Goodman-Rønning-type harmonic univalent functions (Q2755638)

Let \(f(z)=h(z)+\overline{g(z)}\), where \(g\) and \(h\) are analytic functions in \(\Delta= \{z:|z|<1\}\) such that NEWLINE\[NEWLINEh(z)= z+\sum^\infty_{n= 2} a_nz^n, \quad g(z)=\sum^\infty_{n=1} b_nz^n, \quad|b_1 |<1.NEWLINE\]NEWLINE By \(G_H(\gamma)\) we denote the subclass of such functions \(f=h+ \overline g\) that satisfy the condition NEWLINE\[NEWLINE\text{Re} \left\{(1+ e^{i\alpha} {{\partial \over \partial \Theta} f(re^{i\Theta}) \over if(re^{i\Theta})}-e^{i\alpha} \right\} \geq \gamma, \quad 0\leq \gamma<1.NEWLINE\]NEWLINE In this paper some coefficient conditions, extreme points, distortion bounds, convolution conditions and convex combination for the above family of harmonic functions are investigated. In particular it is proved: Theorem. Let \(f=h+ \overline g\) and for some \(\gamma \in [0,1)\) satisfy the condition NEWLINE\[NEWLINE\sum^\infty_{n=1} \left[{2n-1 -\gamma\over 1-\gamma}|a_n|+{2n+1+ \gamma\over 1-\gamma} |b_n|\right]\leq 2.NEWLINE\]NEWLINE Then \(f\) is harmonic univalent in \(\Delta\) and \(f\in G_H (\gamma)\).