Sum and product theorems depending on the (p, q)-th order and (p, q)-th type of entire functions

The main object of the present paper is to obtain new estimates involving the -th order and the -th type of entire functions under some suitable conditions. Some open questions, which emerge naturally from this investigation, are also indicated as a further scope of study for the interested future researchers in this branch of Complex Analysis.


Introduction
A single-valued function of one complex variable, which is analytic in the finite complex plane, is called an entire (integral) function. For example, exp(z), sin z, cos z, and so on, are all entire functions. In the value distribution theory, one studies how an entire function assumes some values and the influence of assuming certain values in some specific manner on a function. In 1926, Rolf Nevanlinna initiated the value distribution theory of entire functions. This value distribution theory is a prominent branch of Complex Analysis and is the prime concern of this paper. Perhaps the Fundamental Theorem of Classical Algebra, which may be stated as follows: If P(z) is a non-constant polynomial in z with real or complex coefficients, then the equation P(z) = 0 has at least one root ABOUT THE AUTHOR Ever since the early 1960s, the first-named author of this paper has been engaged in researches in many different areas of Pure and Applied Mathematics. Some of the key areas of his current research and publication activities include (e.g.) Real and Complex Analysis, Fractional Calculus and Its Applications, Integral Equations and Transforms, Higher Transcendental Functions and Their Applications, q-Series and q-Polynomials, Analytic Number Theory, Analytic and Geometric Inequalities, Probability and Statistics and Inventory Modelling and Optimization. This paper, dealing essentially with the order of growth and the type of entire (or integral) functions, is a step in the ongoing investigations in the value distribution theory (initiated by Rolf Nevanlinna in 1926), which happens to be a prominent branch of Complex Analysis.

PUBLIC INTEREST STATEMENT
The theory of Entire Functions (which are known also as Integral Functions) is potentially useful in a wide variety of areas in Pure and Applied Mathematical, Physical and Statistical Sciences. Indeed, a single-valued function of one complex variable, which is analytic in the finite complex plane, is called an entire (or integral) function. For example, some of the commonly used entire functions include such elementary functions as exp(z), sin z, cos z, and so on. In the value distribution theory, one studies how an entire function assumes some values and the influence of assuming certain values in some specific manner on a function. This investigation is motivated essentially by the fact that the determination of the order of growth and the type of entire functions is rather important with a view to studying the basic properties of the value distribution theory.
is the most well-known value distribution theorem. The value distribution theory deals with the various aspects of the behaviour of entire functions, one of which is the study of comparative growth properties of entire functions. For any entire function f (z) given by we define the maximum modulus M f (r) of f (z) on |z| = r as a function of r by In this connection, for all sufficiently large values of r, we recall the following well-known inequalities relating the maximum moduli of any two entire functions f i (z) and f j (z): and On the other hand, if we consider z r to be a point on the circle |z| = r, we find for all sufficiently large values of r that which implies that In terms of the maximum modulus M f (r) of the function f (z), the order f of the entire function f (z), which is generally useful for computational purposes, is defined by Moreover, with a view to determining (e.g.) the relative growth of two entire functions with the same positive order, the type f of the entire function f (z) of order f 0 < f < ∞ is defined by The determination of the order of growth and the type of entire functions is rather important in order to study the basic properties of the value distribution theory. In this regard, several researchers made extensive investigations on this subject and presented the following useful results.
Theorem 1 (see Holland, 1973) Let f (z) and g(z) be any two entire functions of orders f and g , respectively. Then (1.8) f +g = g when f < g Theorem 2 (see Levin, 1996) Let f (z) and be any two entire functions with orders f and g , respectively. Then and By appropriately extending the notion of addition and multiplication theorems as introduced by Holland (1973) and Levin (1996), our main object in this paper is to give the corresponding extensions of Theorem A and Theorem B. In our present investigation, we make use of index-pairs and the concept of the (p, q)-th order of entire functions for any two positive integers p and q with p ≧ q, which are introduced in Section 2. For the the standard definitions, notations and conventions used in the theory of entire functions, the reader may refer to (e.g. Boas, 1957;Valiron, 1949). Several closely-related recent works on the subject of our present investigation include (e.g. Choi, Datta, Biswas, & Sen, 2015;Datta, Biswas, & Biswas, 2013.

