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Báo cáo toán học: "Uniqueness Theorems for Harmonic and Separately Harmonic Entire Functions on CN"

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 Guelfond.1. Giới thiệu cũng được biết đến định lý cổ điển của Carlson (xem [2, p.153]) 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 B.P 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 1.1.Let 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 1.2. .Let be an entire function on C of exponential type.If. N Z then is a polynomial. Theorem 1.3.Let be an entire function on C an integer greater than one. If Ịg are integers for.and satisfies the inequality .2. . -. 4 .2 where R R satisfies . then is a polynomial. -- 2. Notations and Results . C.N. 1 . . 1 .C. .C. Theorem 2.1. 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 2.2. Let be an entire harmonic function on C of exponential type . If.for.and. then 0 on C Theorem 2.3. 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 2.4. 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 2.5. .Let be an entire harmonic function on R2 C and . Z such that. Suppose that - 0.ỊỊ.-.N - There is a function R R such that .2.and . .2. . . . 4 . where. Then is a polynomial. Theorem 2.6.Let be an entire harmonic function on R2. If satisfies . .N Z-