new subclass of convex functions concerning infinite cone

We introduce a new subclass of convex functions as follows:[  mathcal{k}_{ic}:=left{fin mathcal{a}:{rm  re}left(1+frac{zf’’(z)}{f’(z)}right)>left|f’(z)-1right|,quad  |z|<1right},]where $mathcal{a}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. some properties of this particular class, including subordination relation, integral representation, the radius of convexity, rotation theorem, sharp coefficients estimate and fekete-szeg{o} inequality associated with the $k$-th root transform, are investigated.