A 類半純函數之某些值分佈 - 政大學術集成

全文

(1)國立政治大學應用數學系碩士學位論文. 政治大. 立 Some Value Distribution of Meromorphic. ‧ 國. 學. Functions of Class A. ‧. A 類半純函數之某些值分佈 n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 碩士班學生：陳盈穎撰指導教授：陳天進博士中華民國 100 年 12月.

(2) 謝辭細數我在政大生活的日子，已經六年多了，回憶起過去在大學的日子以及這兩年多在研究所的日子，覺得時間過得好快好快，沒想到，我也到了這個要和政大說再見的時刻，心裡真是有說不盡的感謝。首先，最最要感謝的就是我的指導教授，陳天進老師。謝謝陳老師這些年的教導，從大學時代到現在，老師的課我總是一修再修，從中學到了很多寶貴的知識，在挫折中成長，讓我愈挫愈勇，不害怕失敗。老師的教學熱忱是無人能比的，謝謝老師為了教導我，時常空出自己寶貴的時間，講述一些有趣的數學知識，只為了讓我懂得更多。. 政治大陳老師對學生的關懷是眾所皆知的，當我在課業上有問題或是心情不好立的時候，老師都會主動關心，讓我非常感動。老師是我心目中的偶像，讓我. ‧ 國. 學. 十分的敬佩也是我學習的典範─在教學上，老師認真的態度，還有敬業的精神；在生活上，老師無私的奉獻，還有灑脫的性格。除此之外，更有許許多. ‧. 多與老師一同吃飯的快樂時光，讓身為學生的我們都感受到老師的親切與關. sit. y. Nat. 懷。. er. io. 另外，謝謝蔡炎龍老師與余屹正老師，在我的論文上給予許多幫忙與協. al. iv n C 見。謝謝張宜武老師、符聖珍老師以及劉明郎老師，有你們的關心與鼓勵， hengchi U n. 助。還有謝謝遠道而來的賴恆隆老師，在口試時給予我在論文上寶貴的意. 讓我感受到政大應數的溫暖。謝謝姜志銘老師與宋傳欽老師，讓我在升碩一的暑假參與機率統計研究群，充實了我的暑期生活。謝謝陳政輝老師與陸行老師，碩一時，參與你們的讀書會，讓我學習到很多實用的能力。研究生的日子多半是在研究室度過的，謝謝游竣博，和我一同口試一起畢業的好同學，教甄加油！謝謝林澤佑，在我修改論文時，幫我修正英文的文法，論文加油！謝謝賴哥、兵兵、潘靜儒、潘丞偉、理理人、小強、治陞，很高興跟你們成為同學，這兩年多來一起學習，一起玩樂，祝福大家都可以順利畢業喔！還有謝謝所有研究室的同學以及學弟妹，讓我在研究室的生活不孤單，而且還很歡樂。. i.

(3) 謝謝江泰緯，謝謝你一直以來的陪伴，在我緊張的時候幫我加油打氣；在我難過的時候，帶我去吃好吃的東西，一起看好笑的影片，還會搞笑給我看。在我使用 LaTeX 遇到困難時，總是會停下手邊的事情，立刻幫我解決。謝謝你包容我在論文排版上的吹毛求疵，不管我提出怎樣的要求，你總是不吭一聲的努力達到，真的真的很感謝你。因為有你，讓我的論文順利的完成。謝謝系辦美麗的助教們，琬婷和偉慈，感謝妳們的辛勞，幫我處理很多論文相關的事情。在我口試前，還幫我加油打氣，消除我的緊張，真的很謝謝妳們。. 政治大. 謝謝我最最親愛的爸媽，謝謝你們在背後的支持，雖然你們不能時時陪. 立. 在我的身邊，但是我一直都感受得到你們的關心。謝謝你們讓我無憂無慮的. ‧ 國. 學. 度過了大學以及研究所的求學生涯，即使台北的生活費如此昂貴，你們仍然二話不說讓我到遙遠的台北生活。還記得離開的第一年，思念爸媽的我時常. ‧. 因為想念而流淚，如今，我已漸漸獨立了，也完成了研究所的學業，這都要歸功於你們對我的關懷與愛護。爸爸媽媽，謝謝你們，謝謝你們包容任性的. y. Nat. sit. 我，謝謝你們一直以來支持我，鼓勵我，讓我在台北能夠好好加油，用功學. n. al. er. io. 習。我好愛好愛你們！！！. Ch. i Un. v. 最後謝謝許多關心我、鼓勵我的師長、朋友和家人，謝謝你們，我要畢業了！！！. engchi. 此篇論文謹獻給我親愛的家人、師長和朋友們。陳盈穎謹誌于國立政治大學應用數學系中華民國一百年十二月. ii.

(4) Abstract In this thesis, we study the basic theory of value distribution of meromorphic function of class A. We prove that every meromorphic function of class A has at most two multiple values and the result is sharp. Also, we prove that if a meromorphic function f of class A and its derivative f (k) share a non-zero complex value, then f ≡ f (k) .. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. iii. i Un. v.

