Some Criteria for Univalence of Certain Integral Operators

We derive some criteria for univalence of certain integral operators for analytic functions in the open unit disk. 1. Introduction. Let Ꮽ be the class of the functions f (z) which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and f (0) = f (0) − 1 = 0.

We denote by the subclass of Ꮽ consisting of functions f (z) ∈ Ꮽ which are univalent in U. Miller and Mocanu [1] have considered many integral operators for functions f (z) belonging to the class Ꮽ.In this paper, we consider the integral operators for f (z) ∈ Ꮽ and for some α ∈ C. It is well known that F α (z) ∈ for f (z) ∈ * and α > 0, where * denotes the subclass of consisting of all starlike functions f (z) in U.

Preliminary results.
To discuss our integral operators, we need the following theorems.
Theorem 2.1 [3].Let α be a complex number with Re(α) > 0 and f (z for all z ∈ U, then the integral operator is in the class . Theorem 2.2 [4].Let α be a complex number with Re(α) > 0 and f (z) ∈ Ꮽ.If f (z) satisfies (2.1) for all z ∈ U, then, for any complex number β with Re(β) Re(α), the integral operator with Re(α) > 0, we have that Thus the function f (z) satisfies the condition of Theorem 2.2.Therefore, for Re(β) Re(α), is in the class .

Main results
Theorem 3.1.Let α be a complex number with Re(1/α) = a > 0 and the function Then, for the integral operator is in the class .
Proof.Let 1/α = β.Then we have We consider the function Then the function Noting that h(0) = 0 and applying the Schwarz lemma for h(z), we get and hence we obtain (3.12) from (3.11) and (3.7), we have for z ∈ U. From (3.13) and Theorem 2.1, it follows that (2.4) belongs to the class .By means of (2.4) and (3.5), we have that the integral operator F 1/β (z) is in the class , and hence we conclude that the integral operator F α (z) is in the class .
Example 3.2.If we take the function g(z) = ze z and α = 1/a > 0, then Since the function g(z) satisfies the condition of Theorem 3.1, we have is in the class .
Proof.We have We consider the function which is regular in U.The function for all z ∈ U. We consider the function Proof.From Theorem 3.3 for β = 1/α, the condition Re(β) Re(α) > 0 is identical with |α| < 1 and we have F α,β (z) = F α (z).

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: