On the relation between maximum modulus, maximum term, and Taylor coefficients of an entire function
For an entire function f, relations are established between its Taylor coefficients \(a_ n\), its maximum modulus \(M(r)=\max \{| f(z)|;| z| =r\}\) and its maximum term \(m(r)=\max \{| a_ nr^ n|\), \(n\geq 0\}\). Instead of measuring the growth of f by its order, type, or some generalized order, direct estimates of the above quantities are derived since these are more precise. To give some of the results, let \(g\in C^ 2[x_ 1,\infty)\) for some \(x_ 1>0\), g'(x)\(\to \infty\) as \(x\to \infty\), \(g''(x)>0\), and set \(A(r)=\exp \{(g')^{- 1}(\log r)\log r-g((g')^{-1}(\log r))\}\) for \(r>x_ 1\). We consider the assertions (1) \(M(r)=O(A(2r))\), \(r\to \infty\), (2) \(m(r)=O(A(2r))\), \(r\to \infty\), and (3) \(| a_ n| =O(2^ ne^{-g(n)})\), \(n\to \infty\). For general entire functions, including those of order \(\rho\in (0,\infty)\), we have (1)\(\Rightarrow (3)\) and (2)\(\Leftrightarrow (3)\). The implication (3)\(\Rightarrow (1)\) is true in case f does not grow too fast, i.e. under the assumption that log M(r)\(=O((\log r)^ 2)\), \(r\to \infty\) (g(x) increases at least as fast as \(x^ 2,x\to \infty)\). For part of the remaining f the weaker implication \((3)\Rightarrow M(r)=O([g''((g')^{-1}\log 2r)]^{-1/2}A(2r)),\) \(r\to \infty\), is established. It includes entire functions of order \(\rho\in (0,\infty)\), it is best possible, too, and it is still more precise than known characterizations in terms of type and order. The methods of proof originate in work by Hadamard, Le Roy, Valiron, and Berg.
scientific article; zbMATH DE number 3882976
scientific article