Growth Rates of Meromorphic Functions Focusing Relative Order

A detailed study concerning some growth rates of composite entire and meromorphic functions on the basis of their relative orders (relative lower orders) with respect to entire functions has been made in this paper.

To start our paper we just recall the following definitions.
Definition 1.The order   (the lower order   ) of an entire function  is defined as log [2]   () log  (  = lim inf  → ∞ log [2]   () log  ) . ( If  is a meromorphic function, one can easily verify that where   () is the Nevanlinna's characteristic function of  (cf.[1]).
Similarly, one can define the relative lower order of an entire function  with respect to another entire function  denoted by   () as follows: Extending this notion, Lahiri and Banerjee [6] introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way.

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Chinese Journal of Mathematics Definition 2 (see [6]).Let  be any meromorphic function and let  be any entire function.The relative order of  with respect to  is defined as Likewise, one can define the relative lower order of a meromorphic function  with respect to an entire function  denoted by   () as follows: It is known (cf.[6]) that if () = exp , then Definition 2 coincides with the classical definition of the order of a meromorphic function .
In this paper we wish to prove some results related to the growth rates of composite entire and meromorphic functions on the basis of relative order (relative lower order) of meromorphic functions with respect to an entire function.

Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 1 (see [7]).Let  be meromorphic and let  be entire and suppose that 0 <  <   ≤ ∞.Then for a sequence of values of  tending to infinity, Lemma 2 (see [8]).Let  be meromorphic and let  be entire such that 0 <   < ∞ and 0 <   .Then for a sequence of values of  tending to infinity, where 0 <  <   .
Lemma 3 (see [9]).Let  be a meromorphic function and let  be an entire function such that   <  < ∞ and 0 <   ≤   < ∞.Then for a sequence of values of  tending to infinity, Lemma 4 (see [9]).Let  be a meromorphic function of finite order and let  be an entire function such that 0 <   <  < ∞.

Theorems
In this section we present the main results of the paper.
Theorem 1.Let  be a meromorphic function and let , ℎ be any two entire functions such that for any , ,  satisfying 0 <  < 1,  > 0, ( + 1) > 1, and Proof.From (i) we have for all sufficiently large values of  that and from (ii) we obtain for all sufficiently large values of  that Also  −1 ℎ () is an increasing function of  it follows from ( 13) and ( 14) and Lemma 1 for a sequence of values of  tending to infinity that that is, that is, that is, Since  (> 0) is arbitrary and  ( + 1) > 1, it follows from above that This proves the theorem.
In the line of Theorem 1 and with the help of Lemma 2 one may state the following theorem without its proof.Theorem 2. Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 <   < ∞ and 0 <   .Further suppose that for any , ,  satisfying 0 <  < 1,  > 0, ( + 1) > 1 and 0 <  <   .Then Theorem 3. Let  be a meromorphic function and let , ℎ be any two entire functions such that for any ,  satisfying  > 1, 0 <  < 1,  > 1 and 0 <  <   ≤ ∞.Then Proof.From (i) we have for all sufficiently large values of  that log  −1 ℎ (exp (  )) ≥ (( − ) log [2] ) and from (ii) we obtain for all sufficiently large values of  that As  −1 ℎ () is an increasing function of , it follows from (26) and ( 27) and Lemma 1 for a sequence of values of  tending to infinity that [2] ) −1 log [2] ] ⋅ ( − ) (log [2] ) Since  (> 0) is arbitrary and  > 1,  > 1, the theorem follows from above.Theorem 4. Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 <   < ∞ and 0 <   .Further suppose that for any ,  satisfying  > 1, 0 <  < 1,  > 1, and 0 <  <   .Then We omit the proof of Theorem 4 as it can be carried out in the line of Theorem 3 and with the help of Lemma 2.
Theorem 5. Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 <   ≤ ∞ and  ℎ () > 0.
As  −1 ℎ () is an increasing function of , we get from Lemma 1 for a sequence of values of  tending to infinity that Thus the theorem follows.
In the line of Theorem 5, one can easily prove the following theorem.Theorem 6.Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 < min{  ,   } and  ℎ () > 0. Then Theorem 7. Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 <   ≤ ∞ and  ℎ () > 0. Then Proof.In view of Theorem 5, we obtain that lim sup Thus the theorem follows.
Theorem 8. Let  be a meromorphic function and let , ℎ be any two entire functions such that 0 < min{  ,   } and  ℎ () > 0. Then The proof of Theorem 8 is omitted as it can be carried out in the line of Theorem 7 and in view of Theorem 6. Theorem 9. Let  be a meromorphic function and let ℎ be an entire functions such that 0 <  ℎ () ≤  ℎ () < ∞.Also let  be an entire function with nonzero order.Then for every positive constant  and every real number , Proof.If  is such that 1 +  ≤ 0, then the theorem is trivial.So we suppose that 1 +  > 0.
Since  −1 ℎ () is an increasing function of , we get from Lemma 1 for a sequence of values of  tending to infinity that where we choose 0 <  <   ≤ ∞.
Again from the definition of  ℎ (), it follows for all sufficiently large values of  that Now from ( 38) and (39), it follows for a sequence of values of  tending to infinity that Since   /(log ) 1+ → ∞ as  → ∞, the theorem follows from above.
In the line of Theorem 9 and with the help of Lemma 2, one may state the following theorem without its proof.
Theorem 10.Let  be a meromorphic function with nonzero finite lower order and let  be an entire function with nonzero finite order.Also let ℎ be an entire function such that  ℎ () < ∞ and  ℎ () > 0. Then for every positive constant  and every real number , Theorem 11.Let  be a meromorphic function and let  be an entire function with nonzero order.Also let ℎ be an entire function such that 0 <  ℎ () and  ℎ () < ∞.Then for every positive constant  and every real number , Theorem 12. Let  be a meromorphic function with nonzero finite lower order and let  be an entire function with nonzero finite order.Also let ℎ be an entire function such that 0 <  ℎ () ≤  ℎ () < ∞.Then for every positive constant  and every real number , We omit the proof of Theorems 11 and 12 as those can be carried out in the line of Theorems 9 and 10, respectively.Theorem 13.Let  be a meromorphic function with nonzero finite order and lower order.Also let , ℎ be any two entire functions such that 0 <  ℎ () ≤  ℎ () < ∞.Then for every positive constant  and each  ∈ (−∞, ∞), Proof.If 1 +  ≤ 0, then the theorem is obvious.We consider The proof is omitted as it can be carried out in the line of Theorem 13.Theorem 15.Let  be a meromorphic function with finite order and let  be an entire function with nonzero finite lower order.Also let ℎ be another entire function such that  ℎ () > 0 and  ℎ () < ∞.Then for every positive constant  and each  ∈ (−∞, ∞),