New subclass of analytic functions defined by subordination

By using the subordination relation $"prec"$, we introduce an interesting subclass of analytic functions as follows: begin{equation*}
mathcal{S}^*_{alpha}:=left{fin mathcal{A}:frac{zf'(z)}{f(z)}prec frac{1}{(1-z)^alpha}, |z|<1right},
end{equation*}
where $0<alphaleq1$ and $mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. In the present paper, by the class $mathcal{S}^*_{alpha}$ and by the Nunokawa lemma we generalize a famous result connected to starlike functions of order $1/2$. Also, coefficients inequality and logarithmic coefficients inequality for functions of the class $mathcal{S}^*_{alpha}$ are obtained.