Đối với các chức năng hài hòa và riêng biệt hòa, chúng tôi cho kết quả tương tự như định lý Carlson-Boas. Chúng tôi cung cấp cho hài hòa tương tự như các định lý Polya và . Giới thiệu cũng được biết đến định lý cổ điển của Carlson (xem [2, ]) nói rằng một holomorphic toàn bộ chức năng của loại theo cấp số nhân. | 33 2 2005 183-188 V í e t mi ai m J o u r mi ai l of MATHEMATICS ê VAST 2005 Uniqueness Theorems for Harmonic and Separately Harmonic Entire Functions on C Bachir Djebbar Department of Computer Sciences University of Sciences and Technology M. B of Oran 1505 El M naouer Oran 31000 Algeria Received May 24 2004 Abstract. For harmonic and separately harmonic functions we give results similar to the Carlson-Boas theorem. We give also harmonic analogous of the Polya and Guelfond theorems. 1. Introduction N Theorem be an entire harmonic function on C of exponential type If . for . Then Similarly Ching in showed that the same conclusion holds under the conditions is of exponential type .for. .for all complex . 184 R Theorem . .Let be an entire function on C of exponential . N Z then is a polynomial. Theorem be an entire function on C an integer greater than one. If Ịg are integers satisfies the inequality .2. . -. 4 .2 where R R satisfies . then is a polynomial. -- 2. Notations and Results . . 1 . . 1 .C. .C. Theorem . Let be an entire harmonic function and let. . . be an expansion according to the basis . . 1 Then the growth order 1 of is given as follows _ _ When. the growth type of is given by . . . . C 185 Theorem . Let be an entire harmonic function on C of exponential type . . then 0 on C Theorem . Let be an entire separately harmonic function on C of exponential type with respect to the norm. . with. For . N let - . C . 1. 0 and - . . N for. If 0 on and 1. .-1 z. . 1. . - . . . then 0 on . Corollary . Let be an entire separately harmonic function on C of exponential type If - . . 0 for . 0 - and 1 .-1 . . . 1 . .-1 . . 1 . then 0 on C Theorem . .Let be an entire harmonic function on R2 C and . Z such that. Suppose that - - There is a function R R such that . . .2. . . . 4 . where. Then is a polynomial. Theorem be an entire harmonic function on R2. If satisfies . .N Z-
