove Theem 2.1. Let g(z) be an entire function whose Tayl s series representation is given by (1.1). If α(x) Λ, (x) L 0, then the generalized lower der λ of g(z) satisfies Further, if λ = λ(α,, g) lim n inf { ak } k ψ(n) = max k =n a k, = k + 1 is a non-decreasing function of n then equality holds in (2.1). { log a k 1/n}. (2.1)

3 APPROXIMATION AND GENERALIZED LOWER ORDER 13 Proof. Write Φ = lim inf n { log a k 1/n}. First we prove that λ Φ. The coefficients of an entire Tayl s series satisfy Cauchy s inequality, that is a k S(r, g) r n, k = n. (2.2) Also from (1.2), f arbitrary ε > 0 and a sequence r = r s as s, we have S(r, g) exp[α 1 {λ (log r)}], where λ = λ + ε. Now from (2.2), we get a k r n exp[α 1 {λ (log r)}]. Putting r = exp[ 1 {/λ}] in the above inequality we get a k exp[n n 1 {/λ}], 1 [/λ] 1 1 n {log a k }, { 1 + log a k 1/n} λ. Since (1 + x) (x) as x, proceeding to limits as n, we get Φ = lim inf n { log a k 1/n} λ. Since ε > 0 is arbitrarily small so finally we get Φ λ. Now we prove the reverse inequality i.e., λ Φ. From the assumption on ψ, ψ(n) as n. By the definition given in section 1, if µ(r) = a k r k is the maximum term then f k 1 k < k 2, a k1 r k1 a k r k > a k2 r k2, and f k = n, ψ(n 1) r < ψ(n). Now suppose that a k 1 r k1 and a k 2 r k2 are two consecutive maximum terms. Then k 1 k 2 1. Let k 1 n k 2. Then f ψ( k 1 ) r < ψ( k 1 ), we have ν(r) = k 1 where k 1 = k 1 1. Hence from (1.3), f arbitrary ε > 0 and all r > r 0 (ε), we have { } k 1 = ν(r) > α 1 λ (log r), λ = λ ε, { } k 1 = ν(r) α 1 λ [log{ψ( k 1 ) q}], log ψ( k 1 ) O(1) + 1 {α( k 1 )/λ }, where q is a constant such that { } 0 < q < min 1, [ψ( k 1 ) ψ( k 1 )]/2. Further we have ψ( k 1 ) = ψ( k 1 + 1) =... = ψ(n 1).

4 14 SUSHEEL KUMAR and G. S. SRIVASTAVA Now we can write ψ( k 0 )... ψ( k ) = a k 0 a k where k = n 1 and n k 0 [ψ( k ] n k0, log a k 1 n log ψ( k 1 ) + O(1) n 1 {α( k 1 )/λ } + O(1), 1 n log a k [ 1 {α( k 1 )/λ }][1 + o(1)], 1 n log a k [ 1 {/λ }][1 + o(1)], λ { log a k 1/n}[1 + o(1)]. Now taking limits as n, we get λ Φ. Hence the Theem 2.1 is proved. Next we prove Theem 2.2. Let K C N be a compact set such that Φ K is locally bounded in C N. If α(x) Λ and (x) L 0 then the function f, defined and bounded on K, is a restriction to K of an entire function g of generalized lower der λ(α,, g) if and only if λ = λ(α,, g) lim n sup ; s = 1, 2, 3. (2.3) [log {En(f, s 1/n] K)} Also if E s n(f, K)/E s n+1(f, K) is a non-decreasing function of n, then equality holds in (2.3). Proof. First we assume that f has an entire function extension g which is of generalized der ρ = ρ(α,, g). We write θ s = lim n sup ; s = 1, 2, 3. [log {En} s 1/n] Here E s n stands f E s n (g K, K), s = 1, 2, 3. Following Winiarski [8], we have and where n = we get n and ( n + N n E 1 n E 2 n (n + 2)E 1 n n 0, (2.4) En, 3 2(n + 2) En 1, 1 n 1, (2.5) ). Using Stirling fmula f the approximate value of n! nn N! f all large values of n. Hence f all large values of n, we have E 1 n E 2 n nn N! {1 + o(1)}e1 n, E 3 n 2 nn N! {1 + o(1)} E1 n.

