A Class of Univalent Harmonic Functions Defined by Multiplier Transformation
نویسندگان
چکیده
A continuous function f(x+ iy) = u(x, y) + iv(x, y) defined in a domain D ⊂ C (Complex plane) is harmonic in D if u and v are real harmonic in D. Clunie and Shiel-Small [3] showed that in a simply connected domain such functions can be written in the form f = h+g, where both h and g are analytic. We call h the analytic part and g, the co-analytic part of f . Let w(z) = g ′(z) h′(z) be the dilatation of f = h+g. The mapping f is sense-preserving and locally oneto-one in the open unit disc E = {z : |z| < 1}, if and only if the Jacobian of the mapping, Jf (z) = |h′(z)|2 − |g′(z)|2, is positive. So, the condition for f to be sense-preserving and locally one-to-one is that |h′(z)| > |g′(z)| or equivalently, |w(z)| < 1 in E. We denote by SH the class of harmonic, sense preserving and univalent functions in the unit disc E, normalized by the conditions f(0) = 0 and fz(0) = 1. So, a harmonic mapping in the class SH has the representation f = h + g, where
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