APPROXIMATION OF ENTIRE FUNCTIONS OF SLOW GROWTH ON COMPACT SETS. G. S. Srivastava and Susheel Kumar

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1 ARCHIVUM MATHEMATICUM BRNO) Tomus ), APPROXIMATION OF ENTIRE FUNCTIONS OF SLOW GROWTH ON COMPACT SETS G. S. Srivastava and Susheel Kumar Abstract. In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized der and generalized type of entire functions of slow growth have been obtained in terms of approximation and interpolation errs. 1. Introduction The concept of generalized der and generalized type f entire transcendental functions was given by Seremeta [5] and Shah [6]. Hence, let L 0 denote the class of functions hx) satisfying the following conditions: i) hx) is defined on [a, ) and is positive, strictly increasing, differentiable and tends to as x, h[{1 + 1/ψx)}x] ii) lim = 1 f every function ψx) such that ψx) as x hx) x. Let Λ denote the class of functions hx) satisfying conditions i) and hcx) iii) lim = 1 f every c > 0, that is hx) is slowly increasing. x hx) F an entire transcendental function fz) = b n z n, Mr) = max fz) and n=1 z =r functions αx) Λ, βx) L 0, the generalized der is given by ρα, β, f) = lim sup α[log Mr)] βlog r) Further, f αx), β 1 x) and γx) L 0, generalized type of an entire transcendental function fz) is given as σα, β, ρ, f) = lim sup α[log Mr)] β[{γr)} ρ ], Mathematics Subject Classification: primary 30B10; secondary 30D20, 32K05. Key wds and phrases: entire function, Siciak extremal function, generalized der, generalized type, approximation errs, interpolation errs. Received June 17, 2008, revised May Edit O. Došlý.

2 138 G. S. SRIVASTAVA AND S. KUMAR where 0 < ρ < is a fixed number. Let g : C N C, N 1, be an entire transcendental function. F z = z 1, z 2,..., z N ) C N, we put Sr, g) = sup { gz) : z z z N 2 = r 2}, r > 0. Then we define the generalized der and generalized type of gz) as and ρα, β, g) = lim sup σα, β, ρ, g) = lim sup α[log Sr, g)] βlog r) α[log Sr, g)] β[{γr)} ρ ] Let K be a compact set in C N and let K denote the sup nm on K. The function Φ K z) = sup pz) 1/n : p polynomial, deg p n, p K 1, n N ), 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, we put f n = 1, 2,... E 1 nf, K) = f t n K ; E 2 nf, K) = f l n K ; E 3 n+1f, 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 polynomial f f with nodes at extremal points of K see [2] and [3]). The generalized der of an entire function of several complex variables has been characterized by Janik [3]. His characterization of der in terms of the above errs has been obtained under the condition 1.1) dβ 1 [cαx)]) b ; x a. dlog x) Clearly 1.1) is not satisfied f αx) = βx). Thus in this case, the cresponding result of Janik is not applicable. In the present paper we define generalized der and generalized type of entire functions of several complex variables in a new way. Our results apply satisfactily to entire functions of slow growth and generalize many previous results. Let Ω be the class of functions hx) satisfying conditions i) and iv) there exist a function δx) Λ and constants x 0, c 1 and c 2 such that 0 < c 1 d{hx)} d{δlog x)} c 2 < f all x > x 0. Let Ω be the class of functions hx) satisfying i) and d{hx)} v) lim x dlog x) = c 3, 0 < c 3 <..

