• V K Krishnan

      Articles written in Proceedings – Mathematical Sciences

    • On the relation of generalized Valiron summability to Cesàro summability

      V K Krishnan

      More Details Abstract Fulltext PDF

      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).

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