  In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the
question is the maximum number of copies of $H$ in an $F$-free graph of order
$n$. In this paper, we study the number of double stars $S_{k,l}$ in
triangle-free graphs. We also study an opposite version of this question: what
is the maximum number edges/triangles in graphs with double star type
restrictions, which leads us to study two questions related to the extremal
number of triangles or edges in graphs with degree-sum constraints over
adjacent or non-adjacent vertices.
