The Generalized Relative Gol‘dberg Order and Type: Some Remarks on Functions of Complex Variables

In this chapter, we discussed about the introductory parts connected to the entire functions of n complex variables. In this connection, we add some preliminary defin- itions related to di¤erent Gol`dberg growth indicators such as Gol`dberg order, Gol`dberg type etc.

In this chapter, first we introduce the definitions of generalized Gol`dberg order (α,β), generalized hyper Gol`dberg order (α,β) generalized logarithmic Gol`dberg order (α,β); generalized Gol`dberg type (α,β) and generalized Gol`dberg weak type (α,β) of entire functions of several complex variables and then using these growth indicators, we discuss of some related growth properties of entire functions of n complex variables, where α,β are continuous non-negative functions defined on (-∞,+∞).

The aim of the chapter is to introduce the concepts of generalized relative Gol`dberg order (α,β); generalized relative hyper Gol`dberg order (α,β), and generalized relative logarithmic Gol`dberg order (α,β) of an entire function of several complex vari- ables with respect to another entire function of several complex variables, where α,β are continuous non-negative functions defined on (-∞,+∞). Then we discuss some growth analysis of entire functions of several complex variables. Also we established some integral representations of the above growth indicators.

In this chapter, Some inequalities using generalized Gol`dberg order (α,β), generalized Gol`dberg lower order (α,β), generalized relative Gol`dberg order (α,β) and generalized relative Gol`dberg lower order (α,β) of entire functions of several complex variables are established, where α,β are continuous non-negative functions defined on (-∞,+∞).

In this chapter, we develop some growth properties of entire functions of n complex variables relating to generalized relative Gol`dberg order (α,β); generalized rel- ative Gol`dberg type (α,β) and generalized relative Gol`dberg weak type (α,β):We also establish integral representations of generalized relative Gol`dberg type and weak type (α,β) of entire function of several complex variables and derive some interesting results, where α,β are continuous non-negative functions defined on (-∞,+∞).

In this chapter,we establish some important relations relating to generalized relative Gol`dberg type and weak type (α,β) with generalized Gol`dberg type and weak type (α,β) of entire functions of n complex variables, where α,β are continuous non- negative functions defined on (-∞,+∞).

In this chapter, we intend to and out generalized relative Gol`dberg order (α,β), generalized relative Gol`dberg type (α,β) and generalized relative Gol`dberg weak type (α,β) of an entire function f of several complex variables with respect to another entire function g of several complex variables when generalized relative Gol`dberg or- der ( γ,β); generalized relative Gol`dberg type (γ,β) and generalized relative Gol`dberg weak type ( γ,β) of f and generalized relative Gol`dberg order (γ,α), generalized relative Gol`dberg type ( γ,α) and generalized relative Gol`dberg weak type (γ,α) of g with re- spect another entire function h of several complex variables are given, where α,βγ are continuous non-negative functions defined on (+∞,-∞).

In this chapter, we proved some results about sum and product theorems de- pending on the generalized relative Gol`dberg order (α,β) ;generalized relative Gol`dberg lower order (α,β), generalized relative Gol`dberg type (α,β) and generalized relative Gol`dberg weak type (α,β) of entire function of n complex variables with respect to another entire function of n complex variables, where α,β are continuous non-negative functions de…ned on (+∞,-∞).