Definitions, notations, and preliminaries
Let f (z) be an entire function defined in the complex z-plane ℂ. Also let M f (r) denote the maximum modulus of f (z) for |z| = r (0 < r < ∞) as defined by (1.2). In our investigation, we use the following definitions, notations, and conventions: and Sato (1963) introduced a more general concept of the order and the type of an entire function than those given by (1.8) and (1.9).
Definition 1 (see Sato, 1963) Let l ∈ ℕ ⧵ {1}. The generalized order [l] f of an entire function f (z) is defined by Definition 2 (see Sato, 1963) Let l ∈ ℕ ⧵ {1}. The generalized type [l] f of an entire function f (z) of the generalized order [l] f is defined by Remark 1 When l = 2, Definitions 1 and 2 coincide with the Equations (1.8) and (1.9), respectively.
More recently, a further generalized concept of the (p, q)-th order and the (p, q)-th type of an entire function f (z) was introduced by Juneja et al. (see Juneja, Kapoor, & Bajpai, 1976, 1977 as follows. (2.1) [l] (2.2) [l] Definition 3 (see Juneja et al., 1976) Let p, q ∈ ℕ (p ≧ q). The (p, q)-th order f (p, q) of an entire function f (z) is defined by Definition 4 (see Juneja et al., 1977) where the parameter b is given by Remark 2 By comparing Definitions 3 and 4 with Definitions 1 and 2, respectively, it is easily observed that See also Remark 1 above.
Next, in connection with the above developments, we also recall the following definition.
Definition 5 (see Juneja et al., 1976) An entire function f (z) is said to have the index-pair (p, q) is not a nonzero finite number, where b is given by (2.5). Moreover, if then and The following proposition will be needed in our investigation.
Proposition 1 Let f i (z) and f j (z) be any two entire functions with the index-pairs p i , q i and p j , q j , respectively. Then the following conditions may occur: The following definition will also be useful in our investigation.

A set of Lemmas
Here, in this section, we present three lemmas which will be needed in the sequel.

Then, for any integer n ∈ ℕ and for all sufficiently large values of r,
Lemma 3 (see Levin, 1980, p. 21) Let the function f (z) be holomorphic in the circle |z| = 2eR (R > 0) with f (0) = 1. Also let be an arbitrary positive number not exceeding 3e 2 . Then, inside the circle |z| = R, but outside of a family of excluded circles, the sum of whose radii is not greater than 4 R,

Main results
In this section, we state and prove the main results of this paper.
Theorem 3 Let f i (z) and f j (z) be any two entire functions with index-pairs p i , q i and p j , q j , respectively, where p i , p j , q i , q j ∈ ℕ are constrained by Then where Equality in (4.1) holds true when any one of the first four conditions of the Proposition in Section 2 are satisfied for i ≠ j.
Proof For the result (4.1) is obvious, so we suppose that Clearly, we can also assume that f k p k , q k is finite for k = i, j.
Now, for any arbitrary > 0, from Definition 3 for the p k , q k -th order, we find for all sufficiently large values of r that that is, so that Therefore, in view of (4.4), we deduce from (1.3) for all sufficiently large values of r that Thus, by applying Lemma 1(a), we find from (4.5) for all sufficiently large values of r that T( ) = 2 + log 3e 2 .
p i ≧ q i and p j ≧ q j . (4.1) (4.5) that is, that is, Therefore, we have Since > 0 is arbitrary, it follows that We now let any one of first four conditions of the Proposition in Section 2 be satisfied for i ≠ j (i, j = 1, 2). Then, since > 0 is arbitrary, from Definition 3 for the p k , q k -th order, we find for a sequence of values of r tending to infinity that Therefore, in view of the first four conditions of the Proposition in Section 2, we obtain for a sequence of values of r tending to infinity that We next consider the following expression: By virtue of the first four conditions of the Proposition of Section 2 and Lemma 1(b), we find from (4.9) that Now, clearly, (4.10) can also be written as follows: where but all of the equalities do not hold true simultaneously. So, from (4.11), we find for all sufficiently large values of r that Thus, from (4.2), (4.8) and (4.12), we deduce for a sequence of values of r tending to infinity that that is, Therefore, from (4.8) and (4.13), and in view of Lemma 1(a) and (1.4), it follows for a sequence of values of r tending to infinity that that is, that is, that is, so that which, for a sequence of values of r tending to infinity, yields that is, Clearly, therefore, the conclusion of the second part of Theorem 1 follows from (4.6) and (4.14). □

