Third-order differential inequalities and subordinations in the complex plane

Differential subordinations and inequalities in the complex plane

Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .

Let α e [0,1]p be an analytic function in the unit disc E q and h be analytic and univalent functions in E, Φ be an analytic function in a certain domain D⊃ q (E). The authors study the differential subordinations and where z⊂E Examples of the application of these results are given.

The goal of this paper is to investigate some of the features of differential subordination of analytic univalent functions in an open unit disc. In addition, it has shed light on geometric features such as coefficient inequality, Hadamard product qualities, and the Komatu integral operator. Some intriguing results for third-order differential subordination and superordination of analytic univalent functions have been installed. Then, using the convolution of two linear operators, certain results of third order differential subordination involving linear operators were reported. As a result, we use features of the Komatu integral operator to analyze and study third-order subordinations and superordinations in relation to the convolution. Finally, several results for third order differential subordination in the open unit disk using generalized hypergeometric function have been addressed using the convolution operator.

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.

We investigate univalent functions f ( z ) = z + a 2 z 2 + a 3 z 3 + … f(z)=z+a_2z^2+a_3z^3+\ldots in the unit disk D \mathbb {D} extendible to k k -q.c.(=quasiconformal) automorphisms of C \mathbb {C} . In particular, we answer a question on estimation of | a 3 | |a_3| raised by Kühnau and Niske [Math. Nachr. 78 (1977), pp. 185–192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner’s parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972), pp. 23–43]. Another problem we consider is to find the maximal k ∗ ∈ ( 0 , 1 ] k_*\in (0,1] such that every univalent function f f in D \mathbb {D} having a k k -q.c. extension to C \mathbb {C} with k > k ∗ k>k_* admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that k ∗ > 1 / 6 k_*>1/6 . Moreover, we show that in some cases, Becker’s extension turns out to be the optimal one. Namely, given any k ∈ ( 0 , 1 ) k\in (0,1) , to each finite Blaschke product there corresponds a univalent function f f in D \mathbb {D} that admits a Becker k k -q.c. extension but no k ′ k’ -q.c. extensions to C \mathbb {C} with k ′ > k k’>k .

Key words: In recent years, applications of Bessel functions have been effectively used in the modelling of chemical engineering processes and theory of univalent functions.In this paper, we study a new class of analytic and univalent functions with negative coefficients in the open unit disk defined by Modified Hadamard product with Bessel function. We obtain coefficient bounds and exterior points for this new class.

Convex and alpha-prestarlike subordination chains

A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

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.

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.

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 .

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function.

In this paper, we investigate efficient numerical methods for highly oscillatory integrals with Bessel function kernels over finite and infinite domains. Initially, we decompose the two types of integrals into the sum of two integrals. For one of these integrals, we reformulate the Bessel function Jν(z) as a linear combination of the modified Bessel function of the second kind Kν(z), subsequently transforming it into a line integral over an infinite interval on the complex plane. This transformation allows for efficient approximation using the Cauchy residue theorem and appropriate Gaussian quadrature rules. For the other integral, we achieve efficient computation by integrating special functions with Gaussian quadrature rules. Furthermore, we conduct an error analysis of the proposed methods and validate their effectiveness through numerical experiments. The proposed methods are applicable for any real number ν and require only the first ⌊ν⌋ derivatives of f at 0, rendering them more efficient than existing methods that typically necessitate higher-order derivatives.