© Naim L. Braha. 2011
Received: 5 November 2010
Accepted: 18 February 2011
Published: 10 March 2011
We introduce new sequence space defined by combining an Orlicz function, seminorms, and -sequences. We study its different properties and obtain some inclusion relation involving the space Inclusion relation between statistical convergent sequence spaces and Cesaro statistical convergent sequence spaces is also given.
By , we denote the space of all real or complex valued sequences. If , then we simply write instead of . Also, we will use the conventions that . Any vector subspace of is called a sequence space. We will write , , and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by , we denote the sequence space of all -absolutely convergent series, that is, for . Throughout the article, , , and denote, respectively, the spaces of all, bounded, and -absolutely summable sequences with the elements in , where is a seminormed space. By , we denote the zero element in . denotes the set of all subsets of , that do not contain more than elements. With , we will denote a nondecreasing sequence of positive real numbers such that and , as . The class of all the sequences satisfying this property is denoted by .
2. Definitions and Background
The space is closely related to the space which is an Orlicz sequence space with . An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . It is well known that if is a convex function and , then for all with .
An Orlicz function is said to satisfy the -condition for all values of , if there exists a constant such that (see, Krasnoselskii and Rutitsky ). In the later stage, different Orlicz sequence spaces were introduced and studied by Bhardwaj and Singh , Güngör et al. , Tripathy and Mahanta , Esi , Esi and Et , Parashar and Choudhary , and many others.
Since the proofs of the following theorems are not hard we omit them.
Proof is similar to [14, Theorem 2.5].
4. Statistical Convergence
5. Cesaro Convergence
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