Certain nth Order Differential Inequalities in the Complex Plane

We study the stability of a linear autonomous difference equation with two (generally speaking, complex) coefficients. The starting point of the study is the Schur-Kohn theorem on the location of the roots of the characteristic equation with respect to the unit disk in the complex plane. To construct the domain of exponential stability in the parameter space, we use the D decompodition method, which consists in constructing curves (or surfaces) such that the number of roots of the characteristic equation outside the unit disk changes when passing through the curves; then the area is determined, which corresponds to the zero number of such roots; this is the area of stability. We implement this scheme for the above difference equation: geometric stability criteria are found and the domains of exponential stability in a four-dimensional space of coefficients are described, as well as their three-dimensional, two-dimensional and one-dimensional sections. The Lyapunov stability is studied separately, which is corresponded by the domain of exponential stability supplemented by a part of its boundary. To describe Lyapunov stability exactly we use a "multiplicity curve", which is a line such that all its points correspond to multiple roots of the characteristic equation. In addition, we find and construct a domain of absolute stability with respect to one of the parameters of the initial equation. For this domain, we formulate criteria of exponential stability and Lyapunov stability.The results obtained can be applied to the study of processes in physics, technology, economics, biology, which are modeled using discrete models in the form of difference equations.

There are over one hundred articles in the literature dealing with second-order differential inequalities and differential subordinations in the complex plane, and only two articles dealing with very narrow classes of third-order results. This article addresses a much more general class of third-order differential inequalities and subordinations. Let Ω and Δ be given sets in , let p be an analytic in the unit disc with . The authors determine the conditions on an admissible class of operators , so that the third-order differential subordination implies . Applications of these results to third-order differential inequalities, third-order differential subordinations, univalent functions and Bessel functions are presented.

Convex and alpha-prestarlike subordination chains

Recently, complex fuzzy sets have become a powerful tool for expanding the range of fuzzy sets that are located in a unit disk within complex plane. In this article, the complex fuzzy numbers are discussed and implemented to handle complex fractional fuzzy partial differential equation involving complex time-fractional fuzzy heat equation in sense of Hukuhara differentiability. After that, an explicit finite difference scheme known as the forward time center space are implemented to solve complex time-fractional fuzzy heat equations. The uncertainty present in our problem appears in the boundary and initial conditions as well as coefficients in both amplitude and phase terms simultaneously where the Convex normalized triangular fuzzy numbers are extended to unit disk in the complex plane. The properties and benefits of complex fuzzy set theory have been employed in the proposed numerical methods. A numerical experiment is used to showcase the efficiency and practicality of the presented method. The obtained outcomes are found to be consistent with the exact solution and related theoretical concepts.

State space methods for solving tangential interpolation problems for matrix functions in a given class have been developed extensively. These are interpolation problems in which the values in certain directions of the interpolant matrix functions, and several of their derivatives, are prescribed at given points in the complex plane. For the most part, the matrix functions under consideration are rational functions or polynomials (perhaps with symmetries); however, other important classes of matrix functions have been studied as well, notably, the Nevanlinna classes on the unit disk and the upper half-plane, and the Hardy class H/sub 2/ on the unit disk. This development is motivated by many important applications in systems and control. We present some recent results and open problems in this area.

INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.

Differential subordinations and inequalities in the complex plane

In this paper, we obtain some new sufficient conditions for starlikeness of order $\alpha$ of biholomorphic mappings on the unit ball in $C^n$ or a complex Hilbert space $X$ by using differential inequalities. We also obtain a distortion theorem and a covering theorem. As their special case, we obtain some sufficient conditions for starlikeness of order $\alpha$ of analytic functions on the unit disc in the complex plane $C$, which generalize some results of P. T. Mocanu and G. Oros.

