In this paper, we study the family ${ \mathcal C}_{ H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{ D}$ with $f_{ \bar{ z}}(0)=0$. We derive a sufficient condition for $f$ to belong to the class $\CC_{ H}^0$. We take the analytic part of $f$ to be $zF(a,b;c;z)$ or $zF(a,b;c;z^2)$ and for a suitable choice of co-analytic part of $f$, the second complex dilatation $w(z)=\bar{ f_{ \bar{ z}}}/f_z$ turns out to be a square of an analytic function. Hence $f$ is lifted to a minimal surface expressed by an isothermal parameter. Explicit representation for classes of minimal surfaces are given. Graphs generated by using Mathematica are used for illustration.
Categories: math.CV