In this paper we wish to establish the integral representations of generalized relative type and generalized relative weak type as introduced by Datta et al [9]. We also investigate their equivalence relation under some certain conditions.

‎‎for a finite group $g$ and a subgroup $h$ of $g$‎, ‎the relative commutativity degree of $h$ in $g$‎, ‎denoted by $d(h,g)$‎, ‎is the probability that an element of $h$ commutes with an element of $g$‎. ‎let $mathcal{d}(g)={d(h,g):hleq g}$ be the set of all relative commutativity degrees of subgroups of $g$‎. ‎it is shown that a finite group $g$ admits three relative commutativity degrees if a...

‎in this short note‎, ‎we present some inequalities for relative operator entropy which are generalizations of some results obtained by zou [operator inequalities associated with tsallis relative operator entropy‎, ‎{em math‎. ‎inequal‎. ‎appl.}‎ ‎{18} (2015)‎, ‎no‎. ‎2‎, ‎401--406]‎. ‎meanwhile‎, ‎we also show some new lower and upper bounds for relative operator entropy and tsallis relative o...