A Few Inequalities Established by Using Fractional Calculus and Their Applications to Certain Multivalently Analytic Functions

By making use of different techniques given in Miller and Mocanu (2000) (and also in Jack (1971)), some recent results consisting of certain multivalently analytic functions given both in Irmak (2005) and in Irmak (2010) are firstly restated and some of their applications are then pointed out. 1. Introduction, Definitions, and Notations Let denote the class of functions of the form: which are analytic and multivalent in the open unit disk , where is the set of complex numbers. For some useful implications of the main results, there is a need to recall certain well-known definitions relating to geometric function theory. As is known, a function belonging to the general class is said to be multivalently starlike function, multivalently convex function, and multivalently close-to-convex function (with respect to the origin ( )), if it satisfies , , and in the open unit disk , respectively. For the details of the definitions above, one may look over the works in [1–3]. Here and also throughout this paper, the symbol denotes an operator of fractional calculus (i.e., that fractional derivative(s)), which is defined as follows (cf., e.g., [4] and see (also) [2, 5, 6]). Definition 1. Let be an analytic function in a simply connected region of the -plane containing the origin. The fractional integral of order is defined by and the fractional derivative of order is defined by where the multiplicity of involved in (2) and that of in (2) are removed by requiring to be real when . Definition 2. Under the hypotheses of Definition 1, the fractional derivative of order is defined by The aim of this investigation is first to restate some recent results relating to certain multivalently analytic functions and then to point a number of their applications out. For proofs of the main results, the well-known assertions in [7] and [8, p. 33–35] are used (see, also, for similar proofs, [9–11]). The main results include fractional calculus and the main purpose of using fractional calculus is also to extend the scope of the main results and to reveal certain complex inequalities which can be associated with (analytic and) geometric function theory (see, for their details, [1–4, 9]). For certain results determined by fractional calculus, for example, one may refer to the works in [5, 6, 12, 13]. For some results between certain complex inequalities and analytic functions, one can also check the papers, for instance, in the references in [2, 12, 13]. 2. Main Results and Certain Consequences The following assertions (Lemmas 3 and 4 below) will be required for proving the main