© Hindawi Publishing Corp. ON A CLASS OF HOLOMORPHIC FUNCTIONS DEFINED BY THE RUSCHEWEYH DERIVATIVE

By using the Ruscheweyh operator D m f ( z ) , z ∈ U , we will introduce a class of holomorphic functions, denoted by 
 M n m ( α ) , and obtain some inclusion relations.


Introduction and preliminaries.
Denote by U the unit disc of the complex plane ( Let Ᏼ(U) be the space of holomorphic functions in U .We let with A 1 = A.
We let Ᏼ[a, n] denote the class of analytic functions in U of the form If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or f (z) ≺ g(z), if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for any z ∈ U , such that f (z) = g(w(z)), for z ∈ U.
If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (U) ⊂ g(U).
the class of normalized convex functions in U .We use the following subordination results.Lemma 1.1 (Miller and Mocanu [2,page 71]).Let h be a convex function with h(0) = a and let γ ∈ C * be a complex with Re γ ≥ 0. If p ∈ Ᏼ[a, n] and The function g is convex and is the best (a, n) dominant.
Lemma 1.2 (Miller and Mocanu [1]).Let g be a convex function in U and let where α > 0 and n is a positive integer.If and this result is sharp.
Definition 1.3 [4].For f ∈ A and m ≥ 0, the operator D m f is defined by where * stands for convolution.

Main results
Definition 2.1.If α < 1 and m, n ∈ N, let M m n (α) denote the class of functions f ∈ A n which satisfy the inequality Re D m f (z) > α. (2.1) where (2.3) . By using the properties of the operator D m f (z), we have (2.4) Differentiating (2.4), we obtain which is equivalent to (2.8) By using Lemma 1.1, we have where The function g is convex and is the best dominant.
For n = 1, this result was obtained in [3].
Theorem 2.3.Let g be a convex function, g(0) = 1, and let h be a function such that

.13)
If f ∈ A n and verifies the differential subordination (2.15) we obtain (2.17) If we let p(z) = (D m f ) (z), then we obtain and (2.14) becomes
Proof.A simple application of the differential subordination technique [1,2] shows that the function g is convex.From we obtain (2.26)

.30)
If f ∈ A n and verifies the differential subordination (2.32) Proof.We let p(z) = D m f (z)/z, z ∈ U, and we obtain By differentiating, we obtain

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.