The Journal of the Indian Mathematical Society

V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, Annamalainagar, IN

Abstract

Special cases of the general functional equation

α[β(z)] = γ(z) α(z) (1)

have been considered in earlier papers ([1], [2], [3]). In [1], the case β(z) = z + b was considered in detail. In [2], we have discussed the case when β(z) = az + b, a ≠ 1.

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, Annamalainagar, IN

Abstract

This paper consists of two parts. In the first part, we consider the structure of the closed ideals in the space of integral functions endowed with an algebraic structure based on Hadamard composition. The second part consists of an introduction to the study of classes of functions of finite order and type endowed with suitable complete metric topologies. The notations and definitions are the same as those found in [1], [2], [3], [4].

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, Annamalainagar, IN

Abstract

About a decade ago, I discussed the equation

f(z + a) = g(z)f{z),       (1)

where f and g are integral functions and ∝a is a fixed complex number [see "Translation numbers of Integral Functions", J. Indian Math. Soc. (2), 10 (1946), 17-28].

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V. Ganapathy Iyer

Abstract

Let a = Σa_nzⁿ and β = Σb_nzⁿ be two integral functions, a o β = Σa_nb_nZⁿ and let (a)_n denote a o a…o a, n times. In a recent paper [1], I have proved the following result.

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

The object of this paper is to study the relation

f(z+λ)=g(z)f(z) (1)

from two points of view, f(z) and g(z) being integral functions. In section I, I study the properties of integral functions f(z) possessing one or more numbers λ satisfying an equation of form (I).

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

The object of this paper is to discuss some properties of functions of a real variable possessing derivatives of all orders. If f(x) possesses derivatives of all orders in ((x₀-δ, x₀+δ).

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

It is known that the lattice-points in the complex plane

Z = m+in, m, n = o, ±1, ±2,…                                (I)

have several interesting properties in relation to the values assumed by integral functions and funct ons regular in an angle at these points. The object of this paper is to show that the lattice-points (I) are only a special case of a large class of sequences which may be called pseudolattice points in relation to which also, integral functions and functions regular in an angle possess similar properties.

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

It is obvious that, except in the case when f(z) is a constant of modulus not exceeding unity, the number ρ defined by (I) satisfies the relation 0≤ρ≤∞. In this paper we suppose that ρ ≥ 0 is finite.

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

The object of this paper is to discuss the properties of a special class of integral functions defined as follows. Let g(z) be an integral function and p, d two positive numbers. We shall say that g(z) is a maximal integral function of order p and type d when

lim log|g(z)|/|z|^p=d,                          (1)

as |z|→∞ outside a system of circles the sum of whose radii is finite. We shall denote this fact by saying that g(z) belongs to M (P,d).

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V. Ganapathy Iyer

Abstract

Let z = re^iθ = x + iy. We write R(z) = x, I(z) = y. Let f(z) be an integral function and

M (r,f) = max|f(z)|.

We shall say that f(z) is of order one and finite type when

lim log M(r, f)/r < ∞.                                               (1)

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V. Ganapathy Iyer ¹

Affiliations
1 Annamalai University, IN

Abstract

The equation

P²+Q²=1                                (1)

can be satisfied by means of the integral functions

P=cos(az+b), Q=sin(az+b).       (2)

The question may be asked whether there are other solutions of the equation (1). In this paper I propose to find the most general solution in integral functions of the equation (1), and to discuss certain allied questions.

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V. Ganapathy Iyer ¹

Affiliations
1 University of Madras, IN

Abstract

Let f (z) be an integral function and M(r) the maximum of |f(z)| on |z|=r. The order p and type k of f (z) are defined by the relations

p=lim log logM(r)/log r; k=lim logM(r)/r^p.                                (1)

If p is finite, the function f (z) is said to be of minimal, normal or maximal type according as k vanishes, is a finite positive number, or is infinite. We shall define the number l by the relation

l=lim logM(r)/r^p,                                                                (2)

which might be called the lower type in contrast to k which might be termed the upper type. It is evident that l≤k.

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V. Ganapathy Iyer ¹

Affiliations
1 Madras University, IN

Abstract

Let f(z) be an integral function and M(r) = max |f(z)|. The order p of the function is defined by

p=lim log logM(r)/logr.

If p is finite, the upper type k and the lower type l are defined by

l=lim logM(r)/r^r ≤ lim logM(r)/r^p=k.

The function is said to be of maximal, normal or minimal type according as k=∞, a finite positive number or zero.

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V. Ganapathy Iyer ¹

Affiliations
1 Madras University, IN

Abstract

Let [z_n] be a sequence of distinct complex numbers such that [z_n|→∞ as n→∞. It is supposed that [z_n] are arranged according to non-decreasing moduli, the numbers with the same modulus being arranged according to their amplitudes. We shall refer to the exponent of convergence p of [z_n] as its order. We suppose 0 < p < ∞. Let σ(z) be the canonical product with simple zeros at z_n.

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V. Ganapathy Iyer ¹

Affiliations
1 Madras University, IN

Abstract

All processes which are in use to sum non-convergent sequences are particular cases of linear transformations representable by Toeplitz matrices. So far as I am aware there has been no systematic examination of the scope and limitation of processes representable by these matrices and the relation of these matrices considered as a whole to the class of all sequences on which they operate. In this paper a general study is made of these matrices without going into the special properties of any particular matrix.

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V. Ganapathy Iyer ¹

Affiliations
1 Madras University, IN

Abstract

The following result has been proved by Takenaka:

THEOREM 1. Let f(z) be an integral function of order one and type not exceeding σ. Let [α_n] be a sequence such that

lim |α_n|=L<log2.                                                               (1)

Let f(ⁿ)(α_n)=0, n=0, 1, 2 . . .[f⁽⁰⁾(z)=f(z)]. Then f(z)=0.

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V. Ganapathy Iyer ¹

Affiliations
1 Madras University, IN

Abstract

Let {Φ_n(x)}, n=1, 2, . . . , be a sequence of measurable functions of a real variable x defined almost everywhere in an interval a≤x≤b. Let f(x) be a function belonging to the Lebesgue class L_p(p>0) and let {Φ_n} be such that all the integrals

c_n(f)=∫f(x)Φ_n(x)dx                                                 (1)

exist as L-integrals.

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