Selected Title s i n Thi s Serie s 150 B . Ya. Levin , Lecture s o n entire functions , 199 6 149 Takash i Sakai, Riemannia n geometry , 199 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 199 6 147 S . G . Gindiki n and L . R. Volevich , Mixe d proble m fo r partia l differentia l equation s wit h quasihomogeneous principa l part , 199 6 146 L . Ya. Adrianova , Introductio n t o linea r system s o f differential equations , 199 5 145 A . N. Andriano v an d V. G . Zhuravlev , Modula r form s an d Heck e operators, 199 5 144 0 . V . Troshkin, Nontraditiona l method s i n mathematica l hydrodynamics , 199 5 143 V . A . Malyshev an d E. A. Minlos, Linea r infinite-particl e operators , 199 5 142 N . V. Krylov, Introductio n to the theory o f diffusio n processes , 199 5 141 A . A. Davydov, Qualitativ e theory o f control systems, 199 4 140 Aizi k I. Volpert, Vitaly A . Volpert , an d Vladimir A. Volpert, Travelin g wav e solution s o f parabolic systems , 199 4 139 I . V. Skrypnik , Method s fo r analysi s o f nonlinear ellipti c boundary valu e problems , 199 4 138 Yu . P . Razmyslov, Identitie s o f algebras an d thei r representations , 199 4 137 F . I. Karpelevich an d A. Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 199 4 136 Masayosh i Miyanishi , Algebrai c geometry, 199 4 135 Masar u Takeuchi, Moder n spherica l functions , 199 4 134 V . V. Prasolov , Problem s an d theorem s i n linea r algebra , 199 4 133 P . I. Naumkin and I. A. Shishmarev , Nonlinea r nonloca l equation s i n the theory o f waves, 1994 132 Hajim e Urakawa , Calculu s o f variations an d harmoni c maps , 199 3 131 V . V . Sharko , Function s o n manifolds : Algebrai c an d topologica l aspects , 199 3 130 V . V . Vershinin, Cobordism s an d spectra l sequences , 199 3 129 Mitsu o Morimoto , A n introductio n t o Sato's hype r functions, 199 3 128 V . P. Orevkov , Complexit y o f proof s an d thei r transformation s i n axiomati c theories , 1993 127 F . L. Zak, Tangent s an d secant s o f algebrai c varieties, 199 3 126 M . L . Agranovskii, Invarian t functio n space s o n homogeneou s manifold s o f Li e group s and applications , 199 3 125 Masayosh i Nagata, Theor y o f commutative fields , 199 3 124 Masahis a Adachi , Embedding s an d immersions , 199 3 123 M . A . Akivis an d B. A. Rosenfeld, Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou, Theor y o f entire and meromorphi c functions : Deficien t an d asymptoti c values and singula r directions , 199 3 121 I . B . Fesenk o an d S . V . Vostokov , Loca l fields an d thei r extensions : A constructiv e approach, 199 3 120 lakeyuk i Hid a and Masuyuki Hitsuda, Gaussia n processes , 199 3 119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisso n brackets. Geometry and quantization , 1993 118 Kenkich i Iwasawa, Algebrai c functions , 199 3 117 Bori s Zilber , Uncountabl y categorica l theories , 199 3 116 G . M. Fel'dman, Arithmeti c o f probability distributions , an d characterizatio n problem s on abelia n groups , 199 3 115 Nikola i V. Ivanov , Subgroup s o f Teichmiille r modula r groups , 199 2 114 Seiz o ltd, Diffusio n equations , 199 2 (See the AM S catalo g fo r earlie r titles ) This page intentionally left blank Lectures o n Entire Function s Boris Yakovlevich LEVIN 1906-1993 10.1090/mmono/150 Translations o f MATHEMATICAL MONOGRAPHS Volume 15 0 Lectures o n Entire Function s B. Ya. Levin In collaboration with Yu. Lyubarski i M. Sodin V. Tkachenko Qff l^^,lY & American Mathematical Societ y '5 Providence , Rhode Island B. fl. JIEBH H IIEJIHE ^YHKIIH H ITpn ynacTH H K) . M . JIio6apcKoro , M . JI . Co^HHa , B . A . TKa^eiiK O Translated b y Vadi m Tkachenk o fro m a n origina l Russia n manuscrip t EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s James D . Stashef f (Chair ) ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidlin (Chair ) 2000 Mathematics Subject Classification. Primar y 30Dxx , 30D20 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mmono-150 Library o f Congres s Cataloging-in-Publicatio n Dat a Levin, B . IA. (Bori s IAkovlevich ) Lectures o n entir e function s / B . Ya. Levin ; i n collaboration wit h Yu . Lyubarskii , M . Sodin , V. Tkachenko . p. cm. — (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 150 ) Translated fro m th e author' s Russia n manuscript . Includes bibliographica l reference s an d indexes . ISBN 0-8218-0282- 8 (alk . paper ; har d cover ) ISBN 0-8218-0897- 4 (alk . paper; sof t cover ) 1. Functions, Entire . I . Title. II . Series . QA351.E5L484 199 6 96-31 8 515'.98—<lc20 CI P Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission should b e addressed t o the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 6 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 09 0 8 07 0 6 0 5 Contents Preface x i Introduction x v Part I . Entire Function s o f Finite Orde r 1 Lecture 1 . Growt h o f Entire Function s 3 1.1. Th e growt h scal e fo r entir e functions 3 1.2. Orde r an d typ e o f entire functions 3 1.3. Th e relatio n betwee n th e growt h o f a n entir e functio n an d th e decrease o f the coefficient s o f its power serie s expansion 5 Lecture 2 . Mai n Integra l Formula s fo r Function s Analyti c i n a Dis k 9 2.1. Th e Poisso n formul a an d the Schwar z formul a 9 2.2. Th e Poisson-Jense n formul a 9 2.3. Th e Jense n formul a 1 0 2.4. Th e Nevanlinn a characteristic s 1 1 2.5. Som e corollaries o f the Jensen formul a 1 3 Lecture 3 . Som e Application s o f the Jense n Formul a 1 5 3.1. A theorem o n (J)-quasianalyticit y 1 5 3.2. Th e convergenc e exponen t an d th e uppe r densit y o f the sequenc e of zeros 1 7 3.3. Completenes s o f a system o f exponential function s 1 9 3.4. Completenes s o f a special system o f functions i n countably norme d spaces 2 0 Lecture 4. Factorizatio n o f Entire Functions o f Finite Orde r 2 5 4.1. Th e Weierstrass canonica l product 2 5 4.2. Th e Hadamard theore m 2 6 4.3. Estimate s fo r canonica l products 2 8 Lecture 5 . The Connection betwee n the Growt h o f Entire Functions and th e Distribution o f their Zero s 3 1 5.1. Function s o f noninteger orde r 3 1 5.2. Function s o f integer orde r 3 2 Lecture 6 . Theorem s o f Phragmen an d Lindelo f 3 7 6.1. Function s analyti c insid e an angl e 3 7 6.2.