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The detailed study of horizon structure and the shadow cast by a Kerr-like black hole (BH) is performed. The trajectory of light rays forming the shadow of BH is found using the solutions of geodesic equation for the motion and effective potential of a photon around Kerr-like BH for different values of deviation parameter l in Kerr-like spcetime metric. It is observed that with an increase in the parameter l the size of the shadow of the BH is decreased. Additional, we have consider effect of plasma on BH shadow and the plasma influence on the shadow of Kerr-like BH, the size of observable radius of BH shadow and oblateness are explored with more details.

Testing gravity theories and their parameters using observations is an important issue in relativistic astrophysics. In this context, we investigate the motion of test particles and their harmonic oscillations in the spacetime of non-rotating hairy black holes (BHs) in Hordeski gravity, together with astrophysical applications of quasiperiodic oscillations (QPOs). We show possible values of upper and lower frequencies of twin-peak QPOs which may occur in the orbits from innermost stable circular orbits to infinity for various values of the Horndeski parameter q in relativistic precession, warped disk models, and three different sub-models of the epicyclic resonant model. We also study the behaviour of the QPO orbits and their position relative to innermost stable circular orbits (ISCOs) with respect to different values of the parameter q. It is obtained that at a critical value of the Horndeski parameter ISCO radius takes 6M which has been in the pure Schwarzschild case. Finally, we obtain mass constraints of the central BH of microquasars GRS 1915+105 and XTE 1550-564 at the GR limit and the possible value of the Horndeski parameter in the frame of the above-mentioned QPO models. The analysis of orbits of twin peak QPOs with the ratio of upper and lower frequencies 3:2, around the BHs in the frame of relativistic precession (RP) and epicyclic resonance (ER4) QPO models have shown that the orbits locate close to the ISCO. It is obtained that the distance between QPO orbits and ISCO is less than the error of the observations.