International Mathematical Forum, Vol. 9, 2014, no. 5, HIKARI Ltd,

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1 International Mathematical Forum, Vol. 9, 2014, no. 5, HIKARI Ltd, Applications o Slowly Chanin Functions in the Estimation o Growth Properties o Composite Entire Functions on the Basis o their Maximum Terms and Maximum Moduli Sanjib Kumar Datta Department o Mathematics,University o Kalyani Kalyani, Dist-Nadia,PIN , West Benal, India Tanmay Biswas Rajbari, Rabindrapalli, R. N. Taore Road P.O.- Krishnaar, Dist-Nadia PIN , West Benal, India Azizul Haque Gobarara Hih Madrasah (H.S. P.O.+P.S.- Hariharpara, Dist-Murshidabad PIN , West Benal, India Copyriht c 2014 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the oriinal work is properly cited. Abstract In the paper we prove some comparative rowth properties o composite entire unctions on the basis o their maximum terms and maximum moduli usin eneralised L -order and eneralised L -lower order. Mathematics Subject Classiication: 30D35, 30D30

2 238 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Keywords: Entire unction, maximum term, maximum modulus, composition, rowth, slowly chanin unction, eneralised L -order (eneralised L -lower order 1 Introduction, Deinitions and Notations. Let C be the set o all inite complex numbers and be an entire unction deined in C. The maximum term μ (r, o = a n z n on z = r is deined by μ (r, = max a n r n and the maximum modulus M (r, o on z = r n 0 is deined as M (r, = max (z.we use the standard notations and deinitions in the theory o entire unctions which are available in [11]. In the sequel z =r we use the ollowin ( notation : lo [k] x = lo lo [k 1] x or k =1, 2, 3,... and lo [0] x = x. To start our paper we just recall the ollowin deinitions : Deinition 1 The order ρ and lower order λ o an entire unction are deined as n=0 lo [2] M(r, ρ = sup lo r and λ = in lo [2] M(r,. lo r Extendin this notion, Sato [6] deined the eneralised order and eneralised lower order o an entire unction as ollows : Deinition 2 [6]Let m be an inteer 2. The eneralised order ρ [m] eneralised lower order λ [m] o an entire unction are deined by and ρ [m] respectively. lo [m] M (r, = sup lo r and λ [m] lo [m] M (r, = in lo r For m =2, Deinition 2 reduces to Deinition 1. I ρ < then is o inite order. Also ρ = 0 means that is o order zero. In this connection Datta and Biswas [2] ave the ollowin deinition : Deinition 3 [2]Let be an entire unction o order zero. Then the quantities and λ o are deined by: ρ ρ lo M (r, = sup lo r and λ = in lo M (r,. lo r

3 Applications o slowly chanin unctions 239 Let L L (r be a positive continuous unction increasin slowly i.e., L (ar L (r asr or every positive constant a. Sinh and Barker [7] deined it in the ollowin way: Deinition 4 [7]A positive continuous unction L (r is called a slowly chanin unction i or ε (> 0, 1 k ε L (kr L (r kε or r r (ε and uniormly or k ( 1. I urther, L (r is dierentiable, the above condition is equivalent to rl (r L (r =0. Somasundaram and Thamizharasi [8] introduced the notions o L-order (L-lower order or entire unctions where L L (r is a positive continuous unction increasin slowly i.e.,l (ar L (r as r or every positive constant a. The more eneralised concept or L-order ( L-lower order or entire unction are L -order ( L -lower order. Their deinitions are as ollows: Deinition 5 [8]The L -order ρ L unction are deined as and the L -lower order λ L o an entire ρ L lo [2] M (r, = sup lo [re L(r ] and λ L = in lo [2] M (r,. lo [re L(r ] In the line o Sato [6], Datta and Biswas [2] one can deine the eneralised L -order and eneralised L -lower orderλ [m]l o an entire unction in the ollowin manner : Deinition 6 Let m be an inteer 1. The eneralised L -order eneralised L -lower order λ [m]l o an entire unction are deined as and respectively. lo [m] M (r, = sup lo [re L(r ] and λ [m]l lo [m] M (r, = in lo [re L(r ] Datta, Biswas and Hoque [3] reormulated Deinition 6 in terms o the maximum terms o entire unctions in the ollowin way:

