Distortion Inequalities Research Articles

A Certain Subclass On (P, Q) Sakaguchi Type Functions

In the present article we introduce classes of functions defined by using the concept of (p, q)-derivative and Sak- aguchi functions. In particular we derive coefficient inequalities, distortion inequalities, coefficient estimates etc.

Correction to: On the heterogeneous distortion inequality

Correction to: On the heterogeneous distortion inequality

Pascu-Rønning Type Meromorphic Functions Based on Sălăgean-Erdély–Kober Operator

In the present investigation, we introduce a new class of meromorphic functions defined in the punctured unit disk Δ*:={ϑ∈C:0<|ϑ|<1} by making use of the Erdély–Kober operator Iς,ϱτ,κ which unifies well-known classes of the meromorphic uniformly convex function with positive coefficients. Coefficient inequalities, growth and distortion inequalities, in addition to closure properties are acquired. We also set up a few outcomes concerning convolution and the partial sums of meromorphic functions in this new class. We additionally state some new subclasses and its characteristic houses through specializing the parameters that are new and no longer studied in association with the Erdély–Kober operator thus far.

Existence of Quasiconformal Maps with Maximal Stretching on Any Given Countable Set

Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative Hölder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Hölder continuity. In this work, given any bounded countable set in $$\mathbb {R}^d$$ , we will construct an example of a K-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into $$\mathbb {R}^d$$ .

Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in the open unit disk E is given. Certain properties of this subclass, such as its structural formula, necessary and sufficient conditions, coefficient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results, are investigated. Some new and known consequences of our main results as corollaries are also highlighted.

On certain subclasses of multivalent functions with varying arguments of coefficients

In this paper we introduce and study new classes VM_{p,η}(λ,α,β) and VN_{p,η}(λ,α,β) of multivalent functions with varying arguments of coefficients. We obtain coefficients inequalities, distortion theorems and extreme points for functions in these classes. Also, we investigate several distortion inequalities involving fractional calculus. Finally, results on partial sums are considerd.

On the heterogeneous distortion inequality

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n\] for almost every $x \in \mathbb{R}^n$. Here $K \in [1, \infty)$ is a constant and $\sigma \geq 0$ is a function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover the class of $K$-quasiregular mappings when $\sigma \equiv 0$, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp H\"older continuity estimate for all solutions, provided that $\sigma \in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ for some $\varepsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ d

Abstract Asymptotic properties of the sequences (a) { P j } j = 1 ∞ $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j − 1 ∑ i = 0 j − 1 P i } j = 1 ∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = {f ∈ L 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := ∫ P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for f ∈ L 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j − 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 ∞ $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0 ∈ G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

Remarks on the Scale-Invariant Cassinian Metric

We study the geometry of the scale-invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale-invariant Cassinian metric under Mobius transformations.

Quasiregular curves

We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let $n\le m$ and let $M$ be an oriented Riemannian $n$-manifold, $N$ a Riemannian $m$-manifold, and $\omega \in \Omega^n(N)$ a smooth closed non-vanishing $n$-form on $N$. A continuous Sobolev map $f\colon M \to N$ in $W^{1,n}_{\mathrm{loc}}(M,N)$ is a $K$-quasiregular $\omega$-curve for $K\ge 1$ if $f$ satisfies the distortion inequality $(\lVert\omega\rVert\circ f)\lVert Df\rVert^n \le K (\star f^* \omega)$ almost everywhere in $M$. We prove that quasiregular curves satisfy Gromov's quasiminimality condition and a version of Liouville's theorem stating that bounded quasiregular curves $\mathbb R^n \to \mathbb R^m$ are constant. We also prove a limit theorem that a locally uniform limit $f\colon M \to N$ of $K$-quasiregular $\omega$-curves $(f_j \colon M\to N)$ is also a $K$-quasiregular $\omega$-curve. We also show that a non-constant quasiregular $\omega$-curve $f\colon M \to N$ is discrete and satisfies $\star f^*\omega >0$ almost everywhere, if one of the following additional conditions hold: the form $\omega$ is simple or the map $f$ is $C^1$-smooth.

Certain Subfamily of Harmonic Functions Related to Salagean q-Differential Operator

The theory of q-calculus operators are applied in many areas of sciences such as complex analysis. In this paper we apply S˘al˘agean q-differential operator to harmonic functions and introduce sharp coefficient bounds, extreme points, distortion inequalities and convexity results.

New subclass of <i>q</i>-starlike functions associated with generalized conic domain

In this paper, the concepts of quantum (or q-) calculus and conic regions are combined to define a new domain Ωk, q, γ which represents the generalized conic regions. Then by using a certain generalized conic domain Ωk, q, γ we define and investigate a new subclass of normalized analytic functions in open unit disk E. We also investigate a number of useful properties and characteristics of this subclass such as, structural formula, necessary and sufficient condition, coefficient estimates, Feketo-Szego problem, distortion inequalities, closure theorem, and subordination result. We also highlight some known consequences of our main results as corollaries.

Improved Hölder continuity of quasiconformal maps

Quasiconformal maps in the complex plane are homeomorphisms that satisfy certain geometric distortion inequalities; infinitesimally, they map circles to ellipses with bounded eccentricity. The local distortion properties of these maps give rise to a certain degree of global regularity and H\"older continuity. In this paper, we give improved lower bounds for the H\"older continuity of these maps; the analysis is based on combining the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the extremizers for H\"older continuity are characterized, and some applications are given to solutions to elliptic partial differential equations.

Stretching and rotation sets of quasiconformal mappings

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.

Bi-Sobolev mappings and Kp-distortions in the plane

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms f:D⟶ontoD, f∈Wloc1,1(D;D), where D⊂R2 is the unit disk, such that there exists a constant Kp≥0 for which the following distortion inequality|Df(x)|p≤Kp|Jf(x)|p−1a.e.x∈Dis satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11]. Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with Kp=Kp,f(x) not necessarily constant, we identify the homeomorphism f whose p-distortion function Kp,f(x) is minimal in L1 norm.

Some subclasses of meromorphically multivalent functions associated with the Dziok-Srivastava operator

We introduce two new subclasses Hp,k(?,A,B) and Qp,k(?,A,B) of meromorphically multivalent functions associated with the Dziok-Srivastava operator which is a special case of the Srivastava-Wright operator. Distortion inequalities, partial sums and convolutional theorems for Hp,k(?,A,B) and Qp,k(?,A,B) are obtained.

On Certain Class of Meromorphic Harmonic Concave Functions

In this paper, a class of meromorphic harmonic functions concave in the unit disc is introduced. Coefficient bounds, distortion inequalities, extreme points, geometric convolution, integral convolution for the functions belonging to this class are obtained.

Applications of Fractionalq-Calculus to Certain Subclass of Analyticp-Valent Functions with Negative Coefficients

By making use of the concept of fractionalq-calculus, we firstly defineq-extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class ofq-analogue ofp-valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, theq-extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.

Some Applications of a new class of Univalent Functions Defined by Subordination Property

In this paper, we introduce and study some applications of a new class of univalent functions defined by subordination property. Coefficient inequality, convex linear combinations, growth and distortion inequalities, radii of starlikeness, convexity and close-to- convexity and Hadamard product (or convolution) are given.