Norm inequalities involving upper bounds for certain matrix operators in Orlicz-type sequence spaces

Abstract

In this paper, upper bounds of certain matrix operator norms are estimated in Orlicz-type weighted sequence spaces. Three spaces, namely, weighted Orlicz–Euler $$e_{\lambda , \varphi }^{\alpha }$$ , weighted Orlicz–Fibonacci $$F_{\lambda , \varphi }$$ and weighted Orlicz $$l_\varphi (\lambda )$$ are considered. Denote $$\Vert A\Vert _{X, Y}$$ as the operator norm of the matrix $$A=(a_{n, k})_{n, k\ge 0}$$ which maps X into Y, where X and Y are two normed sequence spaces. Then the evaluation of upper bounds for $$\Vert A\Vert _{l_\varphi (\lambda ), e_{\lambda , \varphi }^{\alpha }}$$ , $$\Vert A\Vert _{l_\varphi (\lambda ),F_{\lambda , \varphi }}$$ and $$\Vert A\Vert _{l_\varphi (\lambda ), l_\varphi (\mu )}$$ , where A is either Hausdorff or Norlund matrices is carried out throughout this paper. Some Hardy type formulas are established in case of Hausdorff matrices. Certain inclusion results are also obtained for each of the three sequence spaces. The results obtained in this work strengthen the results recently presented by Lashkaripour and Foroutannia (Proc Indian Acad Sci (Math Sci) 116(3):325–336, 2006) and Talebi and Dehghan (Linear Multilinear Algebra 62(10):1275–1284, 2014; Linear Multilinear Algebra 64(2):196–207, 2016).