Let \(S\) denote the class of functions \(f\) analytic and univalent in the unit disc \(U\), with \(f(0)= f'(0)-1=0\). In the present paper the authors introduce the classes \(SH(\alpha)\), \(\alpha>0\), \(f\in S\) is said to be in \(SH(\alpha)\) if it satisfies \[ |zf'(z)/f(z)- 2\alpha(\sqrt{2}-1)|< \text{Re} \{\sqrt{2} zf'(z)/f(z)\}+ 2\alpha(\sqrt{2}-1), \qquad z\in U.\tag \(*\) \] Note that \(\Omega(\alpha)= \{zf'(z)/ f(z)\mid z\in U\), \(f\in SH(\alpha)\}\) is the interior of a hyperbola in the right half-plan which is symmetric about the real axis and has vertex at the origin and \(\bigcup_{\alpha>0} SH(\alpha)=S^*\). For the classes \(SH(\alpha)\) an extremal function is determined and some subordination results, sharp growth and distorsion theorems, and sharp estimations for coefficients are given.