Barrow HDE model for Statefinder diagnostic in FLRW Universe

We have analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent $triangle$. Here we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking. To understand the recent accelerated expansion of the universe, considering the Hubble horizon as the IR cut-off. The cosmological parameters, especially the density parameter ($Omega_{_D}$), the equation of the state parameter ($omega_{_D}$), energy density ($rho_{_{D}}$) and the deceleration parameter($q$) are studied in this manuscript and found the satisfactory behaviors. Moreover, we additionally focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_{m}(z)$ to discriminant BHDE model from the $Lambda CDM$ model. Here we determined and plotted the trajectories of evolution for statefinder $(r, s)$, $(r,q)$ and $O_{m}(z)$ diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent $triangle$, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime, and exhibits the phantom-divide line during the cosmological evolution.