GROWTH PROPERTIES OF COMPOSITE ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES IN UNIT POLYDISC FROM THE VIEW POINT OF THEIR NEVANLINNA L -ORDERS

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1 Palestine Journal o Mathematics Vol Palestine Polytechnic University-PPU 209 GROWTH PROPERTIES OF COMPOSITE ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES IN UNIT POLYDISC FROM THE VIEW POINT OF THEIR NEVANLINNA L -ORDERS Sanjib Kumar Datta Tanmay Biswas and Ahsanul Hoque Communicated by Ayman Badawi MSC 200 Classiications: 30D35 30D30 Keywords and phrases: Growth analytic unction o n complex variables composition unit polydisc Nevanlinna n variables based L -order Nevanlinna n variables based L -lower order slowly changing unction in the unit polydisc Abstract In this paper we introduce the idea o Nevanlinna n variables based L -order and Nevanlinna n variables based L -lower order in the unit polydisc Hence we study some growth properties o Nevanlinna s Characteristic unction relating to the composition o two analytic unction in the unit polydisc on the basis o Nevanlinna n variables based L -order and Nevanlinna n variables based L -lower order as compared to the growth o their corresponding let and right actors Introduction Deinitions and Notations A unction analytic in the unit disc U = z : z < is said to be o inite Nevanlinna order [4 i there exist a number µ such that Nevanlinna characteristic unction T r = 2π 2π 0 + re iθ dθ satisies T r < r µ or all r in 0 < r 0 µ < r < The greatest lower bound o all such number µ is called the Nevanlinna order o Thus the Nevanlinna order ρ o is given by ρ = r Similarly Nevanlinna lower order λ o is given by λ = lim in r T r r T r r Somasundaram and Thamizharasi [6 introduced the notions o L-order L-lower order or entire unctions where L L r is a positive continuous unction increasing slowly ie L ar L r as r or every positive constant a In the line o Somasundaram and Thamizharasi [6 one may introduce the notion o Nevanlinna L-order or an analytic unction in the unit disc U = z : z < where L L a unit disc U increasing slowly ie L L in the ollowing manner: Deinition I be analytic in U then the Nevanlinna L-order ρ L ρ L = in µ > 0 : T r < L r is a positive continuous unction in the as r or every positive constant a µ o is deined as or all 0 < r 0 µ < r <

2 GROWTH PROPERTIES OF COMPOSITE ANALYTIC 347 Similarly one may deine λ L the Nevanlinna L-lower order o in the ollowing way: λ = lim in r T r L The more generalised concept o Nevanlinna L-order and the Nevanlinna L-lower order o an analytic unction in the unit disc U are the Nevanlinna L -order and the Nevanlinna L -lower order Their deinitions are as ollows: o an ana- Deinition 2 [2 The Nevanlinna L -order lytic unction in the unit disc U are deined as T r = r expl and Nevanlinna L -lower order T r = lim in r expl and respectively Extending the notion o single variable to several variable let z z 2 z n be a non-constant analytic unction o n complex variables z z 2 z n and z n in the unit polydisc U = z z 2 z n : z j j = 2 n; r > 0 r 2 > 0 r n > 0 Now in the line o Nevanlinna L -order and Nevanlinna L -lower order in this paper we introduce the Nevanlinna n variables based L -order and the Nevanlinna n variables based L -lower order or unctions o n complex variables analytic in a unit polydisc as ollows : and = r r 2 r n = lim in r r 2 r n T r r 2 r n exp L 2 n 2 n T r r 2 r n exp L 2 n 2 n where L L 2 is a positive continuous unction in the unit polydisc U increasing slowly ie L n 2 a a a 2 n L as r or every positive constant a In this paper we study some growth properties o Nevanlinna s Characteristic unction relating to the composition o two analytic unction in the unit polydisc on the basis o Nevanlinna n variables based L -order and Nevanlinna n variables based L -lower order as compared to the growth o their corresponding let and right actors We do not explain the standard deinitions and notations in the theory o entire unctions as those are available in [ [3 and [5 2 Theorems n In this section we present the main results o the paper Theorem 2 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g g < and 0 < vn < then g T g r r 2 r n lim in r r 2 r n T r r 2 r n λ L vn g T g r r 2 r n r r 2 r n T r r 2 r n vnρl g

3 348 Sanjib Kumar Datta Tanmay Biswas and Ahsanul Hoque Proo From the deinition o vn ρ L values o 2 and and vn g we have or arbitrary positive ε and or all large that n T g r r 2 r n g ε exp L 2 r r 2 r n n 2 and T r r 2 r n + ε exp L 2 r r 2 r n Now rom 2 22 it ollows or all suiciently large values o n that T g r r 2 r n T r r 2 r n n g ε exp L 2 n + ε 22 2 and 2 n exp L 2 n 2 n As ε > 0 is arbitrary we obtain that T g r r 2 r n lim in r T r r 2 r n vnλl g 23 Again or a sequence o values o 2 and n tending to ininity T g r r 2 r n g + ε exp L 2 r r 2 r n and or all suiciently large values o 2 and n T r r 2 r n ε n exp L 2 r r 2 r n Combining 24 and 25 we get or a sequence o values o tending to ininity that T g r r 2 r n T r r 2 r n n g + ε exp L 2 n ε and n 2 n exp L 2 n 2 n

