THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

On a conjecture of shah concerning small function

The growth of maximum term of a composite entire function is compared with that of the maximum term of its left and right factors.

/pdf/growth-of-composite-entire-functions-4juom6rs9w.pdf

Growth of composite entire functions

I n this paper we introduce the idea of comparative growth properties of entire functions of two complex variable on the basis relative order, relative -order and relative -order

Growth of entire functions of two complex variable based on relative order

In this paper, the following results are obtained: Let f be a transcendental meromorphic functions withT(r, f)=O((log r)~βexp(log r)~α) 0α1, β0i.e., there exists two positive constants K_1 and K_2 such that K_1≤T(r, f)(log r)~βexp(log r)~α≤K_2; let g_1 and g_2 are transcendental entire functions and the order of g_2 be ρ_ g_2 (0ρ_ g_2 ∞), let a_i(z)(i=1,2,…,n, n≤∞) be entire functions which satisfying T(r, a_i(z))=o(T(r, g_2)) with∑ni=1δ(a_i(z), g_2)=1 δ(a_i(z), g_2)0If T(r, g_1)=o(T(r, g_2)) (r→∞), thenT(r, f(g_1))=o(T(r, f(g_2))) r→∞

Growth of Composite for Entire Functions and Meromorphic Functions with Deficient Functions

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integers p and q, we wish to introduce an alternative definition of relative th order which improves the earlier definition of relative th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

/pdf/on-the-p-q-th-relative-order-oriented-growth-properties-of-2ifs7qtbbp.pdf

On the $(p,q)$th Relative Order Oriented Growth Properties of Entire Functions

Abstract In this paper we study some comparative growth properties of composite entire and meromorphic functions on the basis of their generalized relative order, generalized relative type and generalized relative weak type with respect to another entire function.

/pdf/on-the-growth-measures-of-entire-and-meromorphic-functions-2mi9sgwrjb.pdf

On the growth measures of entire and meromorphic functions focusing their generalized relative type and generalized relative weak type

Relative order of functions measures specifically how different in growth two given functions are which helps to settle the exact physical state of a system. In this paper for any two positive integers $p$ and $q,$ we introduce the notion of relative $(p,q)$ th proximate order of an entire function with respect to another entire function and prove its existence.

/pdf/a-note-on-relative-p-q-th-proximate-order-of-entire-2bmro206on.pdf

A Note on Relative $(p,q)$ th Proximate Order of Entire Functions

In this paper we study some comparative growth properties of composite entire functions in terms of their maximum terms on the basis of their relative L* order ( relative L* lower order ) with respect to another entire function.

http://www.ijmex.com/index.php/ijmex/article/download/382/276

Some Results on the Growth Analysis of Entire Functions Using their Maximum Terms and Relative L -orders

/pdf/on-some-inequalities-concerning-relative-p-q-ph-type-and-4yngytxguw.pdf

On some inequalities concerning relative (p,q)-φ type and relative (p,q)-φ weak type of entire or meromorphic functions with respect to an entire function

Tanmay Biswas

Papers

On the $(p,q)$th Relative Order Oriented Growth Properties of Entire Functions

On the growth measures of entire and meromorphic functions focusing their generalized relative type and generalized relative weak type

A Note on Relative $(p,q)$ th Proximate Order of Entire Functions

Some Results on the Growth Analysis of Entire Functions Using their Maximum Terms and Relative L -orders

On some inequalities concerning relative (p,q)-φ type and relative (p,q)-φ weak type of entire or meromorphic functions with respect to an entire function