Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian (Q2890392)

A harmonic mapping \(f(z)=h(z)+\overline{g(z)}\) in the unit disk \(\mathbb D\) with holomorphic functions \(h\) and \(g\) is locally univalent and sense-preserving if \(J_f(z):=|h'(z)|^2-|g'(z)|^2>0\) in \(\mathbb D\). Let \(L\) denote a family of locally univalent and sense-preserving harmonic functions \(f(z)=\sum_{n=2}^{\infty}a_n(f)z^n+z+\sum_{n=1}^{\infty}a_{-n}(f)\overline z^n\), \(z\in\mathbb D\). For any \(\varphi\in\text{Aut}(\mathbb D)\) and \(\epsilon\in\mathbb C\), \(|\epsilon|<1\), denote NEWLINE\[NEWLINET_{\varphi}(f(z))=\frac{f(\varphi(z))-f(\varphi(0))}{\varphi'(0)h'(\varphi(0))},\;\;\; A_{\epsilon}(f(z))=\frac{f(z)+\epsilon\overline{f(z)}}{1+\epsilon g'(0)}.NEWLINE\]NEWLINE A family \(L\) is called linear invariant if for any \(f\in L\) and \(\varphi\in\text{Aut}(\mathbb D)\), \(T_{\varphi}(f)\in L\). A linear invariant family \(L\) is called affine invariant if for any \(f\in L\) and \(|\epsilon|<1\), \(A_{\epsilon}(f)\in L\). The order of \(L\) is defined as \(\text{ord}\;L=\sup\{|a_2(f)|: f\in L\}\).NEWLINENEWLINE Theorem 2.1: If \(L\) is affine and linear invariant, \(\text{ord}\;L=\alpha\), \(\alpha>0\), then, for \(f\in L\), NEWLINE\[NEWLINE\frac{(1-r)^{2\alpha-2}}{(1+r)^{2\alpha+2}}\leq\frac{J_f(z)}{1-|a_{-1}(f)|^2}\leq \frac{(1+r)^{2\alpha-2}}{(1-r)^{2\alpha+2}},\;\;\;|z|=r<1.NEWLINE\]NEWLINE The equality sign holds for \(f(z)=k_{\alpha}(z)+a_{-1}\overline{k_{\alpha}(z)}\) with the generalized Koebe function \(k_{\alpha}(z)\).NEWLINENEWLINETheorem 2.1 improves the earlier result by Schaubroeck. The inverse statement is given in Theorem 2.2: Assume that \(f\in L\) and the upper bound in Theorem 2.1 holds for some \(\alpha>0\). Then there exists an \(|\epsilon|<1\), such that \(|a_2(f)-{\epsilon\over2}|\leq\alpha\).NEWLINENEWLINEFor \(f\in L\), define the strong order of \(f\) as NEWLINE\[NEWLINE\overline{\text{ord}}\;f=\sup_{\varphi\in\text{Aut}(\mathbb D)}\frac{|a_2(T_{\varphi}(f))- a_{-1}(T_{\varphi}(f))\overline{a_{-2}}(T_{\varphi}(f))|}{1-|a_{-1}(T_{\varphi}(f))|^2}.NEWLINE\]NEWLINE Theorem 3.1: For \(f\in L\), \(\overline{\text{ord}}\;f\leq\alpha\) if and only if for every \(F=F_{\psi}:=T_{\psi}(f)\) and any \(z\in\mathbb D\), NEWLINE\[NEWLINE\frac{(1-r)^{2\alpha-2}}{(1+r)^{2\alpha+2}} \leq\frac{J_F(z)}{J_F(0)}\leq\frac{(1+r)^{2\alpha-2}}{(1-r)^{2\alpha+2}},\quad |z|=r<1.NEWLINE\]NEWLINE Theorem 3.1 implies certain relations between the order and the strong order for linear invariant families of harmonic mappings. The authors prove also that the union \(U_{\alpha}^H\) of all linear invariant families \(L\) of functions whose strong order does not exceed \(\alpha\) is affine and linear invariant.