Summary: Let \({\mathcal A}\) be the class of analytic functions in the unit disk that are normalized with \(f(0)=f'(0)-1=0\) and let \(-1\leq B<A\leq 1\). In this paper we study the class \(G_{\lambda,\alpha}=\{f\in{\mathcal A}:|(1-\alpha +azf''(z)/f'(z)) /zf'(z)/f(z)-(1-\alpha)|<\lambda\), \(z\in{\mathcal U}\}\), \(0\leq \alpha\leq 1\), and give sharp sufficient conditions that embed it into the classes \(S^*[A,B]=\{f\in{\mathcal A}:zf'(z)/f(z)\prec(1+Az)/(1+Bz)\}\) and \(K(\delta)=\{f\in{\mathcal A}:1+zf''(z)/f' (z)\prec(1-\delta)(1+z)/(1-z)+ \delta\}\), where “\(\prec\)” denotes the usual subordination. Also, sharp upper bound of \(|a_2|\) and of the Fekete-Szegö functional \(|a_3-\mu a^2_2|\) is given for the class \(G_{\lambda,\alpha}\).