#### Abstract

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 results.

Lemma 3 (see [7]). *Let be an analytic function in the unit disk with and let . If attains at its maximum value on the circle , then .*

Lemma 4 (see [8]). *Let and suppose that the function satisfies for all and with . If the function is in the class and for all , then .*

By using Lemma 3, we first prove the following theorem.

Theorem 5. *Let , and , and let the function satisfy any one of the following inequalities:**Then
**
where the value of each one of the above complex terms and their certain applications is taken to be its principal value and denotes the set of real numbers.*

*Proof. *First of all, Definition 1 readily provides us with the following fractional derivative formula for a power function:
With the help of (7), let us define the function by
Clearly, the function is analytic in and . Differentiating both sides of the above identity, and if we make use of (8) once again, we easily arrive at
Suppose now that there exists a point such that Applying Jack’s Lemma, we can write Setting and then putting in (9), we obtain
where and . Hence, (10) is a contradiction to the conditions in (5). Consequently, we conclude from definition (8) that
which implies (6). Thus, the proof of the Theorem 5 is completed.

The second theorem (below) is given without proof since its proof is similar to the proof of the Theorem 5.

Theorem 6. *Let , , , , and . If the function in the class satisfies any one of the following conditions:**then
**
where the value of each one of the above complex terms and their certain applications is taken to be its principal value.*

Theorem 7. *Let , , , , , and . If satisfies the following condition:
**
then
*

*Proof. *Define by
It is clear that the function is in the class (with, of course, ) and also it gives us

Let
we then receive that
for all in . Further, for any , and , because of and , we then obtain that
that is, . Therefore, because of the Lemma 4, the function defined in (16) follows the inequality in (15). This completes the desired proof.

Since the proof of the following theorem is similar to the proof of the Theorem 7, the details are here omitted.

Theorem 8. *Let , , , , , and . If the function in the class satisfies the following inequality:**
then
*

It is clear that, by suitably choosing the parameters and also in the all theorems, one may easily receive several interesting and/or important results for analytic and geometric function theory (see, [1, 3], and also the others in the references). Since it is not possible to list all of them, we want to reveal only the useful certain consequences belonging to Theorems 5–8. They are in the following forms.

Letting and —in the Theorem 5, we first get Corollaries 9 and 10, respectively.

Corollary 9. *If a function satisfies the inequality:
**
then
**
that is, is multivalently starlike in .*

Corollary 10. *If a function satisfies the inequality:
**
then
**
that is, is multivalently convex in .*

Setting and —in the Theorem 6, we second obtain Corollary 11 (below).

Corollary 11. *If a function satisfies the inequality:
**
then
**
that is, is multivalently close-to-convex in .*

*Remark 12. *The above three consequences may be comparable with the results given by the papers in [12, 13].

*Remark 13. *Theorems 7 and 8 are equal to Theorems 5 and 6, respectively, when one takes in Theorems 7 and 8.

#### Conflict of Interests

The author declares that there is no conflict of interests.

#### Acknowledgment

The work in this investigation was supported by The Research Project financed by TÜBİTAK (The Scientific and Technological Research Council of TURKEY) under project no. TBAG-U-105T056.