Let H be the class of complex-valued harmonic mappings f=h+g¯ defined in unit disk D:={z?C:|z|<1}, where h and g are analytic functions D with normalization h(0)=0=h?(0)?1 g(0)=0. H0={f=h+g¯?H:g?(0)=0}. PH0(M):={f=h+g¯?H0:Re(zh??(z))>?M+|zg??(z)|,z?DandM>0}. univalent [Ghosh N, Allu V. On some subclasses mappings. Bull Aust Math Soc. 2020;101:130–140.]. In this paper, we obtain sharp Bohr–Rogos...

Recommended by Narendra Kumar Govil We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions their boundaries are determined explicitly and the proofs rely on Schw...

The purpose of the present paper is to study a new subclass of harmonic univalent functions associated with fractional calculus operator. We obtain coefficient conditions, distortion bounds and extreme points for this class and discuss a class preserving integral operator. We also show that the class studied in this paper is closed under convolution and convex combination. The results obtained ...

a function is said to be bi-univalent on the open unit disk d if both the function and its inverse are univalent in d. not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. in this paper we use the faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subord...

Many authors have obtained some inclusion properties of certain subclasses univalent and functions associated with distribution series, such as Pascal distribution, Binomial Poisson Mittag–Leffler-type Geometric distribution. In the present paper, we obtain relations harmonic class H(α,δ) classes SH* starlike KH convex functions, also for TNHβ TRHβ operator Υ defined by applying convolution reg...

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, , such that f = h+g where h(z) = z+ P 1 2 akz , g(z) = P 1 1 bkz . S H will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n 2) and prove that the 2-slit mappings are in SH while for n 3 the mappings are in S H . Finally we show that these mappings establish the sharpne...