Testing Strong Gravitational Lensing Effects of Supermassive Black Holes with String-Inspired Metric, EHT Constraints and Parameter Estimation

We examine and compare the gravitational lensing, in the strong field limit, for the spherically symmetric string-inspired Euler-Heisenberg black holes, characterized by additional parameters Q² and α-β, representing magnetic charge and coupling constant, respectively. Our analysis reveals a reduction in the photon sphere radius xₚₛ, critical impact parameter uₚₛ and angular position θ_∞ with increasing magnitude of Q² and α-β. Consequently, the value of these quantities is consistently lower than that of its GR equivalents. Further, the ratio rmag of the flux of the first image to all others decreases with Q² and α-β. Unlike Schwarzschild black holes, string-inspired Euler-Heisenberg black holes have a smaller deflection angle αD, which decreases even more as Q² increases. Moreover, the time delay for Sgr A* and M87* can reach up to $~11.302$ and $~17085.1$ minutes, respectively, at Q²=0.1 and α-β=-1, deviating from Schwarzschild black holes by $~0.194$ and $~293.6$ minutes which are not very significant. For Sgr A* and M87*, we determine θ_∞ to range within (23.81, 26.28)~μ as and (17.89, 19.78)~μ as respectively, with angular separations s ranging from $(3.33-5.67)~nas$ for Sgr A* and $(2.51-4.26)~nas$ for M87*. EHT bounds on the θₛₕ of Sgr A* and M87* within the 1σ region, bound the parameters Q² and α-β as: for Sgr A* 0.29278 ≤ Q² ≤ 0.60778 and for M87* 0 < Q² ≤ 0.08473, but in both the cases we found no bound on the parameter α-β. We also estimate the parameters α-β and Q² associated with string-inspired Euler-Heisenberg black holes using the EHT observation results of Sgr A* and M87*.