The authors introduce a new subclass $\mathcal{UH}(\alpha,\beta,\gamma,\delta,\lambda,\kappa)$ of functions which are analytic in the open disc $\Delta=\{z\in\mathbb C:|z|<1\}$. Various results studied include the coefficient estimates and distortion bounds, radii of close-to-convexity, starlikeness and convexity and integral means inequalities for functions belonging to the above class. Relevances of the main results are also briefly indicated.