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

Abstract

The main object of the present paper is to obtain new estimates involving the $(p,q)$-th order and the $(p,q)$-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.

Keywords:

• entire functions
• maximum modulus
• (p; q)-th order and (p; q)-th type
• value distribution theory
• relative growth of entire functions
• property (A)

2010 Mathematics Subject classiffications:

• Primary 30D35
• Secondary 30D30

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 $exp(z)$, $sinz$, $cosz$, 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.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

H.M. Srivastava

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.