In the present paper a new subclass of the starlike functions $f$ analytic in the open unit disc $\Delta$, denoted by $\mathcal{S}^*_p$ normalized by $f(0)=0=f'(0)-1$ and satisfying \begin{equation*} \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z-tz^2}{(1-z)^2}\quad(t\in[0,1]), \end{equation*} where $"\prec"$ is the subordination relation, are introduced. Some properties of the class $\mathcal{S}^*_p$ including the radius of starlikeness and convexity of functions $f\in\mathcal{S}^*_p$, majoriziation problem, initial coefficient estimates, Fekete-Szeg\"{o} problem and estimation of initial logarithmic coefficients are investigated.
