On Differential Subordinations for a Class of Analytic Functions Defined by a Linear Operator

We obtain several results concerning the differential subordination between analytic functions and a linear operator defined for a certain family of analytic functions which are introduced here by means of these linear operators. Also, some special cases are considered. 1. Introduction. Let Ꮽ 0 be the class of normalized analytic functions f (z) with f (0) = 0 and f (0) = 1 which are defined in the unit disk ∆ := {z ∈ C : |z| < 1}. Let

(1.7) For two functions f and g analytic in ∆, we say that the function f (z) is subordinate to g(z) in ∆, and write if there exists a Schwarz function w(z), analytic in ∆ with such that In particular, if the function g is univalent in ∆, the above subordination is equivalent to Over the past few decades, several authors have obtained criteria for univalence and starlikeness depending on bounds of the functionals zf (z)/f (z) and 1+zf (z)/f (z).See [4,5,7] and the references in [7].In [2,6], certain results involving linear operators were considered.In this paper, we obtain sufficient conditions involving for functions to satisfy the subordination Also, we obtain sufficient conditions involving for functions to satisfy the subordination , where D n f (z) is the Ruscheweyh derivative of f (z), our results can be specialized to the Ruscheweyh derivative and we omit these details.Note that the Ruscheweyh derivative of order δ is defined by or, equivalently, by (1.17) In our present investigation, we need the following result of Miller and Mocanu [3] to prove our main results.
2. Main results.We begin with the following.
Proof.Define the function p(z) by Then, clearly, p(z) is analytic in ∆.Also, by a simple computation, we find from (2.4) that By making use of the familiar identity (1.7) in (2.5), we get By using (2.4) and (2.6), we obtain (2.7) In view of (2.7), the subordination (2.2) becomes and this can be written as (1.19),where (2.9) Note that ϑ(w), ϕ(w) are analytic in C. Since β ≠ 0, we have ϕ(w) ≠ 0. Let the functions Q(z) and h(z) be defined by (2.10) In light of hypothesis (2.1) stated in Theorem 2.1, we see that Q(z) is starlike and The result of Theorem 2.1 now follows by an application of Theorem 1.1.

Note that
(2.12) By taking a = c = 1 in Theorem 2.1 and after a change in the parameters, we have the following.
Corollary 2.2.Let α be a real number, 1 + α > 0, and let q(z) be univalent in ∆, and let it satisfy and q(z) is the best dominant.
If we take (2.17) then p(z) ≺ q(z) and q(z) is the best dominant.
By using Lemma 2.3, or from Theorem 2.1, we have the following.
By using Theorem 1.1, we obtain the following.
By taking a = c = 1 in Theorem 2.5 and after a suitable change in the parameters, we have the following.

Proof. Define the function p(z) by
(2.36) Then, clearly, p(z) is analytic in ∆.Also, by a simple computation together with the use of the familiar identity (1.7), we find from (2.36) that (2.37) Therefore, it follows from (2.36) and (2.37) that ( (2.41) In light of hypothesis (2.33) stated in Theorem 2.7, we see that Q(z) is starlike and Since ϑ and ϕ satisfy the conditions of Theorem 1.1, the result follows by an application of Theorem 1.1.
By taking a = c = 1 in Theorem 2.7 and after a suitable change in the parameters, we have the following.
If we take (2.16) and γ = 1 in Corollary 2.8, we obtain a recent result of Singh [7, Theorem 1(iii), page 571] and, by setting Theorem 2.9.Let α ≠ 0 and γ be real numbers, (a + 1)αγ < 0. Let q(z) ∈ Ꮽ be univalent in and let it satisfy the following condition for z ∈ : and q(z) is the best dominant.
The proof of this theorem is similar to that of Theorem 2.1 and hence it is omitted.By taking a = c = 1 in Theorem 2.9 and after a suitable change in the parameters, we have the following.
Corollary 2.10.Let 0 ≤ α ≤ 1 and q(z) be univalent in ∆ and let them satisfy then (2.15) holds and q(z) is the best dominant. Let (2.51) After a change of variable in (2.51), we get and q(z) is the best dominant.
The proof of this theorem is similar to that of Theorem 2.1 and therefore it is omitted.By taking a = c = 1 in Theorem 2.11 and after a suitable change in the parameters, we have the following.Corollary 2.12.Let 0 ≤ α ≤ 1 and q(z) be univalent in ∆ and let them satisfy (2.49) and q(z) is the best dominant.

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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