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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 -, representing magnetic charge and coupling constant, respectively. Our analysis reveals a reduction in the photon sphere radius xₒ, critical impact parameter uₒ and angular position _ with increasing magnitude of Q² and -. Consequently, the value of these quantities is consistently lower than that of its GR equivalents. Further, the ratio r₌₀₆ of the flux of the first image to all others decreases with Q² and -. Unlike Schwarzschild black holes, string-inspired Euler-Heisenberg black holes have a smaller deflection angle D, 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 -=-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 _ to range within (23. 81, 26. 28) ~ as and (17. 89, 19. 78) ~ 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 ₒ₇ of Sgr A* and M87* within the 1 region, bound the parameters Q² and - 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 -. We also estimate the parameters - and Q² associated with string-inspired Euler-Heisenberg black holes using the EHT observation results of Sgr A* and M87*.
Vachher et al. (Fri,) studied this question.