On Strongly Starlike Functions Related to the Bernoulli Lemniscate

Let S^L(λ) be the class of functions f, analytic in the unit disc Δ=:|z|<1\, with the normalization $f(0)=f'(0)-1=0$, which satisfy the condition{equation*}zf'(z)/f(z) (1+z)λ,{equation*}where is the subordination relation. The class S^L(λ) is a subfamily of the known class of strongly starlike functions of order λ. In this paper,the relations between S^L(λ) and other classes geometrically defined are considered. Also, we obtain some characteristics such as, bounds for coefficients, radius of convexity, the Fekete-Szeg\"{o} inequality, logarithmic coefficients and the second Hankel determinant inequality for functions belonging to this class. The univalent functions f which satisfy the condition{equation*}\{1+zf''(z)/f'(z)\}<1+λ/2,(z ∈ Δ){equation*}are also considered here.