On composite entire functions and their central indexes

In this paper, we present a few relations on the composition of two entire functions in terms of their central indexes. Furthermore, we investigate various growth properties of composite entire functions.


Introduction
Let us consider an entire function f (z) = ∞ n=0 a n z n . Then, the maximum modulus, maximum term, and central index of the entire function f (z) are defined as It follows immediately that M f (r ), μ f (r ) and ν f (r ) are real-valued increasing continuous functions of r. The properties of M f (r ), μ f (r ) and ν f (r ) can be found in [4,5], so we do not discuss them in details. For the necessity of our paper, we first recall the following definitions. Recently, Biswas [1] used central index to prove some results of growth of entire functions with respect to L * -order. Being motivated, we investigate some results of growth of composite entire functions using their central indexes. In this paper, we present a few relations on the composition of two entire functions in terms of their central indexes. Furthermore, we investigate various growth properties of composite entire functions.
In 1970, Clunie [2] discussed the behaviour of the ratios log M f (r ) and log M g (r ) for a meromorphic function f (z) and an entire function g (z). Considering the fact that central index is much weaker than the maximum modulus in some sense, we investigate the behaviour of the ratios ν f (r ) and ν g (r ) along with few other conditions for the growth analysis of composite entire functions, where f (z) and g(z) are two entire functions of nonzero finite order.
In 1984, Song and Yang [6] discussed the behaviour of the ratios log log M f •g (r ) log log M f (r ) and log log M f •g (r ) log log M g (r ) , where both f (z) and g(z) are entire. We also study the nature of the ratios log ν g (r ) , where in the first ratio, f (z) is an entire function of nonzero finite order and g(z) is an entire function of nonzero finite lower order, and in the second case, f (z) is an entire function of nonzero finite lower order and g(z) is an entire function of nonzero finite order.
Furthermore, under some suitable conditions, for an entire function f (z) of nonzero finite order and an entire function g(z) of nonzero finite lower order, we investigate the behaviour of the ratio Throughout this paper, we use the notations log 1 r = log r, log i+1 r = log log i r for i = 1, 2, . . . and for sufficiently large r , also exp 1 r = e r , exp i+1 r = exp exp i r for 0 ≤ r < ∞ and i = 1, 2, . . ..

Preliminary lemmas
In this section, we present following known lemmas which are useful to prove the main results. Lemma 2.1 [6] Let λ (g) < ∞. Then, for any ε > 0 and sufficiently large r (2.1)

Lemma 2.3 [3]
Let f (z) = ∞ n=0 a n z n be an entire function. Then (i) for a 0 = 0, Lemma 2.4 [1] Let f (z) be an entire function with nonzero finite order. Also, let g (z) be an entire function with nonzero finite lower order. If 0 < α < λ g , then for all sufficiently large values of r Lemma 2.5 [1] Let f (z) be an entire function with nonzero finite lower order. Also, let g (z) be an entire function with nonzero finite order. If 0 < α < λ g , then for all sufficiently large values of r

Main results
In this section, we prove our main results.
Proof For an entire function f (z) = ∞ n=0 a n z n , let us assume that |a 0 | = 0. Now, from (i) of Lemma 2.3, we have that is Again, using Cauchy inequality, we have Hence, using (3.1) and (3.2), we get The above implies ν f (r ) ≤ c 2 log M f (2r ) + c 3 , for some suitable constants c 2 and c 3 .
In view of the above, we get for all sufficiently large values of r Again, for any given ε > 0 and for all sufficiently large values of r, by Definition 1.2, for the entire function f (z) in terms of central index, we have If there exists a constant β, such that β > ρ g , then (3.4) implies From (3.3) and (3.5), we get Using (3.7) in (3.6), we thus have for sufficiently large r and for any ε > 0 As it is given β > ρ g , we can choose ε > 0, such that β > ρ g + ε.
Using this condition in (3.8), we get that (3.9) Therefore, by (3.9), for sufficiently large values of r and for a small constant A (> 0) , we obtain which implies that This completes the proof.

Corollary 3.2 Let f (z) and g (z)
be two entire functions with nonzero finite order. If ρ g < β, then for all sufficiently large values of r Definition 3. 3 We introduce a new function ζ f (r ) for an entire function f (z), which is defined as Theorem 3.4 Let f (z) be an entire function of nonzero finite order and g(z) be an entire function of nonzero finite lower order. Furthermore, let lim sup r →∞ ζ f (r ) = ∞ with the condition 0 < α < λ g , and then Proof For any ε > 0 and for sufficiently large r Using Lemma 2.4, the above definition implies since exp (r α ) is increasing, continuous, and unbounded in r for α > 0.

