New subclasses of Ozaka’s convex functions

Let S∗L (λ) and CVL (λ) be the classes of functions f, analytic in the unit disc ∆ = {z : |z| 1}, with the normalization f( 0) = f´(0) − 1 = 0, which satisfies the conditions zf´(z)/f (z) ≺ (1 + z)λ and (1 + zf′′(z)/f′(z) ≺ (1 + z)λ (0 λ ≤ 1), where ≺ is the subordination relation, respectively. The classes S∗L (λ) and CVL(λ) are subfamilies of the known classes of strongly starlike and convex functions of order λ. We consider the relations between S∗L (λ), CVL(λ) and other classes geometrically defined. Also, we obtain the sharp radius of convexity for functions belonging to S∗L (λ) class. Furthermore, the norm of pre-Schwarzian derivatives and univalency of functions f which satisfy the condition ℜ {1 + zf′′(z)/f′(z)} 1 + λ/2 (z ∈ ∆), are considered.