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Analytic functions of bounded index. (English) Zbl 0980.30020
Mathematical Studies Monograph Series. 6. Lviv: VNTL Publishers. 141 p. (1999).
Let $$l$$ be a positive and continuous function on $$[0,\infty)$$. An entire function $$f$$ is a function of bounded $$l$$-index if there exists $$n\in \mathbb{N}$$ such that $\left|{f^{(n)}(z)\over n!}\right|l^{-n} \bigl(|z|\bigr) \leq\max\left \{\left|{f^{(k)}(z)\over k!}\right |l^{-k} \bigl(|z|\bigr): 0\leq k\leq\mathbb{N}\right\}$ for all $$n\in\mathbb{N}$$ and $$z\in \mathbb{C}$$. This monograph is devoted to study the functions of bounded $$l$$-index. The book consists of 7 chapters. In Chapter 1-4 the author presents various facts of the behavior and properties of such functions. In Part 5 there are investigated properties of analytic solutions of linear differential equations that coefficienties are functions of bounded $$l$$-index. The question of existence of an entire function of bounded $$l$$-index is considered in the next chapter. And in the last part of this book there are studied the functions of bounded index, i.e. the case $$l\equiv 1$$.

##### MSC:
 30D20 Entire functions of one complex variable (general theory) 30D10 Representations of entire functions of one complex variable by series and integrals 30D15 Special classes of entire functions of one complex variable and growth estimates 34A30 Linear ordinary differential equations and systems
##### Keywords:
entire function; bounded index; differential equation