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.

Certain Properties of Komatu Integral Transform with Negative Coefficients with Reference to Uniform Starlike and Uniform Convex Functions

The authors obtained a new subclass about strongly starlike and strongly convex functions with respect to Komatu integral transforms and the inclusion properties of these classes such as 𝓢𝒑𝑻(𝝀, 𝑰𝝂 𝝆 ) and 𝓤𝓒𝓥𝓣(𝝀, 𝑰𝝂 𝝆 ) were discussed . Furthermore, a new subclass about uniformly starlike functions along uniformly convex functions including negative coefficients defined by the Komatu integral transforms are introduced. The various properties about these classes are obtained here including (for instance) coefficient estimates, extreme points, distortion and covering theorems. Mathematics Subject Classification: Primary 30C45

Harmonic univalent functions defined by post quantum calculus operators

We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Distortion and covering theorems of pluriharmonic mappings

The linear-invariant families of analytic functions make it possible to obtain well-known results to broader classes of functions, and are often helpful in obtaining simpler proofs along with new results. Based on this classical approach due to Pommerenke, properties (such as bounds for the derivative, covering and distortion) of a corresponding class of locally quasiconformal and planar harmonic mappings are established by Starkov. Motivated by these works, in this paper, we mainly investigate distortion and covering theorems on some classes of pluriharmonic mappings.

The Distortion Theorems for Harmonic Mappings with Analytic Parts Convex or Starlike Functions of Orderβ

Some sharp estimates of coefficients, distortion, and growth for harmonic mappings with analytic parts convex or starlike functions of orderβare obtained. We also give area estimates and covering theorems. Our main results generalise those of Klimek and Michalski.