Construction of entire function of arbitrary order with given asymptotic properties (Q1086386)

Let \(M_ f(r)=\max \{| f(z)|:| z| =r\}\). For any entire function \(f(z)\), \(z\in {\mathbb{C}}\), the function L(z) is constructed with following properties: the growth of \(L(z)\) is close in some sense to the growth of \(f(z)\), and if \(\lambda_ n\) are zeros of the function \(L(z)\) then \[ | \lambda_ nL'(\lambda_ n)| =M_ f((1+o(1))\lambda_ n),\quad n\to \infty. \]