Comparative growth analysis of Wronskians in the light of their relative orders

Abstract In this paper we study the comparative growth properties of a composition of entire and meromorphic functions on the basis of the relative order (relative lower order) of Wronskians generated by entire and meromorphic functions.


Introduction, definitions and notations
Let C be the set of all finite complex numbers. Also let f be a meromorphic function and g be an entire function defined in C. The maximum modulus function relating to entire g is defined as M g (r) = max{|g(z)| : |z| = r}. For a meromorphic function f , M f (r) cannot be identified as f is not analytic. In this case one may characterize another function T f (r) known as Nevanlinna's characteristic function of f , playing the same role as the maximum modulus function in the following way where the function N f (r, a) resp. N f (r, a), known as counting function of a-points (distinct a-points) of meromorphic f is defined as follows  In addition, we represent by n f (r, a) (n f (r, a)) the number of a-points (distinct a-points) of f in |z| ≤ r and an ∞-point is a pole of f . In many occasions N f (r, ∞) and N f (r, ∞) are symbolized by N f (r) and N f (r), respectively. [136]

S.K. Datta, T. Biswas and A. Hoque
On the other hand, the function m f (r, ∞) alternatively indicated by m f (r), known as the proximity function of f , is defined as m f (r) = 1 2π 2π 0 log + |f (re iθ )| dθ, where log + x = max(log x, 0) for all x 0.
Also we may imply m(r, 1 f −a ) by m f (r, a). If f is entire, then the Nevanlinna's characteristic function T f (r) of f is defined as Moreover, M f (r) and T f (r) are both strictly increasing and continuous functions of r when the entire function f is non-constant. Also their inverses In this connection we immediately remind the following definition which is relevant.

Definition 1.1 ([2])
A non-constant entire function f is said have the Property (A) if for any σ > 1 and for all sufficiently large r, For the examples of functions with or without the Property (A), one may see [2].
However, in the case of any two entire functions f and g, the ratio as r → ∞ is illustrated as the growth of f with respect to g in terms of their maximum moduli. Analogously, while f and g are both meromorphic functions, the ratio Tg(r) as r → ∞ is illustrated as the growth of f with respect to g in terms of their Nevanlinna's characteristic functions. Also the concept of the growth measuring tools such as order and lower order which are conventional in complex analysis and the growth of entire or meromorphic functions can be studied in terms of their orders and lower orders -normally defined in terms of their growths with respect to the exp function which are shown in the following definition.

Definition 1.2
The order ρ f (resp. the lower order λ f ) of an entire function f is defined as resp.
When f is meromorphic, one may easily prove that Comparative growth analysis of Wronskians in the light of their relative orders resp. (1) .
Both entire and meromorphic functions have the regular growth if their order coincides with their lower orders.
Bernal [1,2] initiated the idea of the relative order of an entire function f with respect to another entire function g, symbolized by ρ g (f ) to keep away from comparing growth just with exp z which is as follows The definition agrees with the classical one [10] if g(z) = exp z. Likewise, one may define the relative lower order of an entire function f with respect to another entire function g symbolized by λ g (f ) in the following way Widening this notion, Lahiri and Banerjee [9] established the definition of the relative order of a meromorphic function with respect to an entire function which is as follows.

Definition 1.3 ([9])
Let f be any meromorphic function and g be any entire function. The relative order of f with respect to g is defined as Similarly, one may define the relative lower order of a meromorphic function f with respect to an entire function g in the following way It is known (cf. [9]) that if g(z) = exp z, then Definition 1.3 coincides with the classical definition of the order of a meromorphic function f .
The following definitions are also well known. [138] S.K. Datta, T. Biswas and A. Hoque Let a 1 , a 2 , . . . , a k be linearly independent meromorphic functions and small with respect to f . We denote by .
is called the Nevanlinna's deficiency of the value a.
From the second fundamental theorem, it follows that the set of values of we say that f has the maximum deficiency sum.
In this paper we wish to prove some newly developed results based on the growth properties of the relative order and the relative lower order of Wronskians generated by entire and meromorphic functions. We do not explain the standard definitions and notations in the theory of entire and meromorphic functions as those are available in [7] and [11].

Lemmas
In this section we present some lemmas which will be needed in the sequel.

Lemma 2.1 ([3])
Let f be meromorphic and g be entire, then for all sufficiently large values of r ).

Lemma 2.2 ([4])
Let f be meromorphic and g be entire and suppose that 0 < µ < ρ g ≤ ∞. Then for a sequence of values of r tending to infinity

Lemma 2.3 ([8])
Let f be meromorphic and g be entire such that 0 < ρ g < ∞ and 0 < λ f . Then for a sequence of values of r tending to infinity where 0 < µ < ρ g .

Lemma 2.5 ([5])
If f be a transcendental meromorphic function with the maximum deficiency sum and g be a transcendental entire function of regular growth having non zero finite order and a =∞ δ(a; g) + δ(∞; g) = 2, then the relative order and relative lower order of L(f ) with respect to L(g) are same as those of f with respect to g, i.e.

