Let \({\mathcal A}\) be the class of functions \(f(z)=z+\sum_{k=2}^\infty a_kz^k\) that are analytic in the unit disc \({\mathcal U}=\{z:| z| <1\}.\) In this paper the authors give criteria that imply starlikeness and convexity of the class \[ G_{\lambda,\gamma}=\left\{f\in {\mathcal A}: \left| \frac{1-\gamma+zf''(z)/f'(z)}{zf'(z)/f(z)}-(1-\gamma)\right| <\lambda, z\in {\mathcal U}\right\}. \] They also give sharp upper bound of \(| a_2| \) and of the Fekete-Szegö functional \(| a_3-\mu a_2^2| \) over this class. Special choices of \(\gamma\) lead to some previously known results and comparison with them is made.