Wiman-Valiron theorem for \(q\)-differences (Q2790823)

The Wiman-Valiron theory gives some growth properties of entire functions \(f(z)\) possessing a number of applications in the study of complex differential equations and complex dynamics. This theory pays attention to the relations between \(f(z)\) and its derivative \(f'(z)\) at the maximum points. Recent developments have shown that similar properties hold for meromorphic functions in a certain domain near infinity and analytic functions in the unit disc. Instead of the derivative, one can consider relations between \(f(z)\) and the difference \(\Delta f(z)=f(z+1)-f(z)\) at the maximal points, which is called the difference analogue [\textit{Y.-M. Chiang} and \textit{S.-J. Feng}, Trans. Am. Math. Soc. 361, No. 7, 3767--3791 (2009; Zbl 1172.30009)]. NEWLINENEWLINENEWLINE In this paper, the authors construct the \(q\)-difference analogue of the Wiman-Valiron theory for entire functions. The authors obtain three theorems. One of them is the following: Suppose that \(m\) is a positive integer and \(q\) is a complex number with \(q^m\in\mathbb C\setminus\{0,1\}\). Let \(f(z)\) be a transcendental entire function of order strictly less than 1/2 and \(F\subset \mathbb R^+\) a set of finite logarithmic measure. Then for any \(0<\delta<1/4\) and any \(z\) with \(|z|=r\not\in F\) satisfying NEWLINE\[NEWLINE|f(z)|>M(r,f)\nu(r,f)^{\delta-1/4},NEWLINE\]NEWLINE it holds NEWLINE\[NEWLINE\frac{f(q^mz)}{f(z)}=e^{(q^m-1)\nu(r,f)(1+o(1))},NEWLINE\]NEWLINE where \(M(r,f)\) denotes the maximum modulus and \(\nu(r,f)\) denotes the central index. There are several methods to establish analogues of the Wiman-Valiron theory. The authors adopt the Poisson-Jensen formula as main tool.