The Journal of the Indian Mathematical Society

C. T. Rajagopal ¹

Affiliations
1 The Anchorage, Agaram Road, Madras-59, IN

Abstract

On summabilities (A_α) and (A_x). P. A. Jeyarajan has proved ([3], Theorem 4) the Tauberian theorem for generalized Abel summability (A_α) appearing as Theorem 1(A_α) in this note.

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C. T. Rajagopal ¹

Affiliations
1 Ramanujan Institute of Mathematics, University of Madras, IN

Abstract

R. D. Lord has generalized a classical theorem of Hardy and Little wood ([1], Theorem 156) and shown ([3], Theorem 5) that a sequence, summable by a method which is implied by Borel's, is summable also by Cesaro's method of a certain nonnegative order under an appropriate Tauberian condition.

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C. T. Rajagopal ¹

Affiliations
1 Ramanujan Institute of Mathematics (Karaikudi), Madras, IN

Abstract

In two successive papers [10,11] † O . Szasz proved certain results on the product of two summability methods which with one exception, are included in Theorems I, II of this paper.

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C. T. Rajagopal ¹

Affiliations
1 Ramanujan Institute of Mathematics, (Karaikudi), Madras, IN

Abstract

My earlier note with the same title [3] was mainly about an extension of a theorem of Karamata on the Laplace- Abel transform to certain regular integral transforms Φ of a real function a(u) satisfying the classical one-sided condition,

a(u) u > - W, W > 0, for u > 1, and a(u) = 0 for u < 1. (1)

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C. T. Rajagopal ¹

Affiliations
1 Ramanujan Institute of Mathematics, Madras, IN

Abstract

The principal object of this note is to generalize a one-sided Tauberian theorem of Karamata [3] for the Abel transform of a series. Implicit in the proof of the generalization there is a similar generalization of another one-sided Tauberian theorem, stated without proof by Vijayaraghavan [7, Theorem 8] for the Abel transform.

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C. T. Rajagopal ¹

Affiliations
1 Tambaram, IN

Abstract

This note suggests a further use for a difference formula of L. S. Bosanquet which has already proved to be of service in dealing with certain problems involving Cesaro and Riesz means. More particularly, it is the object of the note to establish, by means of Bosanquet's formula, the integral analogue of a theorem on series, due to M. S. Macphail, [Theorem 2], which contains as a special case a result of H. L. Garabedian. Macphail's theorem itself can be obtained, as shown in the last section of the note, by an obvious adaptation of the argument used to prove its integral analogue.

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C. T. Rajagopal ¹

Affiliations
1 Madras Christian College, Tambaram, IN

Abstract

The first section of the note centres round the problem of constructing a periodic meromorphic function, with assigned poles and principal parts thereat, in a period strip. The second deals with rational functions of exp (2π^iz/λ), the main theorem of the section giving a necessary and sufficient condition for a periodic meromorphic function to be such a rational function, in terms of its Nevanlinna characteristic function. Throughout it is supposed, for the sake of simplicity, that λ is a positive number.

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C. T. Rajagopal ¹

Affiliations
1 Madras Christian College, IN

Abstract

In the paper cited above, I have based the convergence theory of positive series on

If O = d₀ = D₀, O<d_n=D_n-D_n-1 (n = 1, 2,...), D_n→∞, and if F(x) is positive monotone decreasing, ∫ F(x)dx is convergent, then ∫d_nF(D_n) is convergent.

The first result of this note leads to the more general companion.

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C. T. Rajagopal ¹

Affiliations
1 Christian College, Madras, IN

Abstract

This paper brings together certain theorems which enable 00 us to settle the convergence or divergence of Σ a_n, (a_n > 0) with the help of a suitably chosen integral ∫ F(x) dx, F(x) > 0. The theorems include as particular cases all the convergence criteria, in common use, for series of positive terms, and suggest a simple and not inelegant method of presenting such criteria within the framework of a single idea.

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