4 240 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Deinition 7 [3] The rowth indicators are deined as and λ [m]l or an entire unction lo [m] μ (r, = sup lo [re L(r ] and λ [m]l lo [m] μ (r, = in lo [re L(r ] respectively where m be an inteer 1. Lakshminarasimhan [4] introduced the idea o the unctions o L- bounded index. Later Lahiri and Bhattacharjee [5] worked on the entire unctions o L-bounded index and o non uniorm L-bounded index. In this paper we would like to investiate some rowth properties o composite entire unctions on the basis o their maximum terms and maximum moduli usin eneralised L -order and eneralised L -lower order. 2 Lemmas. In this section we present some lemmas which will be needed in the sequel. Lemma 1 [9] Let and be any two entire unctions with (0 = 0. Then or all suiciently lare values o r, μ (r, 1 ( 1 2 μ 8 μ 4, (0,. Lemma 2 [1] I and are any two entire unctions then or all suiciently lare values o r, ( 1 M 8 M 2, (0, M(r, M (M (r,,. 3 Theorems. In this section we present the main results o the paper. Theorem 1 Let and be any two entire unctions such that 0 <λ [m]l < where m 1 and 0 <λ L ρ L <. Then or every constant A and real number x, lo [m] μ (r, 1+x =. lo [m] μ (r A,

5 Applications o slowly chanin unctions 241 Proo. I x is such that 1+x 0, then the theorem is obvious. So we suppose that 1 + x>0. Now in view o Lemma 1, we et or all suiciently lare values o r that μ (r, 1 ( 1 2 μ 16 μ 2,, ( 1 i.e., lo [m] μ (r, O (1 + lo [m] μ 16 μ 2,, i.e., lo [m] μ (r, O (1 + i.e., lo [m] μ (r, O (1 + (λ [m]l ε [ 1 lo (λ [m]l 16 μ, 2 +L ( 1 ] 16 μ, 2 [ ε lo M 2, + O (1 ( 1 ] +L 16 μ, 2 i.e., lo [m] μ (r, O (1 λ L ( + (λ [m]l ε e 2 L(r ε 1 + O (1 + L 16 μ, (1 2 where we choose 0 <ε<min λ [m]l,λ L. Also or all suiciently lare values o r, we obtain that lo [m] μ A, L( lo r A e L(rA i.e., lo [m] μ A, lo r A e L(rA i.e., lo [m] μ A, 1+x 1+x ( ( A lo r + L r A 1+x. (2 Thereore rom (1 and (2 it ollows or all suiciently lare values o r that lo [m] μ (r, lo [m] μ (r A, 1+x O (1 + (λ [m]l ( ε r 2 e L(r λ L ε ( + O (1 + L 1 μ, 16 2 ( 1+x. (3 (A lo r + L (ra 1+x

6 242 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Thus the theorem ollows rom (3. In the line o Theorem 1, we may establish the ollowin theorem or the riht actor o the composite entire unction : Theorem 2 Let and be any two entire unctions with 0 < λ [m]l < and 0 <λ L ρ L < where m 1. Then or every constant A and real number x, The proo is omitted. lo [m] μ (r, 1+x =. lo [2] μ (r A, Theorem 3 Let and be any two entire unctions such that 0 <λ [m]l < and 0 <λ L ρ L < where m 1. Then or any two positive inteers α and β, lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l(exp (exp (r α =, where K (r, α; L = 0 i r β = o L (exp (exp (r α as r L (exp (exp (r α otherwise. Proo. Takin x = 0 and A = 1 in Theorem 1, we obtain or K>1 and or all suiciently lare values o r that lo [m] μ (r, > Klo [m] μ (r, i.e., lo [m 1] μ (r, > lo [m 1] μ (r, i.e., lo [m 1] μ (r, > lo [m 1] μ (r, i.e., lo [m 1] μ (r, > lo [m 1] μ (r, (4 Thereore rom (4 we et or all suiciently lare values o r that K K lo [m] μ (exp (exp (r α, > lo [m] μ (exp (exp (r α, > i.e., lo [m] μ (exp (exp (r α, (λ [m]l ε. lo exp (exp (r α. exp L (exp (exp (r α