Let Ω be a domain in the complex plane \(\mathbb{C}\) containing the origin and let \({\cal P}_{n}^{0}(\Omega)\) denote the set of all polynomials P of degree ≤ n satisfying the conditions \(P(0)=0\) and \(P(\mathbb{D})\subset \Omega\), where \(\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}\) is the unit disk in the complex plane. The maximal range, denoted by Ω n , is then defined as the union of all sets \(P(\mathbb{D}), P \in {\cal P}_{n}^{0}\) .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

We study the integration operator between the growth spaces $$H_w^\infty $$ and $$H_u^\infty $$ on the complex plane with arbitrary radial weights w and u. Also, we investigate the differentiation operator on the Hardy growth spaces $$H_w^p$$ , $$0<p<\infty $$ , defined on the unit disk or on the complex plane. As in the case $$p=\infty $$ , the log-convex weights w play a special role in the problems under consideration.

In this article, we will discuss the type problem to judge whether a given infinitely sheeted covering surface of the complex sphere determined by the sequence of pasting arcs γ n in the complex plane is parabolic, i.e. , or not, in terms of Γ. Here the sequence of simple arcs γ n is supposed to satisfy the following two conditions: for every and the sequence of end points of γ n converges to ∞, the point at infinity of . Then is constructed by pasting to crosswise along γ n for every with γo = ∅ so that is either the unit disc or the complex plane. Then our type problem specifically asks whether the convergence to zero of the sequence of capacities cap of γ n relative to the region implies the parabolicity of or not. The purpose of this article is to answer affirmatively to the above question under the additional assumption that two end points of each pasting arc γ n is symmetric about the imaginary axis of for every .

In this paper, we give some sufficient and necessary conditions for subordination of the solution of the Briot–Bouquet differential equation: where δ and λ are real numbers with δ≠0, δ+λ>0, and U is the unit disc in the complex plane C. Using these subordinations, we obtain some sufficient and necessary conditions that the integral operators and preserve starlikeness, where α, β, δ, and γ are real numbers with α+δ=β+γ>0, αβ≠0.

The influence of the normality of the coefficient A(z) of the differential equation f(k) + A(z)f = 0 on a solution f and also the influence of the normality of a solution f on A(z) are investigated in the unit disk. In particular, an estimate of P. Lappan is used to determine restrictions on the growth of a meromorphic function A(z) when a solution f is α‐normal.

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \\overline {g} $ where h and g are analytic functions. If $ f \\in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \\in K_H^0 $ . For $ f \\in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\ ilde {*}\\varphi = h*\\varphi + \\overline {g*\\varphi } $ is found. With } a strip mapping, $ f\ ilde {*}\\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \\in K_H^0 $ which take the unit disk to regular n -gons.

Let A be the set of functions regular in the unit disc U and Aq the set of all functions / e A which satisfy /(0) = 1.For V C Aq define the dual set V* = {f£A0\f * g * 0 for all g e V, z e U}, V** = (V*f.Here f * g denotes the Hadamard product.THEOREM.Let V C AQ have the following properties: (i) V is compact, (ii) /G V implies f(xz) e V for all \x\ < 1.77ien X(K) = \(V ) for all continuous linear functionals \ on A. This theorem has many applications to functions in A which are defined by properties like bounded real part, close-to-convexity, univalence etc.Introduction.Let f(z) = 2k=0 akzk be regular for |z | < 7?,, g(z) = 2"=0 bkzk regular for |z | <R2.The Hadamard product of /and g is defined by (/ * g) (z) = 2^=0 akbkzk.This new function is regular for |z | < 7? ,7?2.Let U be the unit disc {|z| < 1}, U its closure.By ^4,^4 we denote the set of functions regular in U and U respectively.Furthermore let A0 be the subset of A whose elements f(z) are normalized by /(0) = 1.In this paper we introduce a certain duality between subsets of A0 by means of the Hadamard product: Definition.For V C A0, we call the set V* = {g G A0 \f * g ¥= 0 for all/£ V.zGU] dual to V.We are interested in relations between V and V** = (V*)*.In general, V** is much greater than V, but a lot of properties of V remain valid in V**.Our main result, although its proof is simple, has many applications to classes of functions which are defined by properties like bounded real part, convexity, starlikeness, close-to-convexity, univalence etc.A special case of our method was already treated in [10].The applications that we give in the second section of this paper are numerous, and some new and perhaps interesting information is given.Nevertheless they are merely thought to be examples illustrating the method, and much work remains to be done.