APPROXIMATION OF ENTIRE FUNCTIONS OF ONE OR SEVERAL COMPLEX VARIABLES

Abstract

An entire function is a function f : C C, which is regular in every finite region of the complex plane. The general theory of these functions originated in the works of Weierstrass [66]; in the beginning it was developed by Picard, Borel, Poincare, Hadamard and others. In the beginning of twentieth century some new concepts were introduced by eminent mathematicians such as Valiron [64], Lindelof, Levin, Wiman, Nevanlinna and Hardy etc. Since then, Whittaker [67], Hayman, Boas [5], Holland [21], Clunie, Titchmarch [58] and others have contributed richly to the theory of entire functions. An entire function f (z) has the representation by a power series of the form f (z) = 00 E zn,lO oci a„1 = 0. n—o n=0 This is the simplest class of analytic functions containing all polynomials. Polynomi-als are classified according to their degree, i.e. according to their growth as 1z1 —+ co. An entire function can grow in various ways along different directions. For a gener-alization of the growth, the function (0.1.1) M(r) = M(r; f) = max if(z)i 1 is introduced. Then M(r) is said to be the maximum modulus of f(z) for Izi = r. It has been established that maximum absolute value of an entire function over a closed disc coincides with the maximum absolute value of that function over its boundary. Blumenthal [4] showed that M(r) is a steadily increasing continuous func-tion of r and is different in adjacent intervals. Further, In M(r) is a convex function of In r and has the representation [64] (0.1.2) In M(r) =-- 1nM(ro) + dx, T > ro, f0 1.r w (x) where W(x) is a positive indefinitely increasing function of x which is continuous in adjacent intervals. M(r) plays a key role in the study of the growth of entire functions. A.P.Singh and Baloria [49] have studied on maximum modulus and maximum term of composition of entire functions. In order to estimate the growth of f (z) precisely, the concept of order was intro-duced. An entire function f (z) is called a function of finite order if M(r) < exp (rk) for some k > 0. The order of an entire function f is the greatest lower bound of those values of k for which this asymptotic inequality is fulfilled. We shall usually denote the order of an entire function f by p. It follows from the definition of the order that erP-t < m (r) < erp+ By taking the logarithm twice we obtain In In M (r) p — E < < p + e, In r Thus the order p of f (z) is given by In In M(r) , (0.1.3) p = lim sup , r 0 < p < 00 . r00 i By convention, a constant function is taken to be of order zero. The concept of type has been introduced to determine the relative growth of two entire functions of same non-zero finite order. Let p be the order of an entire function 2 f (z). The function is said to have a finite type if for some A > 0 the inequality M(r) < eArP is fulfilled. The greatest lower bound for those values of A for which the later asymp-totic inequality is fulfilled is called the type r of the function f (z). It follows from the definition of the type that e(r-e)rP < M(r) < jr+E)7.P. By taking logarithm and dividing by rP, we obtain M(r) c) < < E). rP , Thus, an entire function f (z) of order p (0 < p < co) is said to be of type T if (0.1.4) T = lim sup In M (r), 0 < < -0 00 The function f (z) is said to be minimal, maximal or normal type according as T = 0, T = 00 or 0 < T < oo respectively. An entire function f(z)is said to be of growth (p, r) if its order does not exceed p, and its type does not exceed 7 if it is of order p. The function f (z) is of exponential type r if it is of order less than one,and if of order one, of the type less than or equal to r, r < oo. L.R.Sons [52, 53] have studied on regularity of growth and gaps. If an entire function f (z) is of zero or infinite order then the usual definition of type has no meaning. Hence the comparison of growth of such functions can not be made by confining to the above concepts. To overcome this difficulty, V.G.Iyer [24] introduced the concept of logarithmic order. Thus for an entire function of Order zero, p* is said to be logarithmic order of 1(z) if p* = lirn sup In In M(r) 0 < p* < oo. (0.1.5) lnlnr