7 Applications o slowly chanin unctions 243 > i.e., lo [m] μ (exp (exp (r α, (λ [m]l ε. (exp (r α + L (exp (exp (r α > i.e., lo [m] μ (exp (exp (r α, ( (λ [m]l ε. (exp (r α 1+ L (exp (exp (rα (exp (r α i.e., lo [m+1] μ (exp (exp (r α, > O(1 + lo exp (r α + lo 1+ L (exp (exp (rα (exp (r α i.e., lo [m+1] μ (exp (exp (r α, >O(1 + r α + lo 1+ L (exp (exp (rα (exp (r α i.e., lo [m+1] μ (exp (exp (r α, >O(1 + r α + L (exp (exp (r α lo [exp L (exp (exp (r α ] [ + lo 1+ L (exp (exp ] (rα exp (μr α i.e., lo [m+1] μ (exp (exp (r α, >O(1 + r α + L (exp (exp (r α [ 1 + lo exp L (exp (exp (r α ] L (exp (exp (r α + exp L (exp (exp (r α. exp (r α i.e., lo [m+1] μ (exp (exp (r α, >O(1 + r (α β.r β + L (exp (exp (r α. (5 Aain we have or all suiciently lare values o r that lo [m] μ ( exp β, lo exp β e L(exp(rβ i.e., lo [m] μ ( exp β, lo ( exp r β + L ( exp β i.e., lo [m] μ ( exp β, r β + L ( exp β

8 244 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque i.e., lo [m] μ ( exp β, L ( exp β ( r β. (6 Now rom (5 and (6 it ollows or all suiciently lare values o r that lo [m+1] μ (exp (exp (r α, ( r (α β [ O (1 + lo [m] μ ( exp β, L ( exp β] + L (exp (exp (r α (7 i.e., lo [m+1] μ (exp (exp (r α, L (exp (exp (rα + O (1 lo [m] μ (exp (r β, lo [m] μ (exp (r β, + r(α β 1 L ( exp β lo [m] μ (exp (r β,. (8 Aain rom (7 we et or all suiciently lare values o r that lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l(exp (exp (r α O (1 r (α β L ( exp β lo [m] μ (exp (r β,+l(exp (exp (r α ( r (α β lo [m] μ ( exp β, +ε + lo [m] μ (exp (r β,+l(exp (exp (r α L (exp (exp (r α + lo [m] μ (exp (r β,+l(exp (exp (r α i.e., lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l (exp (exp (r α + ( r (α β +ε 1+ L(exp(exp(rα lo [m] μ(exp(r β, + O(1 r (α β L(exp(r β L(exp(exp(r α lo [m] μ(exp(r β, L(exp(exp(r α lo[m] μ(exp(r β, L(exp(exp(r α Case I. I r β = o L (exp (exp (r α then it ollows rom (8 that lo [m+1] μ (exp (exp (r α, in =. lo [m] μ (exp (r β,. (9

9 Applications o slowly chanin unctions 245 Case II. I r β o L (exp (exp (r α then two sub cases may arise: Sub case (a. I L (exp (exp (r α = o lo [m] μ ( exp β,, then we et rom (9 that in lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l(exp (exp (r α =. Sub case (b. I L (exp (exp (r α lo [m] μ ( exp β, then L (exp (exp (r α lo [m] μ (exp (r β, =1 and we obtain rom (9 that in lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l(exp (exp (r α =. Combinin Case I and Case II we obtain that lo [m+1] μ (exp (exp (r α, lo [m] μ (exp (r β,+l(exp (exp (r α =, 0ir where K (r, α; L = μ = o L (exp (exp (r α as r L (exp (exp (r α otherwise. This proves the theorem. Theorem 4 Let and be any two entire unctions with 0 < λ [m]l < and 0 <λ L ρ L < where m 1. Then or any two positive inteers α and β, lo [m+1] μ (exp (exp (r α, lo [2] μ (exp (r β,+l(exp (exp (r α =, where K (r, α; L = 0 i r β = o L (exp (exp (r α as r L (exp (exp (r α otherwise. 3. The proo is omitted because it can be carried out in the line o Theorem Remark 1 In view o Lemma 2, the results analoous to Theorem 1, Theorem 2, Theorem 3 and Theorem 4 can also be derived in terms o maximum moduli o composite entire unctions.

