S. K. VAISH AND R. CHANKANYAL. = ρ(f), b λ(f) ρ(f) (1.1)

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1 TAMKANG JOURNAL OF MATHEMATICS Volume 35, Number, Witer 00 ON THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION OF ENTIRE FUNCTIONS S. K. VAISH AND R. CHANKANYAL Abstract. We study some growth properties of maximum modulus ad maximum term of compositio of etire fuctios of (p, q)-order as compared to the growth of their correspodig left ad right factors. Some of the results that we obtai here geeralize ad improve the kow results of Sigh ad Baloria, ad, Sog ad Yag. 1. Itroductio Let f(z) = =0 a z be a etire fuctio. The as usual µ(r, f) = max 0 { a r } is called the maximum term of f(z) o z = r ad M(r, f) = max z =r f(z) is called the maximum modulus of f(z) o z = r. The cocept of (p, q)-order ad lower (p, q)-order of f(z) havig idex pair (p, q), (p q 1, p ), was itroduced by Jueja, Kapoor ad Bajpai []. Thus f(z) is said to be of (p, q)-order ρ(f) ad lower (p, q)-order λ(f), if sup log [p] M(r, f) if log [q] r = ρ(f), b λ(f) ρ(f) (1.1) λ(f) where b = 1 if p = q ad zero otherwise ad exp [0] x = log [0] x = x; exp [m] x = log [ m] x = exp(exp [m 1] x) = log(log [ m 1] x); m = ±1, ±,.... Throughout this paper wheever (log [m] x) β (0 < β < ) occurs, it is uderstood that x is such that this expressio is a real umber.. Kow Results I this sectio we state some kow results i the form of Lemmas which will be eeded i the sequel. Lemma 1. (Jueja, Kapoor ad Bajpai []). Let f(z) be a etire fuctio of (p, q)-order ρ(f) ad lower (p, q)-order λ(f), the sup log [p] µ(r, f) = ρ(f) if log [q] r λ(f). (.1) Received May, Mathematics Subject Classificatio. 30D35. Key words ad phrases. Etire fuctio, compositio, (p, q)-order, maximum modulus, maximum term. 93

2 9 S. K. VAISH AND R. CHANKANYAL Lemma (Cluie [1]). Let f(z) ad g(z) be two etire fuctios with g(0) = 0. Let α satisfy 0 < α < 1 ad c(α) = (1 α) /(α). The, for r > 0, M(r, fog) M(c(α)M(αr, g), f). (.) Further, if g(z) is ay etire fuctio with α = 1/, for sufficietly large values of r, M(r, fog) M ( 1 8 M, g g(0), f. (.3) Lemma 3 (Sigh [3]). Let f(z) ad g(z) be two etire fuctios with g(0) = 0. Let α satisfyig 0 < α < 1 ad c(α) = (1 α) /(α). Also, let 0 < δ < 1, the µ(r, fog) (1 δ)µ(c(α)µ(αδr, g), f), (.) ad, if g(z) is ay etire fuctio, the with α = δ = 1/, for sufficietly large values of r, µ(r, fog) 1 ( 1 µ 8 µ, g g(0), f. (.5) I this paper we study some growth properties of maximum modulus ad maximum term of compositio of etire fuctios of (p, q)-order as compared to the growth of their correspodig left ad right factors. Some of the results that we obtai here geeralize ad improve the kow results of Sigh ad Baloria [] ad Sog ad Yag [5]. 3. Mai Results Theorem 1. Let f ad g be two etire fuctios such that 0 < λ(f) ρ(f) < ad 0 < λ(g) ρ(g) <. The for every positive costat γ, p > q ad every real umber x, ad = (3.1) {log [p] M(r γ, f)} =. (3.) {log [p] M(r γ, g)} Proof. If x is such that 1 + x 0, the the theorem is obvious. So, we suppose that 1 + x > 0. Now, for all sufficietly large values of r, we get from (.3) M(r, fog) M ( 1 16 M, g, f.

