HARMONIC CLOSE-TO-CONVEX MAPPINGS

Sufficient coefficient conditions for complex functions to be close-to-convex harmonic or convex harmonic are given. Construction of close-to-convex harmonic functions is also studied by looking at transforms of convex analytic functions. Finally, a convolution property for harmonic functions is discussed. Harmonic, Convex, Close-to-Convex, Univalent.


Introduction
Harmonic functions are famous for their use in the study of minimal surfaces and also play important roles in a variety of problems in applied mathematics.Harmonic functions have been studied by differential geometers such as Choquet [2], Kneser [7], Lewy [8], and Rado [9].Recent interest in harmonic complex functions has been triggered by geometric functions theorists Clunie and Sheil-Small [3].
A continuous function is a in a domain complex-valued harmonic functions if both and are real harmonic in .In any simply connected domain, we can write where and are analytic in .We call the and the of .analytic part co-analytic part A necessary and sufficient conditions (see [3] or [8]) for to be and locally univalent sensepreserving in is that in .Denote by the class of functions of the form (1) that are harmonic univalent and sense-preserving in the unit disk for which .Thus we may write and Note that reduces to , the class of normalized univalent analytic functions if the coanalytic part of is zero.Since for , the function ! is also in .Therefore, we may sometimes restrict ourselves to , the subclass of for which .In [3], it was shown that is normal and ! is compact with respect to the topology of locally uniform convergence.Some coefficient bounds for convex and starlike harmonic functions have recently been obtained by Avci and Zlotkiewicz [1], Jahangiri [5,6], and Silverman [14].
In this paper, we give sufficient conditions for functions in to be close-to-convex harmonic or convex harmonic.We also construct close-to-convex harmonic functions by looking at transforms of convex analytic functions.Finally, we discuss a convolution property for harmonic functions.
In the sequel, unless otherwise stated, we will assume that is of the form (1) with and of the form (2). images of are close-to-convex.Recall that a domain is convex if the linear segment joining any two points of lies entirely in .A domain is called close-to-convex if the complement of can be written as a union of non-crossing half-lines.For other equivalent criteria, see [4].
Clunie and Sheil-Small [3] proved the following results.
Theorem A: If are analytic in with and is close-toconvex for each , , then is harmonic close-to-convex.If is locally univalent in and is convex for some , Theorem B: " , then is univalent close-to-convex.A domain is called convex in the direction if every line parallel to the " line through 0 and has a connected intersection with .Such a domain is close-to-convex.# The convex domains are those convex in every direction.We will also make use of the following result, which may be found in [3].
Theorem Necessary coefficient conditions were found in [3] for functions to be in and .We now give some sufficient condition for functions to be in these classes.But first we need the following results.See, for example, [13].The coefficient bound given in Theorem 3 can also be found in [5] and [14].Remark: However, our approach in this paper is different from those given in [5] and [14].
The well-known results for univalent functions that is convex if and only if Remark: is starlike does not carry over to harmonic univalent functions.See [12].Hence, we cannot conclude from Theorem 3 that (3) is a sufficient condition for to map onto a starlike domain.Nevertheless, we believe this to be the case.See [5,6,14].
We now introduce a class of harmonic close-to-convex functions that are constructed from convex analytic functions.