© Hindawi Publishing Corp. PROPERTIES OF CERTAIN p-VALENTLY CONVEX FUNCTIONS

A subclass Ꮿ p (λ, µ) (p ∈ N, 0 < λ < 1, −λ µ < 1) of p-valently convex functions in the open unit disk U is introduced. The object of the present paper is to discuss some interesting properties of functions belonging to the class Ꮿ p (λ, µ).

A function f (z) ∈ Ꮽ 1 is said to be uniformly convex in U if f (z) is in the class and has the property that the image arc f (γ) is convex for every circular arc γ contained in U with center at t ∈ U. We also denote by ᐁ the subclass of Ꮽ 1 consisting of all uniformly convex functions in U. Goodman [2] has introduced the class ᐁ and given that f (z) ∈ Ꮽ 1 belongs to the class ᐁ if and only if Ma and Minda [3] and Rønning [5] have showed a more applicable characterization for ᐁ.We state this as the following theorem.
In this paper, we investigate the following subclass of Ꮽ p .
Let f (z) and g(z) be analytic in U. Then we say that f (z) is subordinate to g(z) in U, written f (z) ≺ g(z), if there exists an analytic function 2. Subordination properties.Our first result for properties of functions f (z) ∈ Ꮽ p is contained in the following theorem. (2.2) Proof.Let 1+zf (z)/f (z) = w and w = u+iv.Then inequality (1.6) can be written as (2.3) By computing, we find that inequality (2.3) is equivalent to Thus the domain of the values of 1 + zf (z)/f (z) for z ∈ U is contained in D = w = u + iv : u and v satisfy (2.4) and (2.5) . (2.6) In order to prove our theorem, it suffices to show that the function h(z) given by (2.2) maps U conformally onto the domain D.
Consider the transformations where β = arccos λ and w 2 = w 1 + w 2 1 − 1 is the inverse function of It is easy to verify that composite function t = t(w) maps D + defined by D + = w = u + iv : u and v satisfy (2.4), (2.5), and v > 0 (2.9) conformally onto the upper-half plane Im(t) > 0 so that w = p corresponds to t = 1 and With the help of the symmetry principle, this function t = t(w) maps D conformally onto the domain maps U onto G, we see that (2.12) maps U onto D with h(0) = p.Hence the proof of the theorem is completed.
Theorem 2.1 gives the following corollaries.
Proof.Using (2.5) in the proof of Theorem 2.1 and noting that Re 1 as z = Re(z) → −1, we have the corollary.
The bound in (2.15) is sharp with the extremal function f 0 (z) given by (2.13).
Proof.Let the function h(z) be defined by (2.4).Then h(U) = D and an easy calculation yields that for −λ < µ < λ < 1.Therefore, the corollary follows immediately from Theorem 2.1.
Next we derive the following theorem.

.18)
The bound in (2.18) is sharp with the extremal function f 0 (z) given by (2.13).
Proof.Since the function h(z) − p is univalent and starlike (with respect to the origin), by Theorem 2.1 and the result due to Suffridge [6, Theorem 3], we have (2.21) Thus, as the special case of Theorem 2.4, we have that if and the result is sharp.

Coefficient inequalities
Proof.It can be easily verified that where h(z) is given by (2.2).Since it follows from (3.3) and Theorem 2.1 that It is well known that if for g(z) ∈ , then (cf.Duren [1]) we get (3.2).Also the bound in (3.2) is sharp for the function f 0 (z) given by (2.13).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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