Some Lipschitz type inequalities for complex functions

where g satis…es a Lipschitz condition with the constant K: In (1.1) and (1.2) the constant 14 is best possible in the sense that it cannot be replaced by a smaller quantity. The same applies for the constant 1 12 in (1.3). The interested reader may …nd many other inequalities involving Lipschitzian functions in the papers [1]-[11], [13]-[20] and in the references therein. In this paper we establish some Lipschitz type inequalities for complex functions when some convexity properties for powers of the absolute value of the derivative are assumed. The case of functions de…ned by power series is analysed. Some