Coefficient characterizations and sections for some univalent functions

Let (α) denote the class of locally univalent normalized analytic functions f in the unit disk |z| < 1 satisfying the condition Re( 1 + zf''(z) f'(z) ) < 1 + α /2for|z| < 1 and for some 0 < α ≤ 1. We firstly prove sharp coefficient bounds for the moduli of the Taylor coefficients a n of f ∈ (α). Secondly, we determine the sharp bound for the Fekete-Szegö functional for functions in (α) with complex parameter λ. Thirdly, we present a convolution characterization for functions f belonging to (α) and as a consequence we obtain a number of sufficient coefficient conditions for f to belong to (α). Finally, we discuss the close-to-convexity and starlikeness of partial sums of f ∈ (α). In particular, each partial sum s n (z) of f ∈ (1) is starlike in the disk |z| ≤ 1/2 for n ≥ 11. Moreover, for f ∈ (1), we also have Re(s′ n (z)) > 0 in |z| ≤ 1/2 for n ≥ 11.