In this work, we shall study k-inflation theories with non-minimal coupling of the scalar field to gravity, in presence only a higher-order kinetic term form $$\sim \hbox {const}\times X^{\mu }$$ , $$X=\frac{1}{2}\partial _{\mu } \phi \partial ^{\mu }\phi $$ . The will be focused cases where potential is included or absent, and evolution assumed satisfy slow-roll constant-roll condition. case m...