Theorems
In this section we present the main results of the paper.
Since T −1 h (r) is an increasing function of r, it follows from Lemma 2.1, Lemma 2.4 and the inequality T g (r) ≤ log M g (r) (cf. [7]) for all sufficiently large values of r that i.e.
Again, for all sufficiently large values of r, we get in view of Lemma 2.5 that [140]
In view of Theorem 3.1, the following theorem can be carried out.
The proof is omitted.

Remark 3.4
If we take in Theorem 3.3 the condition ρ h (g) > 0 instead of λ h (g) > 0, the theorem remains true with "limit" replaced by "limit inferior".

Theorem 3.5
Let f be a transcendental meromorphic function with a =∞ δ(a; f ) + δ(∞; f ) = 2 and g be an entire function with λ g < µ < ∞. Also let h be any transcendental entire function of the regular growth having non zero finite order with the maximum deficiency and satisfies Property (A) and Then for a sequence of values of r tending to infinity Proof. Let us consider δ > 1. Since T −1 h (r) is an increasing function of r, it follows from (1) that for a sequence of values of r tending to infinity Comparative growth analysis of Wronskians in the light of their relative orders

[141]
Now, (3) and (5), for a sequence of values of r tending to infinity, yield As λ g < µ, we can choose ε (> 0) in such a way that Thus, from (6) and (7), we obtain that From (8), we obtain for a sequence of values of r tending to infinity and also for . Thus the theorem follows.
In the line of Theorem 3.5, we may state the following result without its proof.

Theorem 3.6
Let g be any transcendental entire function with a =∞ δ(a; g) + δ(∞; g) = 2, λ g < µ < ∞ and h be any transcendental entire function of the regular growth having non zero finite order with the maximum deficiency sum and satisfy the Property (A). Let moreover, f be a meromorphic function with finite relative order with respect to h. Then for a sequence of values of r tending to infinity when λ h (g) > 0.

Theorem 3.7
Let f be a meromorphic function and h, g be any two transcendental entire functions with a =∞ δ(a; h) + δ(∞; h) = 2, a =∞ δ(a; g) + δ(∞; g) = 2, λ h (f ) > 0 and 0 < ρ h (g) < ∞. If h is of the regular growth having non zero finite order, then Proof. Let 0 < µ < µ < ρ g . As T −1 h (r) is an increasing function of r, it follows from Lemma 2.2 for a sequence of values of r tending to infinity that [142]

S.K. Datta, T. Biswas and A. Hoque
Again for all sufficiently large values of r we get in view of Lemma 2.5 that Therefore combining (9) and (10), we obtain for a sequence of values of r tending to infinity that log Since µ < µ , the theorem follows from (11).

Corollary 3.8
Under the assumptions of Theorem 3.7, Proof. In view of Theorem 3.7, we get for a sequence of values of r tending to infinity that (exp r µ ) from which the corollary follows.
Similarly one may state the following theorem and corollary without their proofs as those can be carried out in the line of Theorem 3.7 and Corollary 3.8, respectively.

Theorem 3.9
Let f be a transcendental meromorphic function with the maximum deficiency sum and h be an transcendental entire function of regular growth having non zero finite order with a =∞ δ(a; )h + δ(∞; h) = 2. If h satisfies 0 < λ h (f ) ≤ ρ h (f ) < ∞, then for any entire function g lim sup where 0 < µ < ρ g .
Comparative growth analysis of Wronskians in the light of their relative orders

[143]
Corollary 3.10 Under the assumptions of Theorem 3.9, As an application of Theorem 3.5 and Corollary 3.10, we may state the following result.

Theorem 3.11
Let f be a transcendental meromorphic function with a =∞ δ(a; f ) + δ(∞; f ) = 2 and g be an entire function with λ g < µ < ρ g . Let moreover, h be any transcendental entire function of the regular growth having non zero finite order with the maximum deficiency and satisfying the Property (A) and let .
The proof is omitted. Similarly, in view of Theorem 3.6 and Corollary 3.8, the following theorem can be carried out.

Theorem 3.12
Let h be any transcendental entire function of the regular growth having non zero finite order with the maximum deficiency sum and satisfying the Property (A) and let g be any transcendental entire function with a =∞ δ(a; g) + δ(∞; g) = 2, 0 < λ g < µ < ρ g < ∞ and 0 < λ h (g) ≤ ρ h (g) < ∞. Moreover, let f be a meromorphic function with .
The proof is omitted.
Proof. Let 0 < µ < ρ g . As T −1 h (r) is an increasing function of r, it follows from (9), for a sequence of values of r tending to infinity that log [2] [144]

S.K. Datta, T. Biswas and A. Hoque
Hence, for a sequence of values of r tending to infinity, we get Again in view of Lemma 2.5, we have for all sufficiently large values of r that log [2] T −1 Now combining (12) with (13) we obtain for a sequence of values of r tending to infinity that log [2] T −1 h T f •g (exp r B ) log [2] which completes the proof.
In view of Theorem 3.13, we can state the following result. where 0 < µ < ρ g and B > 0.
The proof is omitted. Proof. Since T −1 h (r) is an increasing function of r, we get from Lemma 2.2 for a sequence of values of r tending to infinity that log T −1 h T f •g (r) ≥ log T −1 h T f (exp(r µ )),