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Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of S H 0 , the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of S H 0 . Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well. [ABSTRACT FROM AUTHOR]

It is known that if the convolution of two suitably normalized planar harmonic mappings from certain families of mappings, such as those mapping into a half-plane or a strip, is locally univalent, the convolution is univalent and convex in one direction. After extending this to convolutions of mappings into a slanted half-plane with those into a slanted asymmetric strip, we prove properties for the dilatation of the convolution of a mapping from a family of slanted generalized right half-plane mappings with mappings into a slanted half-plane or a slanted asymmetric strip with a finite Blaschke product dilatation. The properties lay the foundation for a direct application of polynomial zero distribution techniques in the determination of local univalence of such convolutions. We conclude by producing a family of univalent convolutions convex in one direction between a slanted generalized right half-plane mapping and a mapping into a half-plane with a two-factor Blaschke product dilatation. [ABSTRACT FROM AUTHOR]

In this article, we prove a two-points distortion theorem and obtain sharp coefficient estimates for the families of close-toconvex harmonic mappings whose analytic part is a function convex in one direction. By making use of these results, we determine the radius of univalence of sections of these families in terms of zeros of a certain equation. the lower bound for the radius of univalence has been obtained explicitly for the case α = 1/2. Comparison of radius of univalence of the sections has been shown by providing a table of numerical estimates for the special choices of α.

Let denote the class of all complex-valued harmonic functions f in the open unit disk normalized by f(0) = fz(0) − 1 = = 0, and the subclasses of consisting of univalent and sense-preserving functions and normalized analytic functions, respectively. For φ ∈ , let := {f = h + ḡ ∈ : h – e2αig = φ} be subfamily of. In this paper, we shall determine the conditions under which the analytic function φ with φ ∈ , the linear convex combination tf1 + (1 − t)f2 with fj ∈ , j = 1, 2, and the harmonic convolution f1 ∗ f2 with fj ∈ , j = 1, 2, are univalent and convex in one direction, respectively. Many previous related results are generalized. [ABSTRACT FROM AUTHOR]

Let f = h + g ‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D. In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ ∈ [ 0 , 2 π) such that the function h + e i θ g is convex in D. In this article, we first disprove a more flexible conjecture: " Let f = h + g ‾ be a convex harmonic mapping in the disk D. Then there is a θ ∈ [ 0 , 2 π) such that the function h + e i θ g is starlike in D ". In addition, we present an example to show that there exists a harmonic automorphism f = h + g ‾ of a disk such that the function h + e i θ g is convex in only one direction for θ ≠ 0 , and that the analytic function h + g is not starlike therein. The article concludes with a new conjecture. [ABSTRACT FROM AUTHOR]

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ , $$c>0$$ , and consider the convolution $$\Psi _c * f$$ for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ . We prove it is sufficient for $$h \pm g$$ to be starlike for $$\Psi _c *f$$ to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator. [ABSTRACT FROM AUTHOR]

In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of section s n , n ( f ) is sharp especially when the corresponding mappings have convex range. In this case, each section s n , n ( f ) is univalent in the disk of radius 1 / 4 for all n ≥ 2 , which may be compared with classical result of Szegö on conformal mappings. [ABSTRACT FROM AUTHOR]

Clunie and Sheil-Small ['Harmonic univalent functions', Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 3-25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the coefficient conjecture of Clunie and Sheil- Small for a class of univalent harmonic functions which includes functions convex in some direction. Next, we prove growth and covering theorems and some related results. Finally, we propose two conjectures, an affirmative answer to one of which would then imply, for example, a solution to the conjecture of Clunie and Sheil-Small. [ABSTRACT FROM AUTHOR]

In 1984, Clunie and Sheil-Small constructed the harmonic Koebe functionexpecting it to play the role of extremal function for extremal problems over the class of sense-preserving harmonic univalent functions suitably normalized in the open unit disc. In this paper, we investigate the convolution properties of. In addition, we discuss the geometric properties of-starlike functions defined using the Sălăgean differential operator. Given a-starlike function, the productis shown to be univalent and convex in the direction of the real axis. [ABSTRACT FROM AUTHOR]

Letdenote the class of all complex-valued harmonic functionsin the open unit disc normalized by, and letbe the subclass ofconsisting of normalized analytic functions. For, letandbe subfamilies of. In this paper, we shall determine the conditions under which the harmonic convolutionis univalent and convex in one direction ifand. A similar analysis is carried out ifand. Examples of univalent harmonic mappings constructed by way of convolution are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a criterion for a harmonic function to be convex in one direction. Also, we discuss the class of harmonic functions starlike in one direction in the unit disk $${\mathbb D}$$ and obtain a method to construct univalent harmonic functions convex in one direction. Although the converse of classical Alexander's theorem for harmonic functions was proved to be false, we obtain a version of converse of it under a suitable additional condition. [ABSTRACT FROM AUTHOR]

Abstract: The harmonic function in the open unit disc can be written as a sum of an analytic and an anti-analytic function, , where and are analytic functions in , and are called the analytic part and co-analytic part of , respectively. One of the most important questions in the study of the classes of such functions is related to bounds on the modulus of functions (growth) or modulus of the derivative (distortion), because the growth theorem and distortion theorem give the compactness of the classes of these functions. In this paper we consider both of these questions with the shear construction method. [Copyright &y& Elsevier]

In this paper, we prove that a univalent orientation-preserving harmonic mapping f=h+g¯ defined on the unit disk U with the normalization f(0)=0, h′(0)+g′(0)>0, is a typically real mapping, if f(U) is a starlike domain with respect to the origin or f(U) is convex in one direction. [Copyright &y& Elsevier]