Re

Two new subclasses of uniformly convex and uniformly 
close-to-convex functions are introduced. We obtain inclusion 
relationships and coefficient bounds for these classes.


The class UCC(α). Denote by S the family consisting of functions
that are analytic and univalent in ∆ = {z : |z| < 1} and by C, S * , and K the subfamilies of functions that are, respectively, convex, starlike, and close to convex in ∆.Noor and Thomas [7] introduced the class of functions known as quasiconvex functions.A normalized function of the form (1.1) is said to be quasiconvex in ∆ if there exists a convex function g with g(0) = 0, g (0) = 1 such that for z ∈ ∆, Re zf (z) g (z) > 0. (1.2) Let Q denote the class of quasiconvex functions defined in ∆.It was shown that Q ≺ K, where ≺ denotes subordination, so that every quasiconvex function is close to convex.Goodman [2,3] introduced the classes UCV and UST of uniformly convex and uniformly starlike functions.In [10], Rønning defined the class UCV(α), −1 ≤ α < 1, consisting of functions of the form (1.1) satisfying Geometrically, UCV(α) is the family of functions f for which 1+zf (z)/f (z) takes values that lie inside the parabola Ω = {ω : Re(ω − α) > |ω − 1|}, which is symmetric about the real axis and whose vertex is w = (1 + α)/2.

Since the function
maps ∆ onto this parabolic region, f ∈ UCV(α) if and only if Rønning [10] also defined the family S p (α) consisting of functions zf (z) when f is in UCV(α).In particular, f is in S p (α) if and only if zf (z)/f (z) ≺ q α (z).
More generally, we give the following definition.
Since Re q α (z) > 0, we see that UCC(α) is a subclass of K. To see that UCC(α) also contains the family S p (α), we note for We have thus proved the following inclusion chain.
We next give a sufficient condition for a function to be in UCC(α).

A convolution relation.
We now prove a convolution result for the family UCC(α).But first we need the following lemma.
The above result was proved in [11] for the case α = 0.
and this proves the result.
Since q α (z) is univalent and maps ∆ onto a convex region, we may apply Lemma 3.1.Now we compare the coefficients of z n for the expansion of φ(z) to obtain ... From (3.4), we get and the proof is complete.
In view of the above remark, we obtain from Theorem 1.3 a sufficient coefficient bound for inclusion in the family UQC(α).
We next prove a theorem which shows that every function in UQC(α) is close to convex and hence univalent.We need a result due to Miller and Mocanu [5].Lemma 4.4.Let M(z) and N(z) be regular in ∆ with M(z) = N(z) = 0 and let α be real.If N(z) maps ∆ onto a possibly many-sheeted region which is starlike with respect to the origin, then for z ∈ ∆, an application of Lemma 4.4, with M(z) = zf (z), N(z) = g(z), proves the result.
We close with coefficient estimates for the class UQC(α). ) Proof.Proceeding on the same lines as in the proof of Theorem 3.2, we obtain the result.Remark 4.8.When α = 0, UQC(0) = Q [6] and we see that the bounds are lower than the corresponding bounds for Q in [6].

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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