10 246 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Theorem 5 Let and be any two entire unctions such that 0 <ρ L λ [m]l < where m 1. Then or any β>1, lo [m] μ(r, lo [m] μ (r, K (r, ; L =0, 1 i L (μ (βr, = o r α e αl(r as r where K (r, ; L = and or some α<λ [m]l L (μ (βr, otherwise. < Proo. In view o Lemma 2 and takin R = βr in the inequality μ (r, M (r, R μ (R, c. [10], we have or all suiciently lare values o R r r that μ(r, M(r, M (M (r,, i.e., lo [m] μ(r, lo [m] M (M (r,, i.e., lo [m] μ(r, i.e., lo [m] μ(r, i.e., lo [m] μ(r, [lo ] M (r, e L(M(r, [lo M (r, +L (M (r, ] (10 [ re L(r (ρ L +ε ( ] β + L (β 1 μ (βr, i.e., lo [m] μ(r, [ re L(r (ρ L +ε ] + L (μ (βr,. (11 Also we obtain or all suiciently lare values o r that lo [m] μ (r, (λ [m]l ε lo [ re L(r] i.e., lo [m] μ (r, (λ [m]l ε lo [ re L(r] i.e., lo [m] μ (r, [ re L(r] λ [m]l ε. (12

11 Applications o slowly chanin unctions 247 Now rom (11 and (12 we et or all suiciently lare values o r that (ρ [ [m]l lo [m] re L(r (ρ L +ε ] + L (μ (βr, μ(r, lo [m]. (13 μ (r, λ [re L(r ] [m]l ε Since ρ L <λ [m]l, we can choose ε (> 0 in such a way that ρ L <λ [m]l ε. (14 Case I. Let L (μ (βr, = o r α e αl(r as r and or some α<λ [m]l As α<λ [m]l, we can choose ε (> 0 in such a way that. α<λ [m]l ε. (15 Since L (μ (βr, = o r α e αl(r as r we et on usin (15 that i.e., L (μ (βr, 0as r α e αl(r L (μ (βr, 0as. (16 λ [re L(r ] [m]l ε Now in view o (13, (14 and (16 we obtain that lo [m] μ(r, =0. (17 lo [m] μ (r, Case II. I L (μ (βr, o r α e αl(r as r and or some α<λ [m]l then we et rom (13 that or a sequence o values o r tendin to ininity, lo [m] μ(r, lo [m] μ (r, L (μ (βr, Now usin (14 it ollows rom (18 that [re L(r ] + [re L(r ] re L(r (ρ L λ [m]l ε +ε L (μ (βr,. (18 λ [m]l ε lo [m] μ(r, =0. (19 lo [m] μ (r, L (μ (βr,

12 248 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Combinin (17 and (19 we obtain that where K (r, ; L = lo [m] μ(r, lo [m] μ (r, K (r, ; L =0, Thus the theorem is established. 1iL (μ (βr, = o r α e αl(r as r and or some α<λ [m]l L (μ (βr, otherwise. The ollowin theorem can be carried out in the line o Theorem 5 and thereore its proo is omitted : Theorem 6 Let and be any two entire unctions with 0 <ρ L where m 1. Then or any β>1, where K (r, ; L = in lo [m] μ(r, lo [m] μ (r, K (r, ; L =0, 1 i L (μ (βr, = o r α e αl(r as r and or some α< L (μ (βr, otherwise. < < Replacin maximum term by maximum modulus in Theorem 5 and Theorem 6 we respectively et Theorem 7 and Theorem 8 and thereore their proos are omitted. Theorem 7 Let and be any two entire unctions such that 0 <ρ L λ [m]l where K (r, ; L = < where m 1. Then lo [m] M(r, lo [m] M (r, K (r, ; L =0, 1 i L (M (r, = o r α e αl(r as r and or some α<λ [m]l L (M (r, otherwise. Theorem 8 Suppose and be any two entire unctions with 0 <ρ L < where m 1. Then where K (r, ; L = in lo [m] M(r, lo [m] M (r, K (r, ; L =0, 1 i L (M (r, = o r α e αl(r as r and or some α< L (M (r, otherwise. < <

