Bi-$f$-Harmonic Legendre Curves on $(\alpha,\beta)$-Trans-Sasakian Generalized Sasakian Space Forms

In this study, we consider bi-$f$-harmonic Legendre curves on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form. We provide the necessary and sufficient conditions for a Legendre curve to be bi-$f$-harmonic on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form without any restrictions by a main theorem. Afterward, we investigate these conditions under nine different cases. As a result of these investigations, we obtain the original theorems and corollaries as well as the nonexistence theorems. We perform these investigations according to the $\rho_{2}$ and $\rho_{3}$ functions from the curvature tensor of the $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form, the curvature and torsion of the bi-$f$-harmonic Legendre curve, and finally, the positions of the basis vectors relative to each other.