(5) 中文摘要在這篇論文裡，我們探討 A 類半純函數的值分佈基本理論。我們證明了每一個 A 類半純函數最多有兩個重值，而這個結果是最佳的情形。進而，我們證明若一個 A 類半純函數 f 與其導數 f (k) 共非零的複數值，則 f ≡ f (k) 。. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. iv. i Un. v.

(6) Contents. 謝辭. i. Abstract. iii. 中文摘要. iv. Content. v. 立. 1. 學. ‧ 國. 1 Introduction. 政治大. 2 Basic Theory of Value Distribution. ‧. 3 Meromorphic Functions of Class A. 3. Nat. sit. y. 13. io. n. al. 21. er. 4 Multiple Values of Meromorphic Functions of Class A. Ch. i Un. v. 5 The Unicity of Meromorphic Functions of Class A. 24. References. 27. engchi. v.

(7) 1. Introduction In this thesis, a meromorphic function will mean meromorphic in the whole. complex plane C. Given two non-constant meromorphic functions f and g and a ∈ C∞ , the extended complex plane. We say that f and g share a IM (ignoring multiplicities) if f − a and g − a have the same zeros ignoring multiplicities. We say that f and g share a CM (counting multiplicities) if f − a and g − a have the same zeros with the same multiplicities. In 1929, R. Nevanlinna [8] proved the following remarkable results which play. 政治大 the theory of value distribution became a fascinating topic. Thereafter, more and 立 more people participated in the research of this theory.. an important role in the area of value distribution of meromorphic functions. Thus. ‧ 國. 學. Theorem 1.1 [8] Let f and g be two non-constant meromorphic functions. If f. ‧. and g share five distinct values in C∞ , then f ≡ g.. sit. y. Nat. io. er. Theorem 1.2 [8] Let f and g be two non-constant meromorphic functions. If f and g share four distinct values a1 , a2 , a3 , and a4 CM, then f is a Möbius transfor-. n. al. Ch. i Un. v. mation of g, two of the values, say a1 and a2 , must be Picard exceptional values of. engchi. f and g, and the cross ratio (a1 , a2 , a3 , a4 ) = −1.. After having these results, one question may be arised. What happen if two non-constant meromorphic functions share four values but not all CM ? Does the conclusion in Theorem 1.2 still hold for the other cases? In 1979 and 1983, G. G. Gundersen [4, 5] proved the following results. The results say that two non-constant meromorphic functions sharing either three values CM and one value IM or two values CM and two values IM are not different from sharing all four values CM.. 1.

(8) Theorem 1.3 [4] Let f and g be two non-constant meromorphic functions. If f and g share three values CM and share a forth value IM, then they share all four values CM, and hence the conclusion of Thereom 1.2 holds.. Theorem 1.4 [5] Let f and g be two non-constant meromorphic functions sharing four values a1 , a2 , a3 , and a4 . If f and g share a1 , a2 CM, and a3 , a4 IM, then f and g share all four values CM, and hence the conclusion of Theorem 1.2 holds.. The remaining case, f and g share one value CM and the other three values IM, is still open and is an interesting research problem.. 政治大. In view of Nevanlinna and Gundersen’s results, it is natural to ask what happen. 立. if two meromorphic functions share the number of values less than four.. ‧ 國. 學. In this thesis, we will study some value distribution of meromorphic functions of class A. Some well-known properties will be discussed and some new results will. ‧. be obtained.. y. Nat. sit. The thesis will be divided into five sections. In section 1, we give some intro-. er. io. ductions of the sharing value problems. In section 2, we review some basic theory. al. n. iv n C U h we In section 4, values of meromorphic e nstudy g c hthei multiple. of value distribution. In section 3, we discuss the basic properties of meromorphic functions of class A.. functions of class A. In section 5, we consider the unicity of meromorphic functions of class A. In the end of section 4 and section 5, we get our main results in this thesis.. 2.

(9) 2. Basic Theory of Value Distribution In this section, we introduce and review some basic facts and notations in. complex analysis and value distribution which will be used throughout the rest of the thesis. For the sake of brevity, proofs are omitted because they are standard and can be found in [1, 3, 6, 9, 10]. In Nevanlinna’s value distribution theory, the following Poisson-Jensen’s formula plays a very important role.. 政治大 be the zeros and poles of f in |z| < R, 1 ≤ µ ≤ M ,. Theorem 2.1 (Poisson-Jensen’s formula) Let 0 < R < ∞ and f be meromorphic in |z| < R and aµ and bν. 立. 1 ≤ ν ≤ N , respectively. If z = reiθ , 0 ≤ r < R, and f (z) 6= 0, ∞, then we have. 2π. log |f (Reiϕ )|. 0. R2 − r 2 dϕ R2 − 2Rr cos(θ − ϕ) + r2. ‧. ‧ 國. 學. Z