3 ENTIRE FUNCTIONS OF SLOW GROWTH 139 Kapo and Nautiyal [4] showed that classes Ω and Ω are contained in Λ and Ω Ω = φ. They defined the generalized der ρα, α, f) f entire functions as α[log Mr)] ρα, α, f) = lim sup, αlog r) where αx) either belongs to Ω to Ω. Ganti and Srivastava [1] defined the generalized type σα, α, ρ, f) of an entire function fz) having finite generalized der ρα, α, f) as σα, α, ρ, f) = lim sup 2. Main results α[log Mr)] [αlog r)] ρ. Theem 2.1. Let K be a compact set in C N. If αx) either belongs to Ω to Ω then the function f defined and bounded on K, is a restriction to K of an entire function g of finite generalized der ρα, α, g) if and only if ρα, α, g) = lim sup α { 1 n log } ; s = 1, 2, 3. Es nf, K) Proof. Let g be an entire transcendental function. Write ρ = ρα, α, g) and θ s = lim sup α { 1 n log } ; s = 1, 2, 3. Es n Here E s n stands f E s n g K, K ), s = 1, 2, 3. We claim that ρ = θ s, s = 1, 2, 3. It is known see e.g. [7]) that 2.1) E 1 n E 2 n n + 2)E 1 n, n 0, 2.2) En 3 2n + 2)En 1 1, n 1, where n = ) n+n n. Using Stirling fmula f the approximate value of n! e n n n+1/2 2π, we get n nn N! f all large values of n. Hence f all large values of n, we have and E 1 n E 2 n nn N! [1 + o1)]e1 n E 3 n 2 nn N! [1 + o1)]e1 n. Thus θ 3 θ 2 = θ 1 and it suffices to prove that θ 1 ρ θ 3. First we prove that θ 1 ρ. Using the definition of generalized der, f ε > 0 and r > r 0 ε), we have log S r, g) α 1{ ρ αlog r) }, where ρ = ρ + ε provided r is sufficiently large. Without loss of generality, we may suppose that K B = {z C N : z z z N 2 1}.

4 140 G. S. SRIVASTAVA AND S. KUMAR Then Now following Janik [3, p.324]), we get E 1 n E 1 ng, B). E 1 ng, B) r n Sr, g), r 2, n 0 log E 1 n n log r + α 1{ ρ αlog r) }. Putting r = exp [ α 1 {/ρ} ] in the above inequality, we obtain Taking limits as n, we get log E 1 n n [ α 1 {/ρ} ] + n α { 1 1 n log } ρ. E1 n lim sup α { 1 n log } ρ. E1 n Since ε > 0 is arbitrary small. Therefe finally we get 2.3) θ 1 ρ. Now we will prove that ρ θ 3. If θ 3 =, then there is nothing to prove. So let us assume that 0 θ 3 <. Therefe f a given ε > 0 there exist n 0 N such that f all n > n 0, we have 0 α { 1 n log } θ 3 + ε = θ 3 E3 n E 3 n exp [ nα 1 {/θ 3 } ]. Now from the property of maximum modulus, we have Sr, g) Enr 3 n, n 0 Sr, g) Enr 3 n + Now f r > 1, we have 2.4) Sr, g) A 1 r n0 + n=n 0+1 n=n 0+1 where A 1 is a positive real constant. We take r n exp [ nα 1 {/θ 3 } ]. r n exp [ nα 1 {/θ 3 } ], 2.5) Nr) = α 1 θ 3 α[log{n + 1)r}] ).

5 ENTIRE FUNCTIONS OF SLOW GROWTH 141 Now if r is sufficiently large, then from 2.4) and 2.5) we have Sr, g) A 1 r n0 + r Nr) exp [ nα 1 {/θ 3 } ] 2.6) + n>nr) Sr, g) A 1 r n0 + r Nr) + n>nr) n 0+1 n Nr) r n exp [ nα 1 {/θ 3 } ] n=1 exp [ nα 1 {/θ 3 } ] r n exp [ nα 1 {/θ 3 } ]. Now we have lim sup exp[ nα 1 {/θ 3 }] ) 1/n = 0. Hence the first series in 2.6) converges to a positive real constant A 2. So from 2.6) we get Sr, g) A 1 r n0 + A 2 r Nr) + r n exp [ nα 1 {/θ 3 } ], 2.7) Sr, g) A 1 r n0 + A 2 r Nr) + Sr, g) A 1 r n0 + A 2 r Nr) + Sr, g) A 1 r n0 + A 2 r Nr) + n>nr) n>nr) n>nr) n=1 r n exp [ n log{n + 1)r} ], 1 ) n, N ) n. N + 1 It can be easily seen that the series in 2.7) converges to a positive real constant A 3. Therefe from 2.7), we get Sr, g) A 1 r n0 + A 2 r Nr) + A 3, Sr, g) A 2 r Nr)[ 1 + o1) ], log Sr, g) [ 1 + o1) ] Nr) log r, log Sr, g) [1 + o1)]α 1 θ 3 α [ log{n + 1)r} ] ) log r, log Sr, g) [ 1 + o1) ][ α 1 {θ 3 + δ)α[log{n + 1)r}]} ], where δ > 0 is suitably small. Hence α[log Sr, g)] θ 3 + δ)α[log{n + 1)r}] α [log Sr, g)] α[log r] θ 3 + δ) [ 1 + o1) ].

