Comparison of strong and statistical convergences in some families of summability methods

Abstract

The paper deals with certain families $\{A_\alpha\}$  $(\alpha > \alpha_0)$ of summability methods.   Strong and statistical convergences in Ces\`{a}ro- and Euler--Knopp-type  families  $\{A_\alpha\}$  
are investigated. Convergence of a sequence $x = (x_n)$ with respect to the different strong summability methods $[A_{\alpha + 1}]_t$ (with positive exponents $t = (t_n)$) in a family $\{A_\alpha\}$ 
 is compared, and characterized with the help of statistical convergence.   A convexity theorem for comparison of three  strong summability methods $[A_{\gamma + 1}]_t$, $[A_{\delta + 1}]_t$ and  $[A_{\beta + 1}]_t$ $(\beta > \delta > \gamma > \alpha_0 )$ in a Ces\`aro-type family $\{A_\alpha\}$ is proved. This theorem can be seen as a generalization of some convexity theorems known earlier. Interrelations between strong convergence and  certain statistical convergence  are also studied  and described with the help of  theorems proved here. All the results can be transferred to the particular cases of the family $\{A_\alpha\},$ e.g., to the families of  generalized N\"orlund methods $(N,p_n^\alpha, q_n).$