An interesting area in operator theory is the study of norm inequalities for Hilbert space operators. Many mathematicians have worked on this subject, for example in [2, 3 and 5]. On the other hand, contractive and normaloid operators have been considered separately by [1, 6, 7 and 8]. In this paper, we results on conditions for normaloidity and contractivity of Hilbert space operators. We begin by simple lemmas before we move to main results. Let be a complex Hilbert space with an inner product and the algebra of all bounded linear operators on . denotes the usual operator norm and Dom( ) denotes the domain of