13 Applications o slowly chanin unctions 249 Theorem 9 Let and be any two entire unctions with <, 0 < λ L ρ L < where m is any positive inteer. Then or any β>1, (a I L (μ (βr, = o lo [2] μ (r, then sup lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, ρl and (b i lo [2] μ (r, =o L (μ (βr, then λ L lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, =0. Proo. Takin R = βr in the inequality μ (r, M (r, R μ (R, c. [10] R r and also usin lo 1+ O(1+L(μ(βr, lo μ(βr, values o r we obtain rom (10 that O(1+L(μ(βr,, or all suiciently lare lo μ(βr, lo [m] μ(r, [ ( ] β lo μ (βr,+o(1 + L (β 1 μ (βr, i.e., lo [m] μ(r, [ lo μ (βr, 1+ ] O(1 + L (μ (βr, lo μ (βr, i.e., lo [m+1] μ(r, lo i.e., lo [m+1] μ (r, lo i.e., lo [m+1] μ (r, lo + lo [2] μ (βr, 1+ + lo + lo O(1 + L (μ (βr, lo μ (βr, + ( ρ L lo βre L(βr 1+ O(1 + L (μ (βr, lo μ (βr, + ( ρ L lo βre L(r 1+ + lo O(1 + L (μ (βr, lo μ (βr,

14 250 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque i.e., lo [m+1] μ (r, O (1 + ( ρ L lo βr + L (r + O(1 + L (μ (βr, lo μ (βr, i.e., lo [m+1] μ (r, O (1 + ( ρ L lo r + L (r + ( ρ L lo β + O(1 + L (μ (βr, lo μ (βr,. (20 Aain rom the deinition o L -lower order, we et or all suiciently lare values o r that lo [2] μ (r, ( λ L ε lo [ re L(r] i.e., lo [2] μ (r, ( λ L ε lo [ re L(r] i.e., lo [2] μ (r, ( λ L ε [lo r + L (r] i.e., lo r + L (r lo[2] μ (r, ( λ L ε. (21 Hence rom (20 and (21 it ollows or all suiciently lare values o r that lo [m+1] μ (r, ( ρ L O (1 + ε λ L lo [2] μ (r, + ( ρ L lo β + O(1 + L (μ (βr, lo μ (βr, lo [m+1] μ (r, i.e, lo [2] μ (r, +L (μ (βr, O (1 + ( ρ L lo β lo [2] μ (r, +L (μ (βr, + ( ρ L λ L lo [2] μ (r, ε lo [2] μ (r, +L (μ (βr, O(1 + L (μ (βr, + [ ] lo [2] μ (r, +L (μ (βr, lo μ (βr, i.e, lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, + [ O(1+(ρ L L(μ(βr, +ε lo β + lo [2] μ(r, +1 L(μ(βr, 1 1+ lo[2] μ(r, L(μ(βr, ( ρ L +ε λ L ε 1+ L(μ(βr, lo [2] μ(r, ] lo μ (βr,. (22