5 APPROXIMATION AND GENERALIZED LOWER ORDER 15 Thus θ 3 θ 2 = θ 1. First we prove that θ s λ. Without any loss of generality, we may suppose that Then K B = { z C N : z z z N 2 1 }. E 1 n E 1 n(g, B). Now following Janik ([3], p. 324), we have E 1 n(g, B) r n S(r, g) r 2, n 0, E 1 n r n exp { α 1 [ λ (log r) ]}. Putting r = exp { 1 [ /λ ]} in the above inequality, we get E 1 n exp { n n 1 [ /λ ]}, ( 1 1 n [log {E1 n}] ) λ. Since (1 + x) (x) as x, proceeding to limits as n, we get θ 1 = lim n ( 1 n [log {E1 n}] ) λ. Since ε > 0 is arbitrarily small, we finally get θ 1 λ, θ s λ. Now we will prove that λ θ s. Let ψ(n) = E k s /Es k. Then ψ(n) as n. Now as in the proof of Theem 2.1, here we have log [E s n ] 1 n log ψ( k 1 ) + O(1) n 1 {α( k 1 )/λ } + O(1), 1 n log Es n [ 1 {α( k 1 )/λ }][1 + o(1)], 1 n log Es n λ [ 1 {/λ }][1 + o(1)], + o(1)]. {log [En] s 1/n}[1 Now taking limits as n, we get λ θ s. Now let f be a bounded function defined on K and such that f s = 1, 2, 3 θ s = lim n sup [log {En} s 1/n]. Then f every d 1 >θ s and f sufficiently large value of n, we have [log {En} s 1/n] d 1,

6 16 SUSHEEL KUMAR and G. S. SRIVASTAVA [ { }] 1 0 En s exp n 1. d 1 Proceeding to limits as n, we get lim n [Es n] 1/n = 0. So by the result of Janik ([2], Prop. 3.1), we infer that the function f can be continuously extended to an entire function. Let us put g = l 0 + (l n l n 1 ), n=1 where {l n } is the sequence of Lagrange interpolation polynomials of f as defined earlier. Now we claim that g is the required continuation of f and ρ(α,, g) = θ s. As in the proof of this Theem given above, we have λ θ s. Now using the inequalities (2.4), (2.5) and the proof of first part given above, we have λ(α,, g) = θ s, as claimed. This completes the proof of the Theem 2.2. References [1] S. K. Bose and D. Sharma, Integral functions of two complex variables, Univ. Iagel. Comp.Math., 15 (1963), [2] A. Janik, On approximation and interpolation of entire functions, Zeszyty Nauk. Uniw. Jagiellon. Prace Mat., 22 (1981), [3] A. Janik, On approximation of entire functions and generalized der, Univ. Iagel. Acta Math., 24 (1984), [4] J. G. Krishna, Maximum term of a power series in one and several complex variables, Pacific J. Math., 29 (1969), [5] J. G. Krishna, Probabilistic techniques leading to a Valiron-type theem in several complex variables, Ann. Math. Statist., 41 (1970), [6] S. Kumar and G. S. Srivastava, Maximum term and lower der of entire function of several complex variables, Bull. Math. Anal. Appl., 3 (1) (2011), [7] M. N. Seremeta, On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion, Amer. Math. Soc. Transl., 2 (88) (1970), [8] T. Winiarski, Application of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann. Pol. Math., 28 (1973),

7 APPROXIMATION AND GENERALIZED LOWER ORDER 17 Susheel Kumar Department of Mathematics, Jaypee University of Engineering & Technology Guna , Madhya Pradesh, India. sus83dma@gmail.com G. S. Srivastava Department of Mathematics Jaypee Institute of Infmation Technology A-10, Sect-62, Noida , Uttar Pradesh, India. gs91490@gmail.com