Wiman's inequality for analytic functions in $\mathbb{D}\times\mathbb{C}$ with rapidly oscillating coefficients

Source abstract

Let $\mathcal{A}^2$ be a class of analytic functions $f$ represented by power series of the from $$f(z)=f(z_1,z_2)=\sum^{+\infty}_{n+m=0}a_{nm}z_1^nz^m_2$$ with the domain of convergence $\mathbb{T}=\{ z\in \mathbb{C}^2 \colon |z_1|<1, |z_2|<+\infty \}$ such that $\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}$ and there exists $r_0=(r^0_1, r^0_2)\in [0,1)\times[0,+\infty)$ such that for all $r\in(r^0_1,1)\times(r^0_2,+\infty)$ we have $r_1\frac{\partial}{\partial r_1}\ln M_f(r)+\ln r_1>1, \$ where $M_f(r)=\sum_{n+m=0}^{+\infty}|a_{nm}|r_1^nr_2^m.$ Let $K(f,\theta)=\{f(z,t)=\sum_{n+m=0}^{+\infty}a_{nm}e^{2\pi it(\theta_n+\theta_m)}:t\in \mathbb{R}\}$ be class of analytic functions, where $(\theta_{nm})$ is a sequence of positive integer such that its arrangement $(\theta^*_k)$ by increasing satisfies the condition $$\theta^*_{k+1}/\theta^*_{k}\geq q>1, k>0.$$ For analytic functions from the class $\mathcal{K}(f,\theta)$ Wiman's inequality is improved.