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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.
Győri et al. (Mon,) studied this question.