6 142 G. S. SRIVASTAVA AND S. KUMAR Proceedings to limits as r, since δ is arbitrarily small, we get ρ θ 3. Now let f be a function defined and bounded on K and such that f s = 1, 2, 3 θ s = lim sup is finite. We claim that the function g = l 0 + α { 1 n log } Es n l n l n 1 ) n=1 is the required entire continuation of f and ρα, α, g) = θ s. Indeed, f every d 1 > θ s provided n is sufficiently large. Hence α { 1 n log } d 1 Es n E s n exp [ nα 1 {/d 1 } ]. Using the inequalities 2.1), 2.2) and converse part of theem, we find that the function g is entire and ρα, α, g) is finite. So by 2.3), we have ρα, α, g) = θ s, as claimed. This completes the proof of Theem 2.1. Next we prove Theem 2.2. Let K be a compact set in C N such that Φ K is locally bounded in C N. Set Gx, σ, ρ) = α 1[ {σ αx)} 1/ρ], where ρ is a fixed number, 1 < ρ <. Let αx) Ω and dgx,σ,ρ) d log x = O1) as x f all 0 < σ <. Then the function f defined and bounded on K, is a restriction to K of an entire function g of generalized type σα, α, ρ, g) if and only if σα, α, ρ, g) = lim sup, s = 1, 2, 3. ρ 1 log[es nf, K)] 1/n Befe proving the Theem 2.2 we state and prove a lemma. Lemma 2.1. Let K be a compact set in C N such that Φ K is locally bounded in C N. Set Gx, µ, λ) = α 1[ {µ αx)} 1/λ], where λ is a fixed number, 1 < λ <. Let αx) Ω and dgx,µ,λ) d log x = O1) as x f all 0 < µ <. Let p n ) n N be a sequence of polynomials in C N such that i) deg p n n, n N; ii) f a given ε > 0 there exists n 0 N such that p n K exp λ 1 λ n α 1[{ 1 } 1/λ 1)]) µ αn/λ). Then p n is an entire function and σα, α, λ, p n) µ provided is not a polynomial. p n

7 ENTIRE FUNCTIONS OF SLOW GROWTH 143 Proof. By assumption, we have p n K r n r n exp λ 1 λ nα 1[{ 1 } 1/λ 1)]) µ αn/λ), n n 0, r > 0. If αx) Ω, then by assumptions of lemma, there exists a number b > 0 such that f x > a, we have dgx, µ, λ) < b. d log x Let us consider the function φx) = r x exp λ 1 λ xα 1[{ 1 } 1/λ 1)]) µ αx/λ). Using the technique of Seremeta [5], it can be easily seen that the maximum value of φx) is attained f a value of x given by where Thus x r) = λα 1{ µ[α{log r ar)}] λ 1}, ar) = dgx/λ, 1/µ, λ 1) d log x 2.8) p n K r n exp { bλα 1 [µ{αlog r + b)} λ 1 ] }, n n 0, r > 0. Let us write K r = {z C N : Φ K z) < r, r > 1}, then f every polynomial p of degree n, we have see [3]) p n z) p n K Φ n Kz), z C N. So the series p n is convergent in every K r, r > 1, whence p n is an entire function. Put M r) = sup { p n K r n : n N, r > 0}. On account of 2.8), f every r > 0, there exists a positive integer νr) such that and M r) = p νr) K r νr) M r) > p n K r n, n > νr). It is evident that νr) increases with r. First suppose that νr) as r. Then putting n = νr) in 2.8), we get f sufficiently large r 2.9) M r) exp { bλα 1 [µ{αlog r + b)} λ 1 ] }. Put F r = {z C N : Φ K z) = r}, r > 1.

