Geometrically, a graph is a collection of points (or vertices) together with a set of edges (or curves) each of which joins two distinct vertices of the graph, and no two of which have points in common except possibly end points. Two given vertices of the graph may be joined by no edge or one edge, but may not be joined by more than one edge. From an abstract point of view, a graph G is a collection of elements ﹛x1, x2, …﹜ called points or vertices, together with a second collection of certain pairs (xα, Xβ) of distinct points of G. It is helpful to retain the geometric language, and refer to any pair in as an edge (or a curve) of G that joins the points xα and Xβ.

Abstract. We introduce a new class of normalized functions regular and univalent in the unit disk. These functions, called uniformly convex functions, are defined by a purely geometric property. We obtain a few theorems about this new class and we point out a number of open problems.

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