On Starlike Functions of Negative Order Defined by q-Fractional Derivative

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.

The Approximation of the Nonlinear Singular System with Impulses and Sliding Mode Control via a Singular Polynomial Fuzzy Model Approach

In this paper, the Singular-Polynomial-Fuzzy-Model (SPFM) approach problem and impulse elimination are investigated based on sliding mode control for a class of nonlinear singular system (NSS) with impulses. Considering two numerical examples, the SPFM of the nonlinear singular system is calculated based on the compound function type and simple function type. According to the solvability and the steps of two numerical examples, the method of solving the SPFM form of the nonlinear singular system with (and without) impulse are extended to the more general case. By using the Heine–Borel finite covering theorem, it is proven that a class of nonlinear singular systems with bounded impulse-free item (BIFI) properties and separable impulse item (SII) properties can be approximated by SPFM with arbitrary accuracy. The linear switching function and sliding mode control law are designed to be applied to the impulse elimination of SPFM. Compared with some published works, a human posture inverted pendulum model example and Example 3.2 demonstrate that the approximation error is small enough and that both algorithms are effective. Example 3.3 is to illustrate that sliding mode control can effectively eliminate impulses of SPFM.

On the Differentiation of Henstock and McShane Integrals

It is well known that the derivative of the primitive of 1-dimensional Henstock integral exists almost everywhere. Point-interval pairs used in the derivative are Henstock point-interval pairs, which are consistent with point-interval pairs used in the Henstock integral. Note that “almost everywhere” is a set of points, more precisely, the derivative does not exist on a set of points with measure zero. We can transform a set of Henstock point-interval pairs to a set of points with measure zero because of Vitali’s covering theorem. For 1-dimensional McShane integrals, [Formula: see text]-dimensional McShane and Henstock integrals, covering theorems of Vitali’s type cannot be applied. In this paper, we shall discuss differentiation of [Formula: see text]-dimensional McShane and Henstock integrals.

Uniform Continuity of Fractal Interpolation Function

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

Bohr theorems for slice regular functions over octonions

In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions $\mathbb {O}$ . To this end, we establish the coefficient estimates for self-maps of the unit ball of $\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.

On generalized gamma-Bazilevic functions

In this paper, we define and study the class [Formula: see text] of generalized gamma-Bazilevic functions. Our main focus is to discuss certain problems such as inclusion results, covering theorem and radius problem.

Coefficient body for nonlinear resolvents

This paper is devoted to the study of families of so-called nonlinear resolvents. Namely, we construct polynomial transformations which map the closed unit polydisks onto the coefficient bodies for the resolvent families. As immediate applications of our results we present a covering theorem and a sharp estimate for the Schwarzian derivative at zero on the class of resolvents.

Revisiting Leighton’s theorem with the Haar measure

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.