The geometric function theory of analytic functions has a century of development, but that of harmonic mappings is still rather young. This paper is concerned with a very special subclass of harmonic mappings and the adaptation of classical methods to this class. The characterization of the class by the properties of the coefficients of the series development is a major tool. The methods used in this paper are rather interesting but the paper is marred by many misprints and rather confusing and incomplete proofs. Given a positive integer \(m\) let \(H_m\) be the class of harmonic functions of the form \(f(z) = h(z) + \overline{g(z)}\) where \(h(z) = z^m + \sum_{n=m}^\infty a_n z^n\) and \(g(z) = \sum_{n=m}^\infty b_n z^n\) are analytic for \(z \in U' = \{z:| z| > 1\}\) (except for the pole of \(h(z)\) at infinity). This class eliminates the possible logarithmic singularity at infinity. For a given \(\alpha\) with \(0 \leq \alpha < 1\) this paper introduces the special class \(M_H(m,\alpha)\) of functions in \(H_m\) that satisfy \(\text{Re}\{(1+e^{i\phi})(zh'(z) - \overline {zg'(z)})/(h(z) + \overline{g(z)}) - me^{i\phi}\} \geq m\alpha\) for all (not some!) real \(\phi\). The main results of the paper are concerned with a subclass \(M_{\overline H}(m,\alpha)\) of those functions in \(M_H(m,\alpha)\) with \(a_n \geq 0\), and \(b_n \leq 0\). The author proves Theorem 1: If \(f \in H_m\) and \(\sum_{n=m}^\infty n(| a_n| + | b_n| ) \leq m\) then \(f\) is sense preserving in \(U'\). (Note that this reviewer prefers a different indexing of the sums than the author uses. This helps show the symmetry in some of the results.) Theorem 2 states: If \(f \in H_m\) and (7): \(\sum_{n=m}^\infty \{[2n +m(1+\alpha)]| a_n| + [2n -m(1+\alpha)]| b_n| \} \leq m(1 - \alpha)\) then \(f \in M_H(m,\alpha)\) and \(f\) is sense preserving in \(U'\). The proof of the first assertion is well done, but no proof is given of the second. The result is true, but the proof is not transparent since it requires observing that \(2n - m(1+\alpha) \geq n(1 -\alpha)\) for all \(n \geq m\). Theorem 3 states that if all \(a_n \geq 0\) and all \(b_n \leq 0\) then \(f \in M_{\overline H}(m,\alpha)\) if and only if (7) holds. Theorem 2 proves the \textit{if} part. Unfortunately, the proof of the \textit{only if} part is given in a negative form (in which no proof is offered that a denominator is always positive) despite the fact that a direct proof is easy. This characterization of the class \(M_{\overline H}(m,\alpha)\) in terms of the coefficients allows the determination of the extreme points. Again, this section is marred by many misprints. The closure of the class under convolution is proved and finally, the author makes use of results of Ahuja and Jahangiri (in a preprint) to show that the last class is multivalently starlike.