© Hindawi Publishing Corp. A COEFFICIENT INEQUALITY FOR THE CLASS OF ANALYTIC FUNCTIONS IN THE UNIT DISC

The aim of this paper is to give a coefficient inequality for the 
class of analytic functions in the unit disc D = { z | | z | 1 } .


Introduction.
Let Ω be the family of functions ω(z) regular in the disc D and satisfying the conditions ω(0) = 0 and |ω(z)| < 1 for z ∈ D.
Next, for arbitrary fixed numbers A and B, −1 < A ≤ 1, −1 ≤ B < A, denote by P (A,B) the family of functions regular in D such that p(z) is in P (A,B) if and only if for some function ω(z) ∈ Ω and every z ∈ D. The class P (A,B) was introduced by Janowski [3].Moreover, let S * (A,B,b) (b ≠ 0, complex) denote the family of functions regular in D and such that f (z) is in S * (A,B,b) if and only if for some p(z) in P (A,B) and all z in D.
For the aim of this paper we need Jack's lemma [2]."Let ω(z) be a regular in the unit disc with ω(0) = 0, then if |ω(z)| attains its maximum value on the circle |z| = r at a point z 1 , we can write z 1 ω (z 1 ) = kω(z 1 ), where k is real and k ≥ 1."

Coefficient inequality.
The main purpose of this paper is to give sharp upper bound of the modulus of the coefficient a n .Therefore, we need the following lemma.
Proof.The proof of this lemma is in four steps.
Step 1.Let B ≠ 0 and If we take the logarithmic derivative from equality (2.2), we obtain If we use Jack's lemma [2] in equality (2.3), we get
We note that we choose the branch of (1 (2.14) These bounds are sharp because the extremal function is (2.15) Proof.Let B ≠ 0. If we use the definition of the class S * (A,B,b), then we write (2.16) Equality (2.16) can be written by using the Taylor expansion of f (z) and p(z) in the form Evaluating the coefficient of z n in both sides of (2.17), we get (2.18) on the other hand, which can be written in the form To prove (2.14), we will use the induction principle.Now, we consider inequalities (2.21) and

.22)
The right-hand sides of these inequalities are the same because (i) for n = 2, (2.24) Suppose that this result is true for n = p, then we have (2.25) (2.26) from (2.25), (2.26), and induction hypothesis, we have (2.27) If we write x = |b||A − B| > 0, equality (2.27) can be written in the form. (2.28) After the simple calculation from equality (2.28), we get (2.29) Equality (2.29) shows that the result is valid for n = p + 1.Therefore, we have (2.14).

Corollary 2.3. The first inequality of (2.14) can be rewritten in the form
(2.30) This is the coefficient inequality for the starlike function which is well known.

Corollary 2.4. If
(2.32) This inequality was obtained by Aouf [1].Therefore, by giving the special value to A, B, and b, we obtain the coefficient inequality for the classes S

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.