© Hindawi Publishing Corp. COEFFICIENTS ESTIMATES FOR FUNCTIONS IN Bn(α)

We consider functions f , analytic in the unit disc and of the 
normalised form f ( z ) = z + ∑ k = 2 ∞ a k z k . For 
functions f ∈ B n ( α ) , the class of functions involving 
the Sălăgean differential operator, we give some 
coefficient estimates, namely, | a 2 | , | a 3 | , and | a 4 | .


Introduction.
Let A be the class of functions f which are analytic in the unit disc D = {z : |z| < 1} and are of the form ( For functions f ∈ A, we introduce the subclass B n (α) given by the following definition.
For n = 1, B 1 (α) denotes the class of Bazilević functions with logarithmic growth studied [4,6,7], amongst others.In [2], the author established some properties of the class B n (α) including showing that B n (α) forms a subclass of S, the class of all analytic, normalized, and univalent functions in D. The class B 0 (α) was initiated by Yamaguchi [8].

Preliminary results.
In proving our results, we need the following lemmas.However, we first denote P to be the class of analytic functions with a positive real part in D.

Lemma 2.1. Let p ∈ P and let it be of the form p(
Lemma 2.2 (see [3]).If the functions (2.3)
Inequality (3.4) suggests that there exists p ∈ P such that for z ∈ D, and integration gives Now, repeating the process, we are able to establish the following relation which holds in general for any k = 0, 1, 2,...,n In particular, when n = k, we have On comparing coefficients in (3.9) with f (z) = z + ∞ j=2 a j z j , we obtain (3.10) ) (3.12) Inequality (3.1) follows easily from (3.10) for all α > 0 since Eliminating a 2 in (3.11), we have where we used Lemma 2.1(ii) with First, we consider the case 0 < α < 1/2.Applying the triangle inequality with Lemma 2.1(i) in (3.15) results in the inequality which is the first inequality in (3.3).
For the case 1/2 ≤ α < 1, we use Carathéodory-Toeplitz result which states that for some ε with |ε| < 1, (3.17) Thus, (3.15) becomes (3.18) We then have where and the result follows trivially when using Finally, we consider (3.3) for the case α ≥ 1.Here, we use a method introduced by Nehari and Netanyahu [3] which was also used by Singh [6] and the author in [1].
First, let h and g be defined as in Lemma 2.3, and since p ∈ P , Lemma 2.2 indicates that also belongs to P .Next, it follows from (2.2) that, with g replaced by G, Rewriting (3.15) as and comparing it with (3.23), the required result is easily obtained since, by Lemma 2.3, This however is only true if we can show the existence of functions h and ψ in P where ψ(z) = 1 + g(z).To be simple, we choose ψ(z) = (1+z)/(1−z).Thus, now it remains to construct and show that an h ∈ P .Now since g 1 = g 2 = g 3 = 2, it follows from (3.23) and (3.24) that However, from (2.1), we have

.28)
Solving for h 1 by eliminating γ 1 from (3.25) and (3.27), we obtain Quite trivially, it can be seen that In a similar manner, eliminating γ 2 from (3.26) and (3.28) and using h 1 given by (3.29), we have Next, we construct h by first setting it to be of the form with It is readily seen that for α ≥ 1, both µ 1 and µ 2 are nonnegative and µ 1 +µ 2 = 1.Further, with a little bit of manipulation, it can be shown that |λ| ≤ 1 and the coefficients of z and z 2 in the expansion of h are respectively those given by (3.29) and (3.30).Hence h ∈ P and thus |a 4 | ≤ 2α n−1 /(3 + α) n , the second inequality in (3.3).This completes the proof of Theorem 3.1.

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.