Let P(Z) = 1 + P1Z + P2Z2 +…be an analytic function in the unit disc D. In thispaper, we determine the value of φ(α,β) for whichRe[P(Z) + αZP’(Z)] > β,Z ∈ D,α > 0,β < 1implies that Re{P(Z)} > <φ(α,β) for all Z ∈ D, and this result is sharp. Some of its interesting consequences are also given. In addition, we give a new univalence criterion.
0 IntroductionA cycle (path) in a graph G is called a hamiltonian cycle (path) in G if it con-tains all the vertices of G.A graph is hamiltonian (traceable) if it has a hamiltoniancycle (path).The neighborhood of a vertex v,denoted N(v),is the set of all verticesadjacent to v.We define the distance between two vertices u and v,denoted dist(u,v),as the minmum of the lengths of all u-v paths.Let NC2=min|N(u)∪N(v)|,DC2=min{max{d(u),d(v)}},where the minimum is taken over all pairs of vertices u,vthat are at distance two in the graph.Let δrepresent the minimum degree of G.Referto [5] for other terminology.
