解析函数零点位置的某些结果_英文_

解析函数零点位置的某些结果_英文_ Article ID:1005-3085(2004)05-0785-07 Some Results on the Location of Zeros of Analytic Functions 1 1 2 2LIN Ai-guo, HUANG Bing-jia, CAO Jian-sheng, Robert Gardner (1-University of Petroleum, Shandong Dongying 2-Department of Mathematics East Tennessee State University Johnson City, Tennessee 37614) .nv Abstract: The classical Enestr?om-Kakeya Theorem states that if p(z) = azis a polynomialv u=0 such that 0 ? a? a? a? ? ? ? ? a,then all of the zeros of p(z) lie in the region0 1 2 n |z| 1 in the complex plane. Many generalizations of the Enestrom-Kakeya Theorem exist ? ? which put various conditions on the coefficients of the polynomial (such as mononicity of the moduli of the coefficients). In this paper, we will introduce several results which put conditions on the coefficients of even powers of Z and on the coefficients of odd powers of Z. As a consequence, our results will be applicable to some analytic functions to which these related results are not applicable. Keywords:analytic funetions; lecation of zeres mononicity Classification: AMS(2000) 49K35 CLC number: O174 Document code: A 1 Introduction .nv The classical Enestr?om-Kakeya Theorem states that if p(z) = azis a polynomial v v=0 such that 0 ? a? a? a? ? ? ? ? a,then all of the zeros of p(z) lie in the region |z| ? 1 in 0 1 2 n the complex plane. Many papers (cf.[2?4]) put various conditions of coefficients of a polynomial and obtained several results about the location of zeros of a polynomial by using mononicity of moduli of coefficients of a polynomial, others studied the location of zeros of a polynomial by using mononicity of real and imaginary parts of coefficients of a polynomial. In [1],the authors studied the location of zeros of an analytic function by putting conditions on moduli of coefficients of an analytic function. The following is main result discussed in theorem 6 in [1]. .?π v Theorem 1 Let f (z) = azƒ= 0 be analytic in |z| ? t. If |arga| ? α ? , v j v=02 k k+1 ? 0, 1, 2, 3, ? ? ?} and for finite nonnegative integer k,|a| ? |a| ? ? ? ? ? a? a? ? ? ?j {ttt0 1 k k+1 then (z) does not vanish in f t ..|z| < .? 2 sin α k j(2t|a/a| ? 1) cos α + sin α + t|a|k 0 j 1|a| 0 It is will known that analytic functions such as sine,cosine, exponential and logarithm func- tions have many applications in the practical problems. Finding the locations of zeros of analytic functions is a widely useful topic in complex analysis, since the locations of zeros are the main date: 2003-07-08. LIN in 1964.9),Male, Maiter Degree Senior Eng Received Biography: Aiguo(Born 1 -ineer eathodic protection technology and oil-water treatment technolgy. 786 CHINESE JOURNAL OF ENGINEERING MATHEMATICS VOL. 21 properties of the analytic functions. In this paper, we will put conditions on the coefficients of even powers of Z and on the coefficients of odd powers of Z, and give the example to illustrate that our results will be applicable to some analytic functions to which these related results are not applicable. 2 The Main Results and Applications Motivated by theorem 6 in [1], we put restriction on moduli of odd and even coefficients. .?