Linear differential equations with entire coefficients of small growth
A wellknown theorem due to \textit{S. B. Bank} and the reviewer [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)] for \(\rho(A)<1/2\) and, independently, by \textit{J. Rossi} [Proc. Am. Math. Soc. 97, 61-66 (1986; Zbl 0596.30047)] and \textit{L. Shen} [Kexue Tongbao, Foreign Lang. Ed. 30, 1579-1585 (1985; Zbl 0636.34003)] for \(\rho(A)=\frac{1}{2}\) states that whenever \(A\) is transcendental entire of order \(\rho(A)\leqq 1/2\), the complex differential equation \(w''+A(z)w=0\) cannot have linearly independent solutions \(f_1,f_2\) each with \[ \lambda(f_j)=\limsup_{r\to\infty}\frac{\overset {+}\log N(r,1/f_j)}{\log r}<\infty. \] Here, the author recalls his following conjecture [Result. Math. 20, No. 1/2, 517-529 (1991; Zbl 0743.34039)]: To each integer \(n\geq 2\) it corresponds \(L(n)>0\) such that if \(A_0,\ldots,A_{n-2}\) are entire of order \(<L(n)\) and not all polynomials, then the equation \(w^{(n)}+\sum_{j=0}^{n-2}A_j(z)w^{(j)}=0\) cannot have linearly independent solutions \(f_1,\ldots,f_n\) each satisfying \(\lambda(f_j)<\infty\). By the result above, \(L(2)\geq 1/2\), \textit{S. M. Elzaidi} [Some theorems concerning linear differential equations in the complex domain (Ph.D. thesis, University of Nottingham) (1996), Result. Math. 32, No. 3-4, 291-297 (1997; Zbl 0893.34025)] proved that \(L(3)\geq 1/4\). This paper is now devoted to proving that \(L(n)\geq\frac{1}{2(n-1)}\) for all \(n\geq 3\). The proof is based, not surprisingly, on the \(\cos\pi\rho\) theorem combined with a nice Wronskian determinant analysis for meromorphic functions of type \(W(z)e^{h(z)}\), where \(h\) is entire, \(W(z)\) meromorphic, and both are of finite order in the complex plane.
scientific article; zbMATH DE number 1773427