15 Applications o slowly chanin unctions 251 Since L (μ (βr, = o lo [2] μ (r, obtain rom (22 that as r and ε (> 0 is arbitrary, we sup lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, ρl λ L. (23 Aain i lo [2] μ (r, =o L (μ (βr, then rom (22 we et that lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, =0. (24 Thus the theorem ollows rom (23 and (24. Corollary 1 Let and be any two entire unctions with 0 <ρ L < where m 1. Then or any β>1, (a i L (μ (βr, = o lo [2] μ (r, then < and in lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, 1 and (b i lo [2] μ (r, =o L (μ (βr, then in lo [m+1] μ (r, lo [2] μ (r, +L (μ (βr, =0. We omit the proo o Corollary 1 because it can be carried out in the line o Theorem 7. Remark 2 The equality sin in Theorem 5 and Corollary 1 cannot be removed as we see in the ollowin example: Example 1 Let = = exp z, m =2, β =2and L (r = 1 p exp ( 1 r where p is any positive real number. Then ρ L = λ L = ρ L =1. Now lo μ (r, lo M (r, = exp r, and 2μ (2r, M (r, = exp r.

16 252 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque Also So Hence sup lo μ (r, lo M 2, and μ (r, M (r, = exp r. + O(1 = exp + O(1, 2 L (M (r, = L (exp r = 1 ( 1 p exp exp r lo [3] μ (r, lo [2] μ (r, +L (μ (βr, sup lo r ( lo r + O (1 + 1 exp p. 1 exp r =1 and in Thereore in lo [3] μ (r, lo [2] μ (r, +L (μ (βr, lo [3] μ (r, lo [2] μ (r, +L (μ (βr, in = sup lo r + O(1 ( lo r + 1 exp p 1 exp r =1. lo [3] μ (r, lo [2] μ (r, +L (μ (βr, =1. Theorem 10 Let and be any two entire unctions with 0 <λ L ρ L < where m is any positive inteer. Then (a i L (M (r, = o lo [2] M(r, then < and sup lo [m+1] M (r, lo [2] M (r, +L (M (r, ρl and (b i lo [2] M (r, =o L (M (r, then λ L lo [m+1] M (r, lo [2] M (r, +L (M (r, =0. Corollary 2 Let and be any two entire unctions with 0 <ρ L < where m 1. Then or any β>1, (a i L (M (r, = o lo [2] M (r, then < and in lo [m+1] M (r, lo [2] M (r, +L (M (r, 1

17 Applications o slowly chanin unctions 253 and (b i lo [2] M (r, =o L (M (r, then in lo [m+1] M (r, lo [2] M (r, +L (M (r, =0. We omit the proo o Theorem 10 and Corollary 2 because in view o Lemma 2 it can be carried out in the line o Theorem 9 and Corollary 1 respectively. Remark 3 Considerin = = exp z, m =2and L (r = 1 p exp ( 1 r where p is any positive real number, one can easily veriy that the equality sin in Theorem 10 and Corollary 2 cannot be removed. Acknowledement The authors are thankul to Proessor Huzoor H. Khan, Department o Mathematics, Aliarh Mus University, Aliarh (U.P., India or his Constant encouraement to carry out the paper. Reerences [1] J. Clunie : The composition o entire and meromorphic unctions, Mathematical Essays dedicated to A. J. Macintyre,Ohio University Press (1970, pp [2] S.K. Datta and T. Biswas : On the deinition o a meromorphic unction o order zero, International Mathematical Forum, Vol.4, No. 37(2009 pp [3] S. K. Datta, T. Biswas and Md. A. Hoque : Maxumum modulus and maxumum term based rowth analysis o entire unction in the liht o slowly chanin unction, Investiations in Mathematical Sciences, Vol. 3, No. 1 (2013, pp [4] T.V. Lakshminarasimhan : A note on entire unctions o bounded index, J. Indian Math. Soc., Vol. 38 (1974, pp [5] I. Lahiri and N.R. Bhattacharjee : Functions o L-bounded index and o non-uniorm L-bounded index, Indian J. Math., Vol. 40 (1998, No. 1, pp [6] D. Sato : On the rate o rowth o entire unctions o ast rowth, Bull. Amer. Math. Soc., Vol. 69 (1963, pp [7] S.K. Sinh and G.P. Barker : Slowly chanin unctions and their applications, Indian J. Math., Vol. 19 (1977, No. 1, pp 1-6.

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