STARLIKENESS ASSOCIATED WITH PARABOLIC REGIONS

A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity zf′(z)/f(z) lies in a given parabolic region in the right half-plane. Denote the class of such functions by PS∗(ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS∗(ρ) are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szego coefficient functional and investigate convolution properties for PS∗(ρ).


Introduction
Let A denote the class of analytic functions f in the open unit disk U = {z : |z| < 1} and let f be normalized so that f (0) = f (0) − 1 = 0. In [4], Goodman introduced the class UCV of uniformly convex functions consisting of convex functions f ∈ A with the property that for every circular arc γ contained in U, with center also in U, the image arc f (γ) is a convex arc. He derived a two-variable characterization of functions in UCV, that is, f ∈ A belongs to UCV if and only if for every pair (z,σ) ∈ U × U, Ma and Minda [6] and Rønning [10] independently developed a one-variable characterization that f ∈ UCV if and only if for every z ∈ U, Several authors have studied the classes above, amongst which the authors of [4,6,7,8,9,10,12].
In [9], the class PS * was generalized by looking at functions f ∈ A satisfying (1. 4) In this paper, we continue the investigation of this generalized class but under a slight modification of parameter. For 0 ≤ ρ < 1, let Ω ρ be the parabolic region in the right halfplane (1.5) The class of parabolic starlike functions of order ρ is the subclass Similarly, a function f ∈ A belongs to UCV(ρ) if and only if for every pair (z,σ) in the polydisk U × U, A function f ∈ UCV(ρ) is called an uniformly convex function of order ρ. Thus the classes discussed earlier correspond to UCV = UCV(1/2) and PS * = PS * (1/2). In [5], Lee showed that .
In the present paper, we continue the study of PS * (ρ) realized by Ali and Singh [3], and more recently by Aghalary and Kulkarni [1]. We give examples of functions in the class PS * (ρ), and establish subordination results, which yield sharp growth, covering and distortion theorems. Sharp bounds on the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szegö coefficient functional and examine convolution properties for PS * (ρ).

Preliminary results
From its definition, it is clear that the class PS * (ρ) is contained in the class S * (ρ) of starlike functions of order ρ, that is, (z f (z)/ f (z)) > ρ, z ∈ U. It is also fairly immediate that PS * (ρ) is related to the class of strongly starlike functions, where a function f ∈ A is said to be strongly starlike of order α, We state the relation in the theorem below.
A sufficient condition for a function f to be parabolic starlike of order ρ is given by the following theorem.
Proof. The given condition implies that 3) The following two examples are now easily established from Theorem 2.2. where (λ) n is the Pochhammer symbol defined by Ali and Singh [3] showed that the normalized Riemann mapping function q ρ from U onto Ω ρ is given by (2.6) Here Since the latter sum is bounded above by 1 + (1/2)log(2n − 1) (see [6]) an upper bound for each coefficient is given by However these bounds do not yield sharp coefficient estimates for the class PS * (ρ). We will return to the coefficient problem in the next section.
In [8], Ma and Minda established a general result that leads to the following result.

Equality in (b), (c), and (d) holds for some z = 0 if and only if f is a rotation of k.
Since the function k is continuous in U, −k(−1) = lim r→1 −k(−r) and k(1) = lim r→1 k(r) exist. Rønning [9] established the following corollary.

Coefficient bounds
We first give another sufficient condition for a function f to belong to PS * (ρ).
n=2 (a n /n)z n . In view of (1.8), it suffices to show that g ∈ UCV(ρ). Since We next consider the problem of finding 3) If f (z) = z + a 2 z 2 + a 3 z 3 + ··· ∈ PS * (ρ) and h(z) = z f (z)/ f (z), then there exists a Schwarz function w defined in U with w(0) = 0, |w(z)| < 1, and satisfying Since q ρ is univalent in U and h ≺ q ρ , the function belongs to the class P consisting of analytic functions p in the unit disk U with positive real part such that p(0) = 1 and p(z) > 0, z ∈ U. In other words, While (3.5) gives a n in terms of the coefficients b k , (3.7) expresses the b k 's in terms of the coefficients c m 's and B m 's. It is now easily established that Thus the coefficient estimates for PS * (ρ) may be viewed in terms of nonlinear coefficient problems for the class P.
We now introduce the following functions in PS * (ρ). Define k n ,G,H ∈ A, respectively, by It is clear from (3.4) that k n ,G,H ∈ PS * (ρ), and that k 2 (z) = k(z). Since we find that On the other hand, Ali and Singh [3] proved that which also yields the sharp order of growth |a n | = O(1/n). From a result of Ma and Minda [8], we can also deduce the following solution to the Fekete-Szegö coefficient functional over the class PS * (ρ). We will omit the details.
Theorem 3.2. Let f (z) = z + a 2 z 2 + a 3 z 3 + ··· ∈ PS * (ρ). Then (3.13) The above estimates can be used to determine sharp upper bounds on the second and third coefficients, respectively, which we will state below. In addition, the sharp bound on the fourth coefficient A 4 is determined with the aid of the following lemma. Lemma 3.3 [2]. Let p(z) = 1 + ∞ k=1 c k z k ∈ P. If 0 ≤ β ≤ 1 and β(2β − 1) ≤ δ ≤ β, then (3.14) In particular, When β = 0, equality holds if and only if Then with equality if and only if f = k or its rotations. Further  Proof. In the light of Theorem 3.2, we are left to finding an estimate on the fourth coefficient. The relation (3.8) gives (3.21) We will apply Lemma 3.3 with The conditions on β and δ are satisfied if Thus |a 4 | ≤ 16(1 − ρ)/3π 2 , with equality if and only if the function p in (3.7) is given by . This implies that f = k 4 .

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: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009