Theorem 3.5 Suppose f (z) and g (z) are two transcendental entire functions of nonzero finite order with the condition
Proof For any ε > 0 and for sufficiently large r, from η = lim sup r →∞ ζ f (r ) , we obtain Now, by definition of order of ρ f •g and using Theorem 3.1, we get Since ε > 0 is arbitrary, therefore, we get In view of the Theorem 3.5, one can prove the following theorem; hence, we omit the proof. If η < ∞, then λ f •g ≤ ηβ.

Theorem 3.7 Suppose f (z) is an entire function of nonzero finite order and g(z)
is an entire function of nonzero finite lower order with the condition 0 < α < λ g . Furthermore, let Proof By definition of order of ρ f •g and using Lemma 2.4, we get Hence, the result follows. In view of Theorems 3.5 and 3.7, the following result can be proved. Theorem 3.9 Let f (z) and g (z) be two transcendental entire functions of nonzero finite order. Suppose there exist two constants α and β satisfying 0 < α < λ g ≤ ρ g < β. Furthermore, let Theorem 3.10 Let f (z) be an entire function of nonzero finite order and g(z) be an entire function of nonzero finite lower order. If there exists a positive constant α with the condition 0 < α < λ g , then Proof For any ε > 0 and for sufficiently large r, the lower order of composition of two entire functions in terms of central index is which implies that there exists an α, such that 0 < α < λ g , and by Lemma 2.4, we have Again, by Definition 1.2, for any ε > 0 and for sufficiently large r, we have Using (3.11) in (3.10), we thus have which implies that λ f •g = ∞, since α > 0 and r α log r → ∞ as r → ∞. Therefore, the theorem is proved. Proof For any ε > 0 and for sufficiently large r, we have by given condition (3.12) If there exists a constant α, such that 0 < α < λ g , then by (3.12) Now, for any chosen ε > 0 and for sufficiently large r, by definition of the order of λ f •g , we get (Using (3.13)) . (3.14) Since for chosen ε > 0, a − ε > 0 and αα 1 > 0, hence r αα 1 log r → ∞ as r → ∞. Therefore, by (3.14), we have λ f •g = ∞ and the theorem is proved. In view of Theorem 3.11, one can easily prove the following theorem. Theorem 3.14 Let f (z) be an entire function of nonzero finite order and g(z) be an entire function of nonzero finite lower order. Furthermore, let for any positive number β 1 . Then, λ f •g = ∞.
Proof For any chosen ε > 0 and for sufficiently large r, we have from the given condition If there exists a constant α, such that 0 < α < λ g , then by (3.15) By definition of the order of λ f •g , we get for sufficiently large r (By using (3.16)) .
The following theorem can be proved in view of Theorem 3.14.

Theorem 3.15 Let f (z) be an entire function of nonzero finite lower order and g(z) be an entire function of nonzero finite order. Furthermore, let
for any positive number β 1 . Then, λ f •g = ∞.

Theorem 3.16 Let f (z) be an entire function of nonzero finite lower order and g(z) be an entire function of nonzero finite order. Furthermore, let
for any positive integer k ≥ 1 and α 1 is any positive number. Then, λ f •g = ∞.
Proof For any ε > 0 and for sufficiently large r, by given condition, we have If there exists a constant α, such that 0 < α < λ g , then by (3.17) Now, for any chosen ε > 0 and for sufficiently large r, by definition of the order of λ f •g , we get log ν g (exp (r α )) log r , (By using Lemma 2.5) (Using (3.18)) . Since for chosen ε > 0, a − ε > 0 and αα 1 > 0, hence log r → ∞ as r → ∞. Therefore, by (3.19), we have λ f •g = ∞ and the theorem is proved.

Remark 3.17 If f (z) is an entire function of nonzero finite lower order and g(z)
is an entire function of nonzero finite order. Then, assuming lim sup r →∞ log k ν g (r ) for any positive integer k ≥ 1 and α 1 > 0, we can get ρ f •g = ∞.
In view of Theorem 3.16, one can easily prove the following theorem, and hence, its proof is omitted. Proof For any chosen ε > 0 and for sufficiently large r, we have from the given condition If there exists a constant α, such that 0 < α < λ g , then by (3.20), we have By definition of the order of λ f •g , we get for sufficiently large r (By using (3.21)) .

