A new subclass of analytic functions

In the present paper, we introduce a class $\mathcal{B}_\thetainsert ignore into journalissuearticles values(\alpha,\beta);$ of functions, analytic in $|z|<1$, such that $finsert ignore into journalissuearticles values(0);=0$, $f`insert ignore into journalissuearticles values(0);=1$ and \[\alpha< Re\leftinsert ignore into journalissuearticles values(f`insert ignore into journalissuearticles values(z);+\frac{1+e^{i\theta}}{2}zf``insert ignore into journalissuearticles values(z);\right);<\beta\quad insert ignore into journalissuearticles values(|z|<1);,\] where $\theta\ininsert ignore into journalissuearticles values(-\pi,\pi]$, $0\leq\alpha<1$ and $\beta>1$. Integral representation, differential subordination results and coefficient estimates are considered. Also Fekete-Szegö coefficient functional associated with the $k$--th root transform $[finsert ignore into journalissuearticles values(z^k);]^{1/k}$ for functions in the class $\mathcal{B}_\thetainsert ignore into journalissuearticles values(\alpha,\beta);$ is investigated.