Analytic Functions of Bounded Index

8vo (21.5 cm), 141 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 6. Edited by Michael Zarichnyi. A research monograph in complex analysis by Myroslav Sheremeta of Lviv National University, presenting a systematic account of the theory of analytic functions of bounded index and its subsequent generalizations. The work traces the development of the subject from B. Lepson's 1969 introduction of entire functions of bounded index--characterized by uniform bounds on the normalized derivatives in their Taylor expansions--through later contributions by W. K. Hayman, S. M. Shah, G. Fricke, and others, to the generalized theory of bounded l-index developed by A. D. Kuzyk and M. M. Sheremeta using positive continuous weight functions. The monograph adopts the most general setting of analytic functions on arbitrary complex domains, extending earlier results previously obtained for entire functions and functions analytic in the unit disc. The seven chapters treat: Main Criteria, including the definition of bounded index, estimates for derivatives, maximum and minimum modulus, and Hayman's theorem; Value Distribution, covering logarithmic derivatives, zeros, products and sums of functions of bounded l-index, local valency, and bounded value distribution; Growth of Analytic Functions of Bounded l-Index, applying the Wiman–Valiron method to entire and disc-analytic functions; Analytic Functions of Bounded lM-Index, discussing maximal terms, Taylor coefficients, derivatives, and growth; Properties of Analytic Solutions of Linear Differential Equations, including applications to the Mittag-Leffler function and composite functions; Existence Theorems for Entire Functions of Bounded l-Index, establishing the existence of transcendental functions of bounded l-index and l-regular growth together with appropriate weight functions and associated function spaces; and Entire Functions of Bounded Index, devoted to representation theorems, logarithmic derivatives, and related differential equations. The volume concludes with extensive comments and a bibliography. As noted in the preface, the author emphasizes that many classical results for entire functions do not extend automatically to more general analytic function classes, making the broader theory developed here both technically demanding and mathematically significant. A valuable specialist reference for researchers in complex analysis, particularly those working on the growth, value distribution, and differential equations of analytic and entire functions.