© Hindawi Publishing Corp. TECHNIQUES OF THE DIFFERENTIAL SUBORDINATION FOR DOMAINS BOUNDED BY CONIC SECTIONS

We solve the problem of finding the largest domain D for which, under given ψ and q, the differential subordination ψ(p(z),zp′(z))∈D⇒p(z)≺q(z), where D and q(𝒰) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q(𝒰). Also, we find the best dominant of the differential subordination p(z)

The function p 0 is well known; an explicit form of p 1 is due to Rønning [16] and independently Ma and Minda [11]; the remaining cases belong to Kanas [5] and Kanas and Wiśniowska [8].
Let ᏼ denote the well-known Carathéodory class of functions, analytic in the unit disk with positive real part there, so that ᏼ = {p : p analytic in ᐁ, p(0) = 1, Re p(z) > 0}.Denote by ᏼ(p k ) the subclass of the class ᏼ consisting of all functions p such that p(ᐁ) ⊂ p k (ᐁ).
Suppose that functions g and G are analytic in the unit disc ᐁ.The function g is said to be subordinate to G, written g ≺ G (or with ψ, p being analytic and having appropriate normalization, then the subordination is called the first-order differential subordination.The theory of differential subordinations was developed by Miller and Mocanu in their numerous works (cf., e.g., [12,13,14]) and by Eenigenburg et al. [1] and has wide applications in the theory of univalent functions.
We briefly recall one of the problems that characterizes this theory.For more details the reader should consult [13,14].Assume that p is analytic in ᐁ and such that p(0) = a and p(z) ≡ a.Also, let ψ : C 2 C be analytic and such that ψ(p(0), 0) = a.Given ψ and univalent function q, q(0) = p(0), find the "largest" domain D, such that a ∈ D, and the relation holds.The second important problem of this theory is finding the "smallest" function q, when D and ψ are given, and (1.4) holds.Now, we recall some results from the theory of differential subordinations (cf., e.g., [12,13,14]) useful for further investigation (by ᐁ r we denote the open disk centred at the origin and with the radius r ).Lemma 1.1 (see [13]).Let f be analytic in ᐁ, and let g be analytic and univalent on ᐁ, with f (0) = g(0).If f is not subordinate to g, then there exist points z 0 ∈ ᐁ, ζ 0 ∈ ∂ᐁ, and m ≥ 1 for which (1.5) Lemma 1.2 (see [13]).Let ψ : C 2 → C be an analytic function in a domain D ⊂ C 2 , and let p(z) = a + p n z n +••• be analytic in ᐁ, such that where h is analytic and univalent in ᐁ with ψ(p(0), 0) = h(0).If ) The univalent function q is called a dominant of the solution of the differential subordination ψ p(z), zp (z) ≺ h(z), (1.8) or more simply a dominant if p ≺ q for all p satisfying (1.8).A dominant q that satisfies q ≺ q for all dominants q of (1.8) is said to be the best dominant of (1.8).(Note that the best dominant is unique up to the rotation of ᐁ.) Lemma 1.3 (see [1,12,14]).Let β and γ be complex constants, and let h be convex (univalent) in ᐁ, with Re(βh(z) + γ) > 0. Also, suppose that the differential equation has a univalent solution q, which satisfies q ≺ h.If p is analytic in ᐁ with p(0) = h(0) and satisfies then p ≺ q ≺ h in ᐁ, and q is the best dominant of (1.10).
In fact, the above lemmas appear in a more general form, but the above form is sufficient for our considerations.Lemmas 1.3 and 1.4 characterize properties of subordination, known as the subordination of the Briot-Bouquet type, described in detail by Miller and Mocanu in [12,14] and by Eenigenburg et al. in [1].
Numerous subordination results concerning various choices of function q and the domain D in (1.4) appeared in the literature.Most of functions q map the unit disk onto the right half-plane or more general half-plane, the disk and the sector contained in the right half-plane with the vertex at the origin (cf., e.g., [2,3,12,13,14]).The special case of the subordination related to conical domains did not receive the attention it deserves, but some results in this direction are known (cf.[6,10]).In both papers only cases ψ(p, zp ) = p + γzp ≺ h, where function h maps the unit disk onto the region bounded by parabolas and hyperbolas, were considered although the other forms of the function ψ seem to be more useful.For instance, when we put p(z) = zf (z)/f (z) and ψ(p, zp ) = p + zp /p, then the relation ψ(p(z), zp (z)) ∈ D ⇒ p(z) ≺ q(z), for various choices of functions q and D, gives inclusion relations between subclasses of convex and subclasses of starlike functions.Hence, in this case, the function ψ in the form ψ(p, zp ) = p +γzp is useless.
Our principal goal in this paper is to develop the theory of the first-order differential subordinations towards subordinations in conic domains.Such kind of domains is closely related to k-uniformly convex and k-starlike functions (cf.[7,8,9]).We employ geometric techniques to obtain certain differential inclusions when the functions ψ and q (or ψ and a domain D) are given.Mainly, we discuss the case ψ(p, zp ) = p + zp /p and the function h mapping the unit disk onto a domain bounded by a parabola.
In Section 2, we solve the problem of finding the largest domain D for that, under given ψ and q, the condition (1.4) is satisfied.The shape of the domain D is described by the shape q(ᐁ).In Section 3, we present the best dominant of the differential subordination (1.10).Various applications in the theory of univalent functions are given in Section 4.

