Entire functions with infinite sets of deficient functions. (Q2782535)

The authors constructed an example of entire functions that have the following property. Let \(\rho\) be a real number in \((1/2,+\infty)\) and \(\{a_n\}_{n=0}^{\infty}\) be a sequence of small entire functions in the sense that they satisfy the growth restriction NEWLINE\[CARRIAGE_RETURNNEWLINE \log^+M(r, a_n)=o(r^{\rho}),\qquad r\to +\infty, CARRIAGE_RETURNNEWLINE\]NEWLINE for each \(n\). Then there exists an entire function \(f\) of order \(\rho\), norml type, and positive lower type, such that \(f\) has \(\{a_n\}_{n=0}^{\infty}\) as deficient functions. Moreover, there is a constant \(c>0\), NEWLINE\[CARRIAGE_RETURNNEWLINE \delta(a_n,\, f)>\exp (-cn),\qquad n=1, 2, 3, \dots. CARRIAGE_RETURNNEWLINE\]NEWLINE Here an deficient function \(a\) (with respect to \(f\)) is an entire function such that NEWLINE\[CARRIAGE_RETURNNEWLINE \delta(a_n,\, f)=\liminf_{r\to +\infty}{m(r,1/(f-a))\over T(r, f)}>0. CARRIAGE_RETURNNEWLINE\]NEWLINE Edrei and Fuchs proved that an entire function \(f\) has no deficient value (i.e., when \(a\) is a constant and \(\delta(a,f)=0\)) if its order is less than or equal to half. The result under review extended an earlier result also obtained by the first author [Sov. Math. Dokl. 7, 1303--1306 (1996); translation from Dokl. Akad. Nauk SSSR 170, 999--1002 (1966; Zbl 0153.39603)] when all the deficient functions \(\{a_n\}_{n=0}^{\infty}\) are constants. The method of construction of \(f\) is also based on the method in this earlier paper. The paper under review contains a number of interesting results as a consequences of the method of construction. Please note that reference number [2] for the result by Edrei and Fuchs in the middle of page 12 (Theorem 4.13) should be [3] instead.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].