Remark 4
That the inequality sign in Theorem 1 cannot be removed is evident from Example 1 below.
Example 1 Given any two natural numbers l and m, the functions have their maximum moduli given by respectively. Therefore, the following expressions: are both constants for each k ∈ ℕ ⧵ {1}. Thus, obviously, it follows that but Consequently, we have Theorem 4 Let f i (z) and f j (z) be any two entire functions with index-pairs p i , q i and p j , q j , respectively, where p i , p j , q i , q j ∈ ℕ are constrained by Suppose also that f i p i , q i and f j p j , q j are both non-zero and finite. Then, for (4.14) [k] p i ≧ q i and p j ≧ q j . p = max p i , p j and q = min q i , q j ,

provided that any one of the first four conditions of the Proposition of Section 2 is satisfied for i ≠ j.
Proof First of all, suppose that any one of the first four conditions of the Proposition of Section 2 is satisfied for i ≠ j. Also let > 0 and 1 > 0 be chosen arbitrarily. Then, from Definition 4 for the p k , q k -type, we find for all sufficiently large values of r that Moreover, for a sequence of values of r tending to infinity, we obtain Therefore, from (1.3) and (4.16), we get for all sufficiently large values of r that Now, in light of any one of the first four conditions of the Proposition in Section 2, for all sufficiently large values of r, we can make the factor: which occurs on the right-hand side of (4.18), as small as possible. Hence, for any > 1 + 1 , it follows from Lemma 1 (a) and (4.18) that that is, so that for all sufficiently large values of r. Thus, by using (4.19), we find for all sufficiently large values of r that Therefore, in view of Theorem 1, it follows from (4.20) that, for all sufficiently large values of r, (4.18) that is, Hence, upon letting → 1+ in (4.21), we find for all sufficiently large values of r that that is, Again, from (1.4), (4.16) and (4.17), we see for a sequence of values of r tending to infinity that Now, by virtue of any one of the first four conditions of the Proposition in Section 2, for all sufficiently large values of r, we can make the factor: which occurs on the right-hand side of (4.23), as small as possible. Hence, for any constrained by it follows from Lemma 1(a) and (4.23) that, for a sequence of values of r tending to infinity, that is, so that Therefore, by using (4.24), it follows for a sequence of values of r tending to infinity that (4.21) which, in the limit when → 1+, yields Thus, in view of Theorem 1, we find from (4.25) that that is, Theorem 2 now follows from (4.22) and (4.26). □ Our next result (Theorem 3) provides the condition under which the equality sign in the assertion (4.1) of Theorem 1 holds true in the case of the condition (v) of the Proposition of Section 2.
Theorem 5 Let f 1 (z) and f 2 (z) be any two entire functions such that and Then Proof Under the hypotheses of Theorem 3, if we apply Theorem 1, it is easily seen that Let us consider the case when Then, in view of Theorem 2, we find that which is a contradiction. Consequently, the assertion (4.27) of Theorem 3 holds true. □ Theorem 6 Let f i (z) and f j (z) be any two entire functions with the index-pairs p i , q i and p j , q j , respectively, for p i , p j , q i , q j ∈ ℕ such that (4.27) Then where Equality in (4.28) holds true when any one of the first four conditions of the Proposition of Section 2 is satisfied for i ≠ j. Furthermore, a similar relation holds true for the quotient provided that the function (z) is entire.
Proof Since the result is obvious when we suppose that f i ⋅f j (p, q) > 0 . Suppose also that We can clearly assume that f k p k , q k is finite for k = i, j.
Now, for any arbitrary > 0, we find from () that, for all sufficiently large values of r, We further consider the expression: for all sufficiently large values of r. Thus, for any > 1, it follows from the above expression that, for all sufficiently large values of r ≧ r 1 ≧ r 0 , Next, in view of (4.29) and (1.5), we have for all sufficiently large values of r. Also, by applying Lemma 2, we find from (4.30) and (4.31) that, for all sufficiently large values of r, that is, (4.28) f i ⋅f j (p, q) ≦ max f i p i , q i , f j p j , q j , p = max p i , p j and q = min q i , q j .