8 144 G. S. SRIVASTAVA AND S. KUMAR and { } Mr) = sup p n z) : z F r, r > 1. Now following Janik [3] p.323), we have f some positive constant k ) 2.10) S r, p n Mkr) 2M 2kr). Combining 2.9) and 2.10), we get ) S r, p n 2 exp { bλα 1 [µ{αlog r + b)} λ 1 ] } Since αx) Ω, we get on using v) α[ 1 bλ log{ 1 2 Sr, p n)}] [αlog 2kr + b)] λ 1 µ. lim sup α [log Sr, p n)] [αlog r)] λ µ σα, α, λ, p n ) µ. In the case when νr) is bounded then M r) is also bounded, whence reduces to a polynomial. Hence the Lemma 2.1 is proved. p n Proof of Theem 2.2. Let g be an entire transcendental function. Write σ = σα, α, ρ, g) and η s = lim sup, s = 1, 2, 3. ρ 1 log[es n] 1/n Here E s n stands f E s n g K, K), s = 1, 2, 3. We claim that σ = η s, s = 1, 2, 3. Now following Theem 2.1, here we prove that η 1 σ η 3. First we prove that η 1 σ. Using the definition of generalized type, f ε > 0 and r > r 0 ε), we have S r, g) exp α 1 [σ {αlog r)} ρ ] ), where σ = σ + ε provided r is sufficiently large. Thus following Theem 2.1, here we have E 1 n r n exp α 1 [σ {αlog r)} ρ ] ) 2.11) E 1 n exp [ n log r + α 1 [σ {αlog r)} ρ ]) ]. Let r = rn) be the unique root of the equation [ n log r ] 2.12) α = σ {αlog r)} ρ. ρ

9 ENTIRE FUNCTIONS OF SLOW GROWTH 145 Then 2.13) log r α 1{ 1 σ } 1/ρ 1) = Gn/ρ, 1/σ, ρ 1). Using 2.12) and 2.13) in 2.11), we get [ En 1 exp n Gn/ρ, 1/σ, ρ 1) + n ] ρ Gn/ρ, 1/σ, ρ 1), Proceedings to limits, we get ρ ρ 1 log[e1 n] 1/n α 1{ 1 ) 1/ρ 1)} σ, σ. ρ 1 log[e1 n] 1/n lim sup σ ρ 1 log[e1 n] 1/n η 1 σ. Since ε > 0 is arbitrarily small, we finally get 2.14) η 1 σ. Now we will prove that σ η 3. Suppose that η 3 < σ, then f every µ 1, η 3 < µ 1 < σ, we have µ 1 ρ 1 log[e3 n] 1/n provided n is sufficiently large. Thus En 3 exp ρ 1 ρ n α 1[{ 1 } 1/ρ 1) ]). µ 1 Also by previous lemma, σ µ 1. Since µ 1 has been chosen less than σ, we get a contradiction. Hence σ η 3. Now let f be a function defined and bounded on K such that f s = 1, 2, 3 η s = lim sup is finite. We claim that the function ρ 1 log[es n] 1/n g = l 0 + l n l n 1 ) n=1

10 146 G. S. SRIVASTAVA AND S. KUMAR is the required entire continuation of f and σα, α, ρ, g) = η s. Indeed, f every d 2 > η s d 2 ρ 1 log[es n] 1/n provided n is sufficiently large. Hence En s exp ρ 1 ρ n α 1[{ 1 } 1/ρ 1) ]). d 2 Using the inequalities 2.1), 2.2) and previous lemma, we find that the function g is entire and σα, α, ρ, g) is finite. So by 2.14), we have σα, α, ρ, g) = η s, as claimed. This completes the proof of the Theem 2.2. Acknowledgement. The auths are very thankful to the referee f his valuable comments and observations which helped in improving the paper. References [1] Ganti, R., Srivastava, G. S., Approximation of entire functions of slow growth, General Math. 14 2) 2006), [2] Janik, A., A characterization of the growth of analytic functions by means of polynomial approximation, Univ. Iagel. Acta Math ), [3] Janik, A., On approximation of entire functions and generalized der, Univ. Iagel. Acta Math ), [4] Kapo, G. P., Nautiyal, A., Polynomial approximation of an entire function of slow growth, J. Approx. They ), [5] Seremeta, M. N., 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. Ser ), [6] Shah, S. M., Polynomial approximation of an entire function and generalized der, J. Approx. They ), [7] Winiarski, T., Application of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann. Polon. Math ), Department of Mathematics Indian Institute of Technology Rokee Rokee , India girssfma@iitr.ernet.in