Abstract. In the present paper, the authors investigate the univalence of the functions /, analytic in E, f(0) = 0, /'(0) = 1 and which satisfy where a > 0 and 0 < ¡3 < 1. The univalence of such functions has already been established in the case when a < 0 and /3 = 0 by H. S. Al-Amiri and M. O. Reade in 1975.
It is well known that harmonic convolution of two normalized right halfplane mappings is convex in the direction of the real axis, provided the convolution function is locally univalent and sense-preserving in E = {z : |z| < 1}. Further, it is also known that the condition of local univalence and sense-preserving in E on the convolution function can be dropped when one of the convoluting functions is the standard right half-plane mapping with dilatation −z and other is the right halfplane mapping with dilatation e iθ z n , n = 1, 2, θ ∈ R. This result does not hold for n = 3, 4, 5, . . . . In this paper, we generalize this result by taking the dilatation of one of the right half-plane mappings as e iθ z n ( n ∈ N, θ ∈ R) and that of the other as (a − z)/(1 − az), a ∈ (−1, 1). We shall prove that our result holds true for all n ∈ N, provided the real constant a is restricted in the interval [(n − 2)/(n + 2), 1). The range of the real constant a is shown to be sharp.
In the present paper, we introduce a family of univalent harmonic functions, which map the unit disk onto domains convex in the direction of the imaginary axis. We find conditions for the linear combinations of mappings from this family to be univalent and convex in the direction of the imaginary axis. Linear combinations of functions from this family and harmonic mappings obtained by shearing of analytic vertical strip maps are also studied. *
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