3 THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION 95 This gives, log [p] M ( 1 16 M, g, f ( 1 )) > (λ(f) ε)log [q] 16 M, g { > (λ(f) ε)log [q 1] log 1 ( ( 16 + exp[p ] log [q 1] r )) } λ(g) ε ( ( = (λ(f) ε)exp [p q 1] log [q 1] r )) λ(g) ε + (λ(f) ε)o(1) (3.3) where we choose 0 < ε < mi(λ(f), λ(g)). Also, for all r r 0, log [p] M(r, f) < (ρ(f) + ε)log [q] r. Now, it is possible to choose r sufficietly large so that r γ r 0. Thus {log [p] M(r γ, f)} < (ρ(f) + ε) {log [q] (r γ )}. (3.) From (3.3) ad (3.) it follows that for all sufficietly large values of r, { (λ(f) ε)exp[p q 1] log [q 1] ) } λ(g) ε + (λ(f) ε)o(1) {log [p] M(r γ, f)} >. (ρ(f) + ε) {log [q] (r γ )} (3.5) Sice {exp [p q 1] (log [q 1] (r/))} λ(g) ε /{log [q] (r γ )} as r, statemet (3.1) follows from (3.5). Statemet (3.) follows similarly by usig the followig iequality i plcae of (3.), {log [p] M(r γ, g)} < (ρ(g) + ε) {log [q] (r γ )} for all sufficietly large values of r. This proves the theorem. Note. Theorem 1 eed ot be true if either λ(g) = 0 or λ(f) = 0. For example: Let g(z) = z, x = 0 ad γ = 1, the λ(g) = 0 ad we fid = 1. {log [p] M(r γ, g)} Similarly, if f(z) = z, x = 0 ad γ = 1, the λ(f) = 0 ad = 1. {log [p] M(r γ, g)}

4 96 S. K. VAISH AND R. CHANKANYAL Remark 1. (i) For p =, q = 1 ad x = 0 this theorem is due to Sigh ad Baloria []. (ii) For p =, q = 1, x = 0 ad γ = 1 this theorem is due to Sog ad Yag [5]. Theorem. Let f ad g be two etire fuctios of fiite (p, q)-orders ad λ(f) > 0. The, for h > 0 ad p > q, () log [p] M(exp [p 1] (log [q 1] r) h, f) = 0 where h > (1 + x)ρ(g) if p =, q = 1 ad h > ρ(g) otherwise. Also x (, ). Proof. If 1 + x 0 the theorem is trivial. So, we cosider 1 + x > 0. By the maximum modulus priciple, we have so that for all sufficietly large values of r, M(r, fog) M(M(r, g), f), {} < (ρ(f) + ε) {log [q] M(r, g)} Agai, for all sufficietly large values of r, Hece, for all sufficietly large values of r, < (ρ(f) + ε) {exp [p q 1] (log [q 1] r) ρ(g)+ε }. log [p] M(r, f) > (λ(f) ε)log [q] r. {} log [p] M{exp [p 1] (log q 1] r) h, f} < (ρ(f) + ε) {exp [p q 1] (log [q 1] r) ρ(g)+ε } (λ(f) ε)exp [p q 1] (log [q 1] r) h from which the theorem follows because we ca choose ε such that 0 < ε < mi{λ(f), h ρ(g)} if p =, q = 1 ad 0 < ε < mi(λ(f), h ρ(g)) otherwise. Remark. If we take the coditio 0 < ρ(f) iplace of λ(f) > 0 the theorem remais true with it replaced by it iferior ad i this case improves ad geeralizes Theorem of Sigh ad Baloria []. Theorem 3. Let f ad g be two etire fuctios such that 0 < λ(g) ρ(g) < ad ρ(f) <. The, for h > 0 ad p > q, {} ( ) = 0 log [p] M exp [p 1] (log [q 1] r) h, g where h > (1 + x)ρ(g) if p =, q = 1 ad h > ρ(g) otherwise. Also x (, ).

5 THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION 97 We omit the proof of this theorem because it rus parallel to that of Theorem. Theorem. Let f ad g be two etire fuctios of fiite (p, q)-order with λ(g) > ρ(f) λ(f) > 0. The, for p > q, log [p] M(exp [p 1] (log [q 1] r) ρ(f), g) =. Proof. From (3.3), we have, for all r r 0, ( ( > (λ(f) ε)exp [p q 1] log [q 1] r )) λ(g) ε + (λ(f) ε)o(1). Also, for all r r 0, log [p] M(r, g) < (ρ(g) + ε)log [q] r. Takig r so large that exp [p 1] (log [q 1] r) ρ(f) r 0, the log [p] M(exp [p 1] (log [q 1] r) ρ(f), g) < (ρ(g) + ε)exp [p q 1] (log [q 1] r) ρ(f). Thus, for sufficietly large r, > log [p] M(exp [p 1] (log [q 1] r) ρ(f), g) (λ(f) ε))exp [p q 1] {log [q 1] ) } λ(g) ε + (λ(f) ε)o(1) (ρ(g) + ε)exp [p q 1] (log [q 1] r) ρ(f). Sice, λ(g) > ρ(f) ad we ca choose ε > 0 such that λ(g) ε > ρ(f). Thus, ad the theorem follows. log [p] M(exp [p 1] (log [q 1] r) ρ(f), g) = Remark 3. For p = ad q = 1 this theorem is due to Sigh ad Baloria []. We shall use the techique of the above theorems to get the parallel results o the maximum term of compositio of etire fuctios. We prove: Theorem 5. Let f ad g be two etire fuctios such that 0 < λ(f) ρ(f) < ad 0 < λ(g) ρ(g) <. The, for every positive costat γ, p > q ad every real umber x, log [p] µ(r, fog) = (3.6) {log [p] µ(r γ, f)}