4 GROWTH PROPERTIES OF COMPOSITE ANALYTIC 349 Since ε > 0 is arbitrary it ollows that T g r r 2 r n lim in r T r r 2 r n vnλl g 26 Also or a sequence o values o 2 and n tending to ininity that T r r 2 r n + ε exp L 2 r r 2 r n Now rom 2 and 27 we obtain or a sequence o values o n tending to ininity that T g r r 2 r n T r r 2 r n n g ε exp L 2 n + ε 27 2 and 2 n exp L 2 n 2 n As ε > 0 is arbitrary we get rom above that T g r r 2 r n r T r r 2 r n λ L vn g Also or all suiciently large values o 2 and n T g r r 2 r n g + ε exp L 2 r r 2 r n Now it ollows rom 25 and 29 or all suiciently large values o n that T g r r 2 r n T r r 2 r n Since ε > 0 is arbitrary we obtain that 28 n g + ε exp L 2 n ε T g r r 2 r n r T r r 2 r n Thus the theorem ollows rom and and 2 n exp L 2 n 2 n ρ L vn g 20 The ollowing theorem can be proved in the line o Theorem 2 and so the proo is omitted

5 350 Sanjib Kumar Datta Tanmay Biswas and Ahsanul Hoque Theorem 22 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g g < and 0 < g vn g < then g g T g r r 2 r n lim in r r 2 r n T g r r 2 r n vnλl g g T g r r 2 r n r r 2 r n T g r r 2 r n vnρl g g Theorem 23 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g < and 0 < < then T g r r 2 r n lim in r r 2 r n T r r 2 r n vnρl g ρ L T g r r 2 r n r r 2 r n T r r 2 r n Proo From the deinition o vn we get or a sequence o values o n tending to ininity that T r r 2 r n ε exp L 2 r r 2 r n Now rom 29 and 2 it ollows or a sequence o values o n tending to ininity that T g r r 2 r n T r r 2 r n As ε > 0 is arbitrary we obtain that n g + ε exp L 2 n ε T g r r 2 r n lim in r T r r 2 r n 2 and 2 2 and 2 n exp L 2 n 2 n ρ L vn g 22 Again or a sequence o values o 2 and n tending to ininity T g r r 2 r n g ε exp L 2 r r 2 r n So combining 22 and 23 we get or a sequence o values o n tending to ininity that T g r r 2 r n T h T r T r r 2 r n g ε + ε n 23 2 and exp L 2 n 2 n exp L 2 n 2 n

6 GROWTH PROPERTIES OF COMPOSITE ANALYTIC 35 Since ε > 0 is arbitrary it ollows that T g r r 2 r n r T r r 2 r n Thus the theorem ollows rom 22 and 24 vnρl g 24 The ollowing theorem can be carried out in the line o Theorem 23 and thereore we omit its proo Theorem 24 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g < and 0 < g < then T g r r 2 r n lim in r r 2 r n T g r r 2 r n ρ L vn g ρg L T g r r 2 r n r r 2 r n T g r r 2 r n The ollowing theorem is a natural consequence o Theorem 2 and Theorem 23 Theorem 25 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g g < and 0 < vn < then T g r r 2 r n lim in r r 2 r n T r r 2 r n max min g g vnρl g ρ L vn g T g r r 2 r n r r 2 r n T r r 2 r n The proo is omitted Anaously one may state the ollowing theorem without its proo Theorem 26 I and g be any two non-constant analytic unctions o n complex variables in the unit polydisc U such that 0 < vn g g < and 0 < g vn g < then T g r r 2 r n lim in r r 2 r n T g r r 2 r n max min g g g g vnρl g g vnρl g g T g r r 2 r n r r 2 r n T g r r 2 r n Reerences [ A K Agarwal: On the properties o an entire unction o two complex variables Canadian JMath Vol pp5-57 [2 S K Datta T Biswas and P Sen: Measure o growth properties o unctions analytic in unit disc International J o Math Sci & Engg Appls IJMSEA Vol 8 No IV July 204 pp [3 B A Fuks: Theory o analytic unctions o several complex variables Moscow 963 [4 O P Juneja and G P Kapoor : Analytic unctions-growth aspects Pitman avanced publishing program 985 [5 C O Kiselman: Plurisubharmonic unctions and potential theory in several complex variables a contribution to the book project Development o Mathematics edited by Jean Paul Pier [6 D Somasundaram and R Thamizharasi : A note on the entire unctions o L-bounded index and L-type Indian J Pure Appl Math Vol9 March 988 No 3 pp

7 352 Sanjib Kumar Datta Tanmay Biswas and Ahsanul Hoque Author inormation Sanjib Kumar Datta Department o Mathematics University o Kalyani PO-Kalyani Dist-Nadia PIN West Bengal India sanjib_kr_datta@yahoocoin Tanmay Biswas Rajbari Rabindrapalli R N Tagore Road PO-Krishnagar Dist-Nadia PIN-740 West Bengal India tanmaybiswas_math@redimailcom Ahsanul Hoque Department o Mathematics University o Kalyani PO-Kalyani Dist-Nadia PIN West Bengal India ahoque033@gmailcom Received: November 206 Accepted:October