Subordination results related to conic domains.
Consider first the region bounded by a parabola u = v 2 /2 + 1/2 that is the domain p 1 (ᐁ) with p 1 given by (1.2).Next, we consider the family described by the equality u = v 2 /2 + (2a + 1)/2, a < 1/2, that consists of parabolas with the vertex at w = (2a + 1)/2, symmetric about the real axis (our previous case corresponds to a = 0).The family of domains containing point 1 inside and bounded by those parabolas may be characterized as or equivalently Theorem 2.1.Let p be a function analytic in the unit disk such that p(0 where Ᏸ a is given by (2.1) and a ≥ a 0 = −1/π .
Proof.Suppose, on the contrary, that p ≺ p 1 .Then, by Lemma 1.1, there exist points (2.6) From the above we have (2.7) Observe, by the definition of Ᏸ a , that we will be led to a contradiction if we show that which, after necessary reductions, is equivalent to (2.9) Set t = log 2 x, t ≥ 0. Then the above inequality can be rewritten as if and only if 8/π + 8a ≥ 0, that is, when a ≥ a 0 = −1/π ≈ −0.318309 that leads to the contradiction.The proof is completed.Now, we extend the result of Theorem 2.1 to the case when p(z) + zp (z)/(βp(z) + γ) ∈ Ᏸ a implies p ≺ p 1 .Since the method of the proof is the same as in Theorem 2.1, we omitted the details.
where Ᏸ a is given by (2.1) and where (2.17) Reasoning in the other direction we consider the function that maps the unit disk analytically and univalently onto the parabolic region When b grows, the parabola becomes narrower until it degenerates for b = 1.The function Q b was defined by Rønning in [15].Define now the family of domains, having 1 inside and creating from Ω b by a translation through the vector (a, 0) as follows: Reasoning along the same lines as in Theorem 2.1, in this case, we obtain the following theorem. where (2.22)