6 98 S. K. VAISH AND R. CHANKANYAL ad log [p] µ(r, fog) =. (3.7) {log [p] µ(r γ, g)} Proof. If x is such that 1 + x 0, the the theorem is obvious. So, we suppose that 1 + x > 0. Now, from (.5) for all sufficietly large values of r, µ(r, fog) 1 ( 1 µ 16 µ, g, f. This gives, log [p] µ(r, fog) > 1 ( ( (λ(f) ε)exp[p q 1] log [q 1] r )) λ(g) ε 1 + (λ(f) ε)o(1) because i view of (.1), we have for sufficietly large values of r, Also, from (.1) for all r r 0, ad so for sufficietly large r γ r 0, Thus, as i Theorem 1, we fid µ(r, f) > exp [p 1] (log [q 1] r) λ(f) ε. log [p] µ(r, f) < (ρ(f) + ε)log [q] r {log [p] µ(r γ, f)} < (ρ(f) + ε) (log [q] (r γ )). log [p] µ(r, fog) {log [p] µ(r γ, f)} = sice we ca choose ε such that 0 < ε < mi(λ(f), λ(g)). We omit the proof of (3.7). Remark. The aalogues to Theorem, Theorem 3 ad Theorem may later be filled by the reader. Theorem 6. Let f ad g be two etire fuctios of positive lower (p, q)-order ad of fiite (p, q)-order. The, for every h > 0, ( ) } log [p] µ { exp [q 1] r 1+h sup, fog =. log [p 1] µ(exp [q 1] r, g) Proof. There exists a sequece {r }, = 1,,... such that µ(r, g) > exp [p 1] (log [q 1] r ) ρ(g) ε.

7 THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION 99 Let R = ( log [q 1] r ) 1/(1+h), the µ {exp [q 1] ( R 1+h Now, from (.1) ad (.5), for all r r 0, )} ( ) R > exp [p 1] 1+h ρ(g) ε. log [p] µ(r, fog) > 1 ) (λ(f) ε)log[q] µ, g + 1 (λ(f) ε)o(1). If R r 0, the exp [q 1] ( R1+h ) r 0 ad so the above equatio gives log [p] µ ( R { exp [q 1] 1+h > 1 (λ(f) ε)log[q] µ ) }, fog {exp [q 1] ( R 1+h Usig (.1), we fid for the sequece R r 0, Also, for all r r 0, ) }, g + 1 (λ(f) ε)o(1). ( ) } R log [p] µ { exp [q 1] 1+h, fog > 1 ( ) R 1+h ρ(g) ε (λ(f) ε)exp[p q 1] + 1 (λ(f) ε)o(1). log [p 1] µ(exp [q 1] r, g) < (r) ρ(g)+ε. Thus, for the sequece R ( r 0 ), ( ) } log [p] µ { exp [q 1] R 1+h, fog log [p 1] µ(exp [q 1] R, g) ( ) 1 (λ(f) R 1+h ρ(g) ε ε)exp[p q 1] + 1 (λ(f) ε)o(1) > (R ) ρ(g)+ε as r sice we ca choose ε such that 0 < ε < mi{λ(f), hρ(g) +h }. Remark 5. For p = ad q = 1 this theorem is due to Sigh ad Baloria []. Refereces [1] J. Cluie, The compositio of etire ad meromorphic fuctios, Macityre Memorial Volume, Ohio Uiversity Press, (1970), 75-9.

8 300 S. K. VAISH AND R. CHANKANYAL [] O. P. Jueja, G. P. Kapoor ad S. K. Bajpai, O the (p,q)-order ad lower (p,q)-order of etire fuctio, J. reie agew. Math. 8(1976), [3] A. P. Sigh, O the maximum term of compositio of etire fuctios, Proc. Nat. Acad. Sci. Idia 59(A)I(1989), [] A. P. Sigh ad M. S. Baloria, O the maximum modulus ad maximum term of compositio of etire fuctios, Idia J. pure appl. Math. (1991), [5] G. D. Sog ad C. C. Yag, Further growth properties of compositio of etire ad meromorphic fuctios, Idia J. pure appl. Math. 15(198), Departmet of Mathematics, G. B. Pat Uiversity, Patagar-6315 (U. S. Nagar), Uttarachal, Idia.