Two theorems on factorization of inequalities (Q5932696)

The author continues his research on the factorization of inequalities [Acta Math. Hung. 89, No. 1-2, 103-110 (2000; Zbl 0963.26007)]. The background for these results is offered by \textit{G. Bennet}'s recent monograph [``Factorizing the classical inequalities'', Mem. Am. Math. Soc. 576, 130 p. (1996; Zbl 0857.26009)]. Given a Mulholland function \(\varphi\), the author defines certain sequence spaces \(t(\varphi,H,c)\) and \(t(\varphi,\Lambda,c)\). The two main results of the paper state that a sequence \({\mathbf x}=(x_n)\) is in these sequence spaces if and only if it admits a factorization of the form \(x_n=y_nz_n\), where the sequences \({\mathbf y}=(y_n)\) and \({\mathbf z}=(z_n)\) belong to certain sequence spaces. These results improve and extend that obtained by the author for the function \(\varphi(t)=t^p\) in earlier papers.