On the progress of the comparative growth analysis of entire and meromorphic functions

Abstract

Let S be a nonempty set of points in the complex plane z. If there exists a rule f which maps newlineeach value z = x + iy belonging to S , to one and only one complex number k = u + iv , then this newlinecorrespondence is called a function of the point z to the point k and it is denoted by k = f(z) = u + iv. newlineThe set S represents some region of the z -plane and and#146;functionand#146;denotes a single-valued function. The newlineset S is said to be the domain of deand#133;nition of the function. The functions of complex variables are of newlinedi¤erent types such as polynomial functions, integral functions ,meromorphic functions etc. A single newlinevalued function which is analytic in the and#133;nite complex plane having a pole of order m (m being a positive newlineinteger) at the point at inand#133;nity is called a polynomial function degree m. For example, z3 +3z2 +z +2 is newlinea polynomial function. But if a function of one complex variable is analytic in the and#133;nite complex plane newlineis called an integral function. exp z is an example of an integral function.These functions are also known newlineas entire functions.So a function that is analytic at every point in the complex plane C is said to be an newlineentire function. A function that is analytic at every point of C except at points it has poles is said to be newlinemeromorphic. The meromorphic function having an essential singularity at the point at inand#133;nity is known newlineas transcendental meromorphic function newline