## On some new sequence spaces of invariant means defined by Orlicz functions.(English)Zbl 1010.46006

By combining the concept of invariant means, lacunary convergence and Orlicz functions $$M$$, the authors define some sequence spaces and study their properties. Let $$p=(p_k)$$ be a sequence of positive numbers and $$\theta =(k_r)$$ be a lacunary sequence with $$h_r:=k_r-k_{r-1} \to\infty$$. Denote $[w,M,p]_\sigma: =\{x=(x_k) \mid\lim_n {1\over n+1} \sum^n_{k=0} \left[M\left( {\bigl |t_{km}(x-l) \bigr|\over\rho} \right)\right]^{p_k}=0$
$\text{for some }l\text{ and }\rho>0, \text{ uniformly in }m\}$ and $[w^\theta, M, p]_\sigma: =\{x=(x_k) \mid\lim_r {1\over h_r} \sum^{k_r}_{k=k_{r-1}+1}\left[ M \left({\bigl |t_{km}(x-l)\bigr |\over \rho}\right) \right]^{p_k}=0$
$\text{for some }l\text{ and }\rho>0, \text{ uniformly in }m\},$ where $$t_{km} (x):=(x_m+x_{\sigma(m)} +x_{\sigma^2(m)} +\cdots+x_{\sigma^k(m)})/(k+1)$$ and $$\sigma: \mathbb{N}\to \mathbb{N}$$ is the translation mapping $$n\mapsto n+1$$. The authors show that (a) $$[w,M,p]_\sigma \subset[w^\theta,M,p]_\sigma$$ if $$\lim \inf_r q_r>1$$ and (b) $$[w^\theta,M,p]_\sigma \subset[w,M,p]_\sigma$$ if $$\lim\sup q_r<\infty$$ (here $$q_r:= k_r/k_{r-1}$$, $$r=0,1,\dots)$$.

### MSC:

 46A45 Sequence spaces (including Köthe sequence spaces) 40D25 Inclusion and equivalence theorems in summability theory 40H05 Functional analytic methods in summability 40A05 Convergence and divergence of series and sequences
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