µ Theorem 2 If P (z) = azis an analytic function in |z| ? T and | arg a? β| ? µ j µ=0 π α ? for j ? {0, 1, 2 ? ? ? , } and for some real β and some nonnegative integers k and l and some 2 positive t such that t ? T 2 4 2k 2k+2 |a| ? |a|? |a|? ? ? ? ? |a|? |a|? ? ? ? ? ? ? ? tttt0 2 4 2k 2k+2 2 4 2l?2 2l |a| ? |a|t? |a|t? ? ? ? ? |a 2l+1 |t? |a ?|t? ? ? ? ? ? ? ? 1 3 5 2l1 Then p(z) does not vanish in |z| < Rwhere 1 .. 3 t|a| 0 R= min , t 1 M 1 Here 2 2k+2 2l+1 M= ?|a|t cos α + |a|t(1 ? cos α) + 2(|a|t+ |a ?|t) cos α+ 1 0 1 2k 2l1 ?.2 i (|a|t+ |ai i ?|)tsin α 2 =2 i .?µ Theorem 3 Let P (z) = azbe an analytic function in |z| ? T with real coefficients µ µ=0 such that a? a? a? ? ? ? ? 0 2 4 a? a? a? ? ? ? ? 1 3 5 Then p(z) does not vanish in |z| < min(R, t) where 1 t| a| 0 =R 1 M 1 and M= |a| + 2|a| 1 0 1 We now apply this Theorem 3 to a specific analytic function. 2 4 6 z z z Example 2.1. Consider p(z) = 1 ? + ? + ? ? ? = cos z which is an analytic function2! 4! 6! in |z| ? ?. Then according to Corollary 3, p(z) does not vanish in |z| < 1, that is, all zeros of cos z satisfy |z| ? 1. We can not apply theorem 1 ([1]) to cos z because the coefficients of cos z do not satisfy the condition in theorem 1([1]). In the following theorem, we obtain the following inequality by using generalization of Schwarz’sinequality . 787 NO. 5 LIN Aiguo et al: Some Results on the Location of Zeros of Analytic Functions .?µ Theorem 4 If P (z) = azis an analytic function in |z| ? T and | arg a? β| ? µ jµ=0 π ? 0, 1, 2 ? ? ? , and for some real β , and some nonnegative integers k and thereα ? for j {} l, 2 exists some positive such that ? T andt t 2 4 2k 2k+2 |a| ? |a|? |a|? ? ? ? ? |a|? |a|? ? ? ? ? ? ? ? tttt0 2 4 2k 2k+2 2 4 22 l?2l |a| ? |a|t? |a|t? ? ? ? ? |a 2l+1 |t? |a ?|? ? ? ? ? ? ? ? t 1 3 5 2l1 Then p(z) does not vanish in |z| < Rwhere 1 . .1 4 2 8 2 2 2 3 4 2 t|a|(t|a| ? M) + {t|a|(|a|t? M)+ 4M t|a|} 1 1 0 1 0 1 0 1 , tR= min 1 2 2M 1 Here 2 2k+2 2l+1 M= ?|a|t cos α + |a|t(1 ? cos α) + 2(|a|t+ |a ?|t) cos α+ 1 0 1 2k 2l1 ?.2 i (|a|t+ |a ?|)tsin α i i 2 i=2 As inspired from [2] by putting restriction of real and imaginary parts of analytic function, we get the following theorem. .?µ Theorem 5 Let P (z) = azbe an analytic function in |z| ? T and Re(a) = α, µ j j µ=0 I m(a) = βand for some nonnegative integers k,l,s and q , there exists a positive t such that j j ? T and t 2 4 2k 2k+2 α? αt? αt? ? ? ? ? αt? αt? ? ? ? ? ? ? ?0 2 4 2k 2k+2 2 4 22 l?2l 2l+1 α? αt? αt? ? ? ? ? α ? αt? t? ? ? ? ? ? ? ? 1 3 5 2l1 2 4 2s 2s+2 β? βt? βt? ? ? ? ? βt? βt? ? ? ? ? ? ? ? 0 2 4 2s 2s+2 2 4 2q?2 2q β? βt? βt? ? ? ? ? βt? β2q+1 ? t? ? ? ? ? ? ? ? 1 3 5 2q1 Then p(z) does not vanish in |z| ? Rwhere 1 . . t| a| 0 , tR= min 1 M 1 Here M= (|α| + |β|)t ? (α+ β)t ? (α+ β)+ 1 1 1 1 1 0 0 2s 2q?1 2k 2l?1 t+ β ]t2[αt+ α t+ β2k 2l1 2s2q?1 ? The following inequality of analytic function is obtained by using generalization of Schwarz’s inequality and mononicity of real and imaginary parts. .?