• On the relation of generalized Valiron summability to Cesàro summability

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/090/03/0181-0193

• # Keywords

Generalized Valiron summability; Boral summability; Rajagopal’s theorem

• # Abstract

A family (Vak) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (Bα,γ), (Eρ, (Tα), (Sβ) and (Va) are members of this family. In §3 some properties of the (Bα,γ) and (Vak) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (Vak) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If sn (n ≥ 0) isa real sequence satisfying$$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$, and if sns (Vak) thensn → s (C, 2ρ).

• # Author Affiliations

1. St. Thomas College, Trichur - 680 001, India

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019