CERTAIN CONVEX HARMONIC FUNCTIONS

We define and investigate a family of complex-valued harmonic
convex univalent functions related to uniformly convex analytic
functions. We obtain coefficient bounds, extreme points,
distortion theorems, convolution and convex combinations for this
family.


Introduction.
A continuous complex-valued function f = u + iv defined in a simply connected complex domain Ᏸ ⊂ C is said to be harmonic in Ᏸ if both u and v are real harmonic in Ᏸ.Consider the functions U and V analytic in Ᏸ so that u = U and v = V .Then the harmonic function f can be expressed by where h = (U + V )/2 and g = (U − V )/2.We call h the analytic part and g the coanalytic part of f .If the co-analytic part of f is identically zero then f reduces to the analytic case.
The mapping z f (z) is sense-preserving and locally one-to-one in Ᏸ if and only if the Jacobian of f is positive (see [1]), that is, if and only if Let Ᏼ denote the family of functions f = h+ ḡ which are harmonic, sense-preserving, and univalent in the open unit disk ∆ = {z : |z| < 1} with h(0) = f (0) = f z (0) − 1 = 0. Thus, we may write Also let Ᏼ denote the subclass of Ᏼ consisting of functions f = h + ḡ so that the functions h and g take the form Recently, Kanas and Wisniowska [5] (see also Kanas and Srivastava [4]), studied the class of k-uniformly convex analytic functions, denoted by k-ᐁᏯᐂ, 0 ≤ k < ∞, so that h ∈ k-ᐁᏯᐂ if and only if For real φ we may let ζ = −kze iφ .Then condition (1.5) can be written as Now considering the harmonic functions f = h + ḡ of the form (1.3) we define the family ᏴᏯᐂ(k, α), 0 ≤ α < 1, so that f = h + ḡ ∈ ᏴᏯᐂ(k, α) if and only if Notice that if g ≡ 0 and α = 0 then the family ᏴᏯᐂ(k, α) defined by (1.7) reduces to the class k-ᐁᏯᐂ of k-uniformly convex analytic functions defined by (1.5).If we, further, let k = 1 in (1.5), we obtain the class of uniformly convex analytic functions defined by Goodman [2].A geometric characterization of the general family ᏴᏯᐂ(k, α) is an open question.
In Section 2, we introduce sufficient coefficient bounds for functions to be in ᏴᏯᐂ(k, α) and show that these bounds are also necessary for functions in ᏴᏯᐂ(k, α).In Section 3, the inclusion relation between the classes k-ᐁᏯᐂ and ᏴᏯᐂ(k, α) is examined.Extreme points and distortion bounds for ᏴᏯᐂ(k, α) are given in Section 4. Finally, in Section 5, we show that the class ᏴᏯᐂ(k, α) is closed under convolution and convex combinations.
Here we state a result due to Jahangiri [3], which we will use throughout this paper.
Theorem 1.1.Let f = h + ḡ with h and g of the form (1.3).

Coefficient bounds.
First we state and prove a sufficient coefficient bound for the class ᏴᏯᐂ(k, α).
and hence f is sense-preserving and convex univalent in ∆.Now, we only need to show that if (2.1) holds then (2.2)

Using the fact that (w) ≥ α if and only if |1−α +w| ≥ |1+α −w| it suffices to show that
where 3), we obtain ( The harmonic functions where The functions of the form (2.5) are in Next we show that the bound (2.1) is also necessary for functions in ᏴᏯᐂ(k, α).
Theorem 2.2.Let f = h + ḡ with h and g of the form (1.4).Then f ∈ ᏴᏯᐂ(k, α) if and only if Proof.In view of Theorem 2.1, we only need to show that f ∉ ᏴᏯᐂ(k, α) if condition (2.7) does not hold.We note that a necessary and sufficient condition for f = h+ ḡ given by (1.4) to be in ᏴᏯᐂ(k, α) is that the coefficient condition (1.7) to be satisfied.Equivalently, we must have (2.8) Upon choosing the values of z on the positive real axis where 0 ≤ z = r < 1, the above inequality reduces to If condition (2.7) does not hold then the numerator in (2.9) is negative for r sufficiently close to 1. Thus there exists z 0 = r 0 in (0, 1) for which the quotient (2.9) is negative.This contradicts the required condition for f ∈ ᏴᏯᐂ(k, α) and so the proof is complete.

Inclusion relations.
As mentioned earlier in the proof of Theorem 2.1, the functions in ᏴᏯᐂ(k, α) are convex harmonic in ∆.In the following example we show that this inclusion is proper.
More generally, we can prove the following theorem.
To show that the inclusion is proper, consider the harmonic functions On the contrary, by Theorem 2.2, (3.7)

Extreme points and distortion bounds.
Using definition (1.7), and according to the arguments given in [3], we obtain the following extreme points of the closed convex hulls of ᏴᏯᐂ(k, α) denoted by clcoᏴᏯᐂ(k, α).Theorem 4.1.Let f be the form of (1.4) In particular, the extreme points of ᏴᏯᐂ(k, α) are {h n } and {g n }.
Similarly, follows the distortion bounds for functions in ᏴᏯᐂ(k, α).
If we let r → 1 in the left-hand inequality of Theorem 4.2 and collect the like terms, we obtain the following theorem.

Convolutions and convex combinations. For harmonic functions
we define the convolution of f and F as In the following theorem we examine the convolution properties of the class ᏴᏯᐂ(k, α).
Proof.Express the convolution of f and F as that given by (5.1) and note that |A n | ≤ 1 and |B n | ≤ 1.Now the theorem follows upon the application of the required condition (2.7).
The convex combination properties of the class ᏴᏯᐂ(k, α) is given in the following theorem.
Proof.For i = 1, 2,..., suppose that f i ∈ ᏴᏯᐂ(k, α) where f i is given by f Now, the theorem follows by (2.7) upon noting that

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.