µ Theorem 6 Let P (z) = azbe an analytic function in |z| ? T and Re(a) = α, µ j j µ=0 m(a) βand for some nonnegative integers k ,,s and q , there exists a positive such that I= lt jj t ? T and 2 4 2k 2k+2 α? αt? αt? ? ? ? ? αt? αt? ? ? ? ? ? ? ?0 2 4 2k 2k+2 2 4 2l?2 2l α? αt? αt? ? ? ? ? α t? α2l+1 ? t? ? ? ? ? ? ? ? 1 3 5 2l1 2 4 2s 2s+2 β? βt? βt? ? ? ? ? βt? βt? ? ? ? ? ? ? ? 0 2 4 2s 2s+2 788 CHINESE JOURNAL OF ENGINEERING MATHEMATICS VOL. 21 2 4 2q?2 2q β? βt? βt? ? ? ? ? β2q+1 ?t? β t? ? ? ? ? ? ? ? 1 3 5 2q1 Then p(z) does not vanish in |z| ? Rwhere 1 ..1 2 4 2 2 3 2 2 t|a|(|a| ? M) + {t|a|(|a| ? M)+ 4M t|a|} 1 0 1 1 0 1 0 1 , tR= min 1 2 2M 1 Here M= (|α| + |β|)t ? (α+ β)t ? (α+ β)+ 1 1 1 1 1 0 0 2l?1 2s 2q?1 2k t+ βt+ β t]2[αt+ α 1 2s2k 2l2q?1 ? 3 Proof of the Main Results We need the following lemma which is the generalization of Schwarz’sinequality to prove the main theorems. r Lemma 1 Let f (z) be analytic in |z| < R, f (0) = 0, f = b, and |f (z)| ? M for (z) |z| = R , then for |z| ? R 2 M |z| M |z| + R |b|| (z)| ? f2 R M |z ||b| + The following Lemma is due to Aziz and Mohammad [1]. .nπµ Lemma 2 Let P (z) = azbe analytic in |z| ? such that | arg a? β| ? α ? t µ jµ=0 2 for j ? {0, 1, 2 ? ? ? , n} and for some real β , and positive t and nong ative integer k, 2 k k+1 n|a| ? |a| ? |a|? ? ? ? ? |a|? |a|? ? ? ? ? |a| ttttt0 1 2 k k+1 n Then for ? 1, 2 ? ? ? , n j {} |ta? a| ? (t|a? |a|) cos α + (t|a| + |a|) sin α. 1 1 1 jj?jj?jj? Proof of Theorem 2 Proof Consider the following analytic function g(z) ? .2 2 2 2 2 i 2 g(z) = (t? z)p(z) = ta+ atz + (at? a)zaG(z) = t + 0 1 ii?2 01 i=2 on |z| = t ?. 3 2 i|G(z)| ? |a|t+ |at? a ?|t 1 1 i i 2 i=2 By using Lemma 1 in the above, we obtain ?. 3 2 2 i|G(z)| ? |a|[(||a|? |at+ t1 1 ||) cos α + (|a|t+ |a ?2 iii? |) sin α)]t 2 i i=2 2 3 2k+2 2+1 l? ?|a|tcos α + |a|t(1 ? cos α) + 2(|a|t+ |a ?|t) cos α+ 0 1 2k 2l1 ?.2 ii+ (|a|t+ |a ?|)tsin αt= M i i 12 i=2 789 NO. 5 LIN Aiguo et al: Some Results on the Location of Zeros of Analytic Functions Then follows from Schwarz’slemma, therefore it M| z| 1 |G(z)| ? or |z| ? ft1 t Which implies 2 |g(z)| = |ta+ G(z)| 0 1 2 ? t|a| ? |G(z)| 0 1 M| z| 1 2 ? |a| ? or |z| ? tft0 t 3t |a| 0 Therefore, if |z| ? R= min{ , t}, then g(z) ƒ= 0 and so p(z) ƒ= 0 that is, p(z) does not1 M 1vanish in |z| ? R1 Proof of Theorem 4 Proof Consider the following analytic function ? .2 2 2 2 2 i 2 g(z) = (t? z)p(z) aaz (at? a= t+ t+ )z= ta+ G(z) 0 1 ii?2 01 i=2 on |z| = t ?. 3 2 i|G(z)| ? |a|t+ |at? a ?|t 1 1 ii 2 =2 iBy using the above Lemma 1, we obtain ?. 3 2 2 i|G(z)| ? |a|t+ [(||a|t? |a1 1 ||) cos α (|a||a 2 ii+ t+ i?? |) sin α)]t 2 ii=2 2 3 2k+2 2l+1 ? ?|a|tcos α + |a|t(1 ? cos α) + 2(|a|t+ |a |t) cos α+ ? 0 1 2k 2l1 ?.2 i i + (|a|t+ |a ?|)tsin αt= M ii 2 1i=2 Then follows from Lemma 2, therefore it 4 M|z| M|z| + t|a|1 1 1 |G(z)| ?|z| ? tfor 1 2 2 tMt ||a| 1|z + 1 Which implies 2 |g(z)| = |ta+ G(z)| 0 1 2 ? |a| ? |G(z)| t 0 1 4 M|z| M|z| + t|a| 2 1 1 1 ? t|a| ?for |z| ? t0 2 2tM+ |z||a|t1 1 . . 1 4 2 8 2 2 2 3 4 2 t |a |}t|a|(t?M)+{t|a| (t?M)+4M 1 1 1 1 0 1 Therefore, if |z| ? R= min, t , then g(z) ƒ= 0 and21 2M 1 so p(z) ƒ= 0 that is, p(z) does not vanish in |z| ? R 1Proof of Theorem 5 790 CHINESE JOURNAL OF ENGINEERING MATHEMATICS VOL. 21 Proof We consider the following analytic function ? . 