Generalized planar Tur\'an numbers related to short cycles

Given two graphs H and F, the generalized planar Tur\'an number exP(n,H,F) is the maximum number of copies of H that an n-vertex F-free planar graph can have. We investigate this function when H and F are short cycles. Namely, for large n, we find the exact value of exP(n, Cₗ,C₃), where Cₗ is a cycle of length l, for 4≤ l≤ 6, and determine the extremal graphs in each case. Also, considering the converse of these problems, we determine sharp upper bounds for exP(n,C₃,Cₗ), for 4≤ l≤ 6.