We consider the family of all meromorphic functions f of the form

analytic and locally univalent in the puncture disk D0 := {z ∈ C : 0 < |z| < 1}. Our first objective in this paper is to find a sufficient condition for f to be meromorphically convex of order α, 0 ≤ α < 1, in terms of the fact that the absolute value of the well-known Schwarzian derivative S_f(z) of f is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions g of the form g(z) = z + a₂z² + a₃z³ + ··· analytic and locally univalent in the open unit disk D := {z ∈C : |z| < 1}, and show that g is belonging to a family of functions convex in one direction if |S_g(z)| is bounded above by a small positive constant depending on the second coefficient a₂. In particular, we show that such functions g are also contained in the starlike and close-to-convex family.

Keywords: Meromorphic functions, convex functions, meromorphically convex functions, close-to-convex functions, starlike functions, Schwarzian derivative.