2 2 2 2 2 i 2 g(z) = (t? z)p(z) = ta+ atz + (at? a)zaG(z) = t + 0 1 ii?2 01 i=2 on |z| = t ?. 3 2 i|G(z)| ? |a|t+ |at? a ?|t 1 1 ii 2 =2 i ?. 3 2 i 2 i? (|α| + |β|)t+ (|αt? α1 1 i i i?2 ||βt? β ?t+ |t) 2 i=2 i 2s+2 2q+1 2+1 l 3 3 2 2k+2 t+ βt]t+ β? (|α| + |β|)t? (α+ β)t? (α+ β)t+ 2[αt+ α 1 2s2q?1 ? 1 1 1 1 0 0 2k 2l 2 = tM 1 We apply Schwarz’stheorem [5, p.168] to G(z),we get 1 2 tM|z|1 |G(z)| ?= tM|z|, f or |z| ? t1 1 t Which implies 2 2 2 |g(z)| = |ta+ G(z)| ? t|a| ? |G(z)| ? t|a| ? tM|z| f or|z| ? t 0 1 0 1 0 1 ..t|a| 0 Hence, if |z| ? R= min , t,then g(z) ƒ= 0 and so p(z) ƒ= 0. that is, p(z) does not vanish 1 M 1in |z| ? R1 Proof of Theorem 6 Proof Consider the following analytic function ? . 2 2 2 2 2 i 2 g(z) = (t? z)p(z) = ta+ atz + (at? a)zaG(z) = t + 0 1 ii?2 01 i=2 on |z| = t ?. 3 2 i|G(z)| ? |a|t+ |at? a ?|t 1 1 i i 2 i=2 ?. 3 2 i 2 i ? (|α| + |β|)t+ (|αt? α1 1 2 iii??|t+ |βt? β |t) 2 i i=2 2s+2 2q+1 2l+1 3 3 2 2k+2 t+ βt]t+ β? (|α| + |β|)t? (α+ β)t? (α+ β)t+ 2[αt+ α 2s1 2q1 ?? 1 1 1 1 0 0 2k 2l 2 = tM 1 We apply Lemma 2 to G(z),we get 1 2 M|z|(M|z| + t|a|1 1 1 |G(z)| ?|z| ? t), for 1 M+ |z ||a| 1 1 Which implies 2 M|z|(M|z| |a|+ t1 1 1 2 2 2 |g(z)| = |ta+ G(z)| ? t|a| ? |G(z)| ? t|a| ?|z| ? tfor0 1 0 1 0 M+ |z||a| 1 1 . . 1 2 4 2 2 3 2 2 t|a|(|a|?M)+{t|a|(|a|?M)+4M 1 0 1 1 0 1 0, t ,then g(z) Hence, if |z| ? R= min=ƒ 0 and21 M 2 1 t |a |}1 so p(z) ƒ= 0. That is, p(z) does not vanish in |z| ? R1 791 NO. 5 LIN Aiguo et al: Some Results on the Location of Zeros of Analytic Functions References: [1] Aziz A, Mohammad Q G. On the zeros of a certain class of polynomials and related analytic functions[J]. J Math Appl, 1980;75:495-502 [2] Robert B, Gardner N K. Govil, On the location of the zeros of a polynomial[J]. 1994;78(2):286-292 [3] Atif Abueida, Robert B. Gardner Some results on the location of zeros of a polynomial,Submitted, consequence of conditions [4] Cao Jiansheng, Robert Gardner. Restriction of the Zeros of a polynomial as a on the coefficients of even powers and odd power of the variables, Submitted. [5] Titchmarsh E C. The theory of functions[M]. 2nd ed OxfordUniv Press, London. ?????,,? :?? %$1 1 2 ?ι?, ??}, ?%?Robert 2Gardner (1-??$?C?fi3,ftfifi?; 2-East Tennessee State University) .n iaz%~?,? 0 ? a? a? a?? :? Enestrom-Kakeya ??p(z) þ%??%$:?= i 0 1 2 =0 i ? ? ? ? a%fi?fi,? p(z) %?$?A??? |z| ? 1 %???:??ofi?fi%fi??Lfi??? n ,? C?,fi?%%???3,???%fi% Enestr?om-Kakeya ‰f%??o$@$,???{? L Z %?? ??? ? ?%fi? ,??? %~:??o }??: ??? ?:?A,?:??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? (1?796 ) Forced Oscillation of Systems of Nonlinear Neutral Parabolic Partial Functional Differential Equations YANG Jun, WANG Chun-yan, LI Jing (Department of Mathematics, Yanshan University, Qinhuangdao 066004) Abstract: This paper studies the systems of nonlinear neutral parabolic partial functional differential equations with continuous distributed deviating arguments, Sufficient conditions are obtained for the forced oscillation of solutions of the systems. Keywords: parabolic partial functional equations; nonlinear neutral type; continuous distributed deviating arguments; systems; the forced oscillation