Radius problems in the class SL∗

Let SL∗ denote the class of all analytic functions f in the unit disc U with the normalization f(0)=f′(0)-1=0 and satisfying the conditionzf′(z)/f(z)2-1<1,z∈U,thus zf′(z)/f(z) is in the interior of the right half of the lemniscate of Bernoulli (x2+y2)2-2(x2-y2)=0. The relations between SL∗ and others classes geometrically defined are considered. The radii of α-convexity, of α-starlikeness (and some of others) of f∈SL∗ are determined.