Since for chosen
Therefore, λ f •g = ∞ and the theorem is proved.
The following theorem can be proved in view of Theorem 3.19, and hence, its proof is omitted.

Theorem 3.21 Let f (z) be an entire function of nonzero finite lower order and g(z) be an entire function of nonzero finite order. Furthermore, let
for any positive integer k ≥ 1 and β 1 is any positive number. Then, λ f •g = ∞.
Proof Case (I ) : In view of Theorem 3.16, first part of the theorem can easily be proved. Case (I I ) : Using the same method as in (3.18) , for sufficiently large r and for any chosen ε > 0, we get from condition (I I ) By definition of the lower order of λ f •g , we get for any chosen ε > 0 and for sufficiently large r log ν g (exp (r α )) log r , (By using Lemma 2.5) (By using (3.22)) . (3.23) Since for chosen ε > 0, a 2 − ε > 0 and α 2 − 1 > 0, hence log r → ∞ as r → ∞. Therefore, by (3.23), we have λ f •g = ∞ and second part of the theorem is proved.
The following theorem can be proved in view of Theorem 3.22, and hence, its proof is omitted.
Proof Case (I ) : In view of Theorem 3.19, first part of the theorem can easily be proved. Case (I I ) : Using the same method as in (3.20) , for sufficiently large r and for any chosen ε > 0, we get from condition (I I ) If there exists a constant α such that 0 < α < λ g , then by (3.24), we have By definition of the lower order of λ f •g , we get for sufficiently large r (By using (3.25)) .

Since for some chosen
Therefore, λ f •g = ∞ and the second part of the theorem is proved.

Remark 3.25
If f (z) be an entire function of nonzero finite order and g(z) be an entire function of nonzero finite lower order. Then, ρ f •g = ∞, if we consider one of the following two conditions: for any two positive numbers β 1 , β 2 (> 1) and k a positive integer.
In view of Theorem 3.24, one can easily prove the following theorem.
log k r for any two positive numbers β 1 , β 2 (> 1) and k a positive integer.
Next, two theorems are about ratios ν f (r ) and ν g (r ) . Theorem 3.27 Let f (z) and g(z) be two entire functions of nonzero finite order along with the condition λ g ≥ ρ f . Then Proof By Definition 1.2, for any ε > 0 and for sufficiently large r, we get log ν f (r ) < ρ f + ε log r. (3.32) Again, if there exists an α, such that 0 < α < λ g , then by (3.31) , we have In view of (3.32) and (3.33) , we obtain Since α > 0, the ratio r α log r → ∞ as r → ∞. Hence, the theorem follows. Example 3.34 For two nonzero entire functions f (z) = cosh (z) and g(z) = exp(az) (a = 0) , we have ρ f = λ f = 1 and λ g = 1.
Therefore, using Definition 1.2, we obtain the ratio log ν f (r ) > r 1 2 log r and the right-hand side tends to ∞ as r → ∞. Hence

Theorem 3.35 Let f (z) be an entire function of nonzero finite lower order and g(z) be an entire function of nonzero finite order. Then
Proof By Definition 1.2, for any ε > 0 and for sufficiently large r, we get log ν g (r ) < ρ g + ε log r. (3.34) Again, if there exists an α, such that 0 < α < λ g , then by Lemma 2.5, we have By Definition 1.2 In view of (3.34) and (3.35) , we obtain Since α > 0, the ratio r α log r → ∞ as r → ∞. Hence, the theorem is proved. Example 3.36 For two nonzero entire functions f (z) = cos a √ z (a = 0) and g(z) = sin z, we have ρ f = λ f = 1 2 and ρ g = λ g = 1. Since, 0 < α < λ g , take α = 1 3 , so that for f • g(z) = cos a √ sin z using Lemma 2.5, we get log ν f •g (r ) > r Therefore, using Definition 1.2, we obtain the ratio log ν f •g (r ) log ν g (r ) > r 1 3 log r and the right-hand side tends to ∞ as r → ∞. Hence lim r →∞ log ν f •g (r ) log ν g (r ) = ∞.

Conclusion
We have proved several results of growth of composite entire functions in terms of their central indexes. There is huge scope of further study on the growth of entire functions by different growth measures in terms of their central indexes. The growth results of entire functions with respect to their generalized order may be investigated using their central indexes. Also, the relative order of an entire function with respect to another entire function may be defined alternatively in terms of their central indexes and the corresponding results may be investigated. Once we get these, we may implement them to the composite entire functions. Furthermore, some of these results may be applied to complex linear differential equations.
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