Let Co(p), \(p\in (0,1]\) be the class of all meromorphic univalent functions \(\varphi \) defined in the open unit disc \({\mathbb {D}}\) with normalizations \(\varphi (0)=0=\varphi '(0)-1\) and having simple pole at \(z=p\in (0,1]\) such that the complement of \(\varphi ({\mathbb {D}})\) is a convex domain. The class Co(p) is called the class of concave univalent functions. Let \(S_{H}^{0}(p)\) be the class of all sense preserving univalent harmonic mappings f defined on \({\mathbb {D}}\) having simple pole at \(z=p\in (0,1)\) with the normalizations \(f(0)=f_{z}(0)-1=0\) and \(f_{\bar{z}}(0)=0\). We first derive the exact regions of variability for the second Taylor coefficients of h where \(f=h+\overline{g}\in S_{H}^{0}(p)\) with \(h-g\in Co(p)\). Next we consider the class \(S_{H}^{0}(1)\) of all sense preserving univalent harmonic mappings f in \({\mathbb {D}}\) having simple pole at \(z=1\) with the same normalizations as above. We derive exact regions of variability for the coefficients of h where \(f=h+\overline{g}\in S_{H}^{0}(1)\) satisfying \(h-e^{2i\theta }g\in Co(1)\) with dilatation \(g'(z)/h'(z)=e^{-2i\theta }z\), for some \(\theta \), \(0\le \theta <\pi \).

让钴（p），\（P \在（0,1] \）是该类的所有亚纯单叶函数\（\ varphi \）在打开的单元盘定义\（{\ mathbb {d}} \）与归一化\（\ varphi（0）= 0 = \ varphi'（0）-1 \）并在\（z = p \ in（0,1] \）处具有简单极点，使得\（\ varphi（ {\ mathbb {D}}）\）是一个凸域，类Co（p）被称为凹单价函数类，令\（S_ {H} ^ {0}（p）\）为凸类。所有在\（{\ mathbb {D}} \）上定义的保留单价谐波映射f在\（z = p \ in（0,1）\）处具有简单极点，归一化为\ {f（0）= f_ {z}（0）-1 = 0 \）和\（f _ {\ bar {z }}（0）= 0 \）。我们首先导出h的第二个泰勒系数的可变性的确切区域，其中\（f = h + \ overline {g} \ in S_ {H} ^ {0}（p）\）与\（hg \ in Co（p ）\）。下一步，我们考虑类\（S_ {H} ^ {0}（1）\）的所有感测保存单叶调和映射˚F在\（{\ mathbb {d}} \）具有简单的极\（Z = 1 \ ），具有与上述相同的规格化。我们得出h系数的确切可变性区域，其中\（f = h + \ overline {g} \在S_ {H} ^ {0}（1）\）中满足\（he ^ {2i \ theta} g \在Co（1）\）中满足\（g' （z）/ h'（z）= e ^ {-2i \ theta} z \），对于某些\（\ theta \），\（0 \ le \ theta <\ pi \）。