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Ganapathy Iyer, V.
- On a Functional Equation (III)
Authors
1 Annamalai University, Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 26, No 1-2 (1962), Pagination: 69-75Abstract
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.
- On the Space of Integral Functions, V
Authors
1 Annamalai University, Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 24, No 1-2 (1960), Pagination: 269-278Abstract
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].- On a Functional Equation
Authors
1 Annamalai University, Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 20, No 1-3 (1956), Pagination: 283-290Abstract
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].
- A Note on the Linear $pace Generated by a Sequence of Integral Functions
Authors
Source
The Journal of the Indian Mathematical Society, Vol 17, No 4 (1953), Pagination: 183-185Abstract
Let a = Σanzn and β = Σbnzn be two integral functions, a o β = ΣanbnZn and let (a)n denote a o a…o a, n times. In a recent paper [1], I have proved the following result.- On the Translation Numbers of Integral Functions
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 10 (1946), Pagination: 17-28Abstract
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).
- On Singular Functions
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 8 (1944), Pagination: 94-108Abstract
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).- The Influence of Zeros on the Magnitude of Functions Regular in an Angle
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 7 (1943), Pagination: 1-16Abstract
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.
- A Property of the Maximum Modulus of Integral Functions
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 6 (1942), Pagination: 69-80Abstract
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.- On Maximal Integral Functions
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 4 (1940), Pagination: 97-115Abstract
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).
- On the Average Radial Increase of a Certain Class of Integral Functions of Order One and Finite Type
Authors
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 87-95Abstract
Let z = reiθ = 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)
- On Certain Functional Equations
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 312-315Abstract
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.
- On Integral Functions of Order One and of Finite Type
Authors
1 University of Madras, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 1-12Abstract
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)/rp. (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)/rp, (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.
- On Integral Functions of Finite Order Bounded at a Sequence of Points
Authors
1 Madras University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 53-66Abstract
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)/rr ≤ lim logM(r)/rp=k.
The function is said to be of maximal, normal or minimal type according as k=∞, a finite positive number or zero.
- On Integral Functions of Finite Order and Minimal Type
Authors
1 Madras University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 131-140Abstract
Let [zn] be a sequence of distinct complex numbers such that [zn|→∞ as n→∞. It is supposed that [zn] 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 [zn] as its order. We suppose 0 < p < ∞. Let σ(z) be the canonical product with simple zeros at zn.- On Summation Processes in General
Authors
1 Madras University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 222-238Abstract
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.- A Property of the Zeros of the Successive Derivatives of Integral Functions
Authors
1 Madras University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 289-294Abstract
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 [αn] be a sequence such that
lim |αn|=L<log2. (1)
Let f(n)(αn)=0, n=0, 1, 2 . . .[f(0)(z)=f(z)]. Then f(z)=0.
- Some Uniqueness Theorems for Functions of Class Lp
Authors
1 Madras University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 324-331Abstract
Let {Φn(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 Lp(p>0) and let {Φn} be such that all the integrals
cn(f)=∫f(x)Φn(x)dx (1)
exist as L-integrals.