Problem of the best dominant.
The second problem of the theory of differential subordinations, mentioned in the introduction, is the problem of finding the "smallest" function q such that (1.9) and (1.10) imply p ≺ q ≺ h.This problem was posed by Miller and Mocanu in [13], repeated among others in [14] and completely solved with reference to Briot-Bouquet differential subordination, although, an explicit form of the best dominant was found only for the functions mapping the unit disk onto a half-plane or a disk (given in term of hypergeometric functions).In this section, employing Miller and Mocanu results, we provide the form of the best dominant of Briot-Bouquet differential subordination in the case when the function h ≡ p k .
In this instance we have the following proposition.
and q is an analytic solution of then q is univalent, p ≺ q ≺ p k , and q is the best dominant of (3.1).
Proof.Since for each k ∈ [0, ∞) the function p k is convex in ᐁ, and Re(βp k (z) + γ) > 0, the assumptions of Lemma 1.4 are satisfied, that guarantees the univalence of a solution of the differential equality (3.2).Moreover, the solution q is the best dominant of the subordination (3.1).Such a solution is given by or equivalently (cf. [12,14]).Thus, by making use of Lemma 1.3, we have p ≺ q ≺ p k with q being the best dominant.
In particular, for β = 1 and γ = 0, one obtains or equivalently In the case when k > 1 we may approximate the domain, being an elliptical domain, by a circular domain.An ellipse must be contained in the disk which is tangent to the ellipse at points common with the real axis.The case of subordinations related to disks is completely solved in the literature.The first attempt was given by Jakubowski and Kamiński [3], but a final result is due to Miller and Mocanu (cf., e.g., [14, page 110]).Since the ellipse intersects the real axis at points k/(k + 1) and k/(k − 1), then such a circle K(S, R) has the center at S = k 2 /(k 2 −1) and a radius R = k/(k 2 −1) and contains the point z = 1 inside.Then the function ϕ : ᐁ → K(S, R) has a form ϕ(z) = k/(k − z), and therefore Proposition 3.1 with β = 1 and γ = 0 yields the following proposition.Proposition 3.2.Let 1 < k < ∞ and let p be analytic in ᐁ, p(0) = 1, and p satisfies (3.1) Proof.Since p satisfies (3.1) and for each fixed and the assertion follows.

Applications.
Functions p k and the class ᏼ(p k ) were constructed with reference to the subclass of the class of univalent functions, called k-uniformly convex, and denoted k-ᐁᏯᐂ (cf.[7,8,9]), so that every domain appeared in Section 2 is strictly related with the class ᏼ(p k ) and, obviously, with k-ᐁᏯᐂ.It should be mentioned that the class k-ᐁᏯᐂ was defined pure geometrically as the class of such functions that map every circular arc, with the center One of the most important cases of k-ᐁᏯᐂ functions is the uniformly convex functions case which corresponds to the choice k = 1.In this case, condition (1.1), when setting u + iv = 1 + zf (z)/f (z), z ∈ ᐁ, means that the range of the expression 1 + zf (z)/f (z) is the region bounded by a parabola u = v 2 /2 + 1/2.
Applying Theorem 2.2 and proving similarly as above, we obtain the following theorem.
Making use of Theorem 2.3 by a simple substitution, like in the proof of Theorem 4.1, we have the following theorem.(2.19) and a ≥ a 1 (the value a 1 is described by (2.21)).Then zf (z)/f (z) ≺ Q b (z) with Q b given by (2.18).
Conversely to the presented applications we may assume that and from this, by making use of results of Section 3, we conclude the containment results about k--.
On applying Proposition 3.1 (for β = 1 and γ = 0) and Proposition 3.2, we have the following relations between the classes k-ᐁᏯᐂ and k-below.
, where q is given by (3.6).
We presented applications of results of Sections 3 and 4 only for k-ᐁᏯᐂ and k-functions.However, by setting p and so forth, in the propositions and the theorems of the mentioned sections, we may obtain many more results related to conical domains.

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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onto a convex arc.With reference to the class k-ᐁᏯᐂ, Kanas and Wiśniowska defined the class k--= {g ∈ Ᏼ : g = zf , f ∈ k-ᐁᏯᐂ}, called the class of k-starlike functions.Here Ᏼ denotes the class of functions analytic in the unit disk and such that f (0) = f (0) − 1 = 0.