In this manuscript, we elucidate essential connections associated with the ( α , k ) − gamma and ( α , k ) − beta functions, initially introduced by Sarikaya et al. in their work as referenced in [14]. Our investigation includes the establishment of numerous conformable fractional integral inequalities, extending those previously established for the k−gamma and k−beta functions. Furthermore, we provide rigorous proofs affirming the log-convex nature of both the ( α , k ) − gamma and ( α , k ) − beta functions.