On some properties of solutions of the biharmonic equation

摘要

In the present paper, the properties of the linear complex operator L(f)=zfz-z¯fz¯, which is defined on the class of complex-valued C1 functions in the plane, are investigated. It is shown that harmonicity and biharmonicity are invariant under the linear operator L. Results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered. The operator L can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way.