Classes of Uniformly Starlike and Convex Functions

Some classes of uniformly starlike and convex functions are introduced. The geometrical properties of these classes and their behavior under certain integral operators are investigated. 1. Introduction. Let A denote the class of functions of the form f (z) = z+ ∞ n=2 a n z


Introduction.
Let A denote the class of functions of the form f (z) = z+ ∞ n=2 a n z n which are analytic in the open unit disk U = {z : |z| < 1}.A function f in A is said to be starlike of order β, 0 ≤ β < 1, written as f ∈ S * (β), if Re[(zf (z))/(f (z))] > β.A function f ∈ A is said to be convex of order β, or f ∈ K(β), if and only if zf ∈ S * (β).
Let SD(α, β) be the family of functions f in A satisfying the inequality We note that for α > 1, if f ∈ SD(α, β), then zf (z)/f (z) lies in the region G ≡ G(α, β) ≡ {w : Rew > α|w −1|+β}, that is, part of the complex plane which contains w = 1 and is bounded by the ellipse Using the relation between convex and starlike functions, we define KD(α, β) as the class of functions f ∈ A if and only if zf ∈ SD(α, β).For α = 1 and β = 0, we obtain the class KD(1, 0) of uniformly convex functions, first defined by Goodman [1].Rønning [3] investigated the class KD(1,β) of uniformly convex functions of order β.For the class KD(α, 0) of α-uniformly convex function, see [2].In this note, we study the coefficient bounds and Hadamard product or convolution properties of the classes SD(α, β) and KD(α, β).Using these results, we further show that the classes SD(α, β) and KD(α, β) are closed under certain integral operators.

Main results.
First we give a sufficient coefficient bound for functions in SD(α, β).
For the right-hand side and left-hand side of (2.1) we may, respectively, write and similarly Now, the required condition (2.1) is satisfied, since (2.4) The following two theorems follow from the above Theorem 2.1 in conjunction with a convolution result of Ruscheweyh and Sheil-Small [5] and the already discussed relation between the classes SD(α, β) and KD(α, β).
From Theorem 2.3 and the fact that we obtain the following corollary upon noting that given by (2.5).
Similarly, the following corollary is obtained for given by (2.6).
We observed that if α > 1 and if f ∈ SD(α, β), then (zf (z)/f (z)) z∈U ⊂ E, where E is the region bounded by the ellipse 2 with the parametric form where h is given by the normalized function and w is given by (2.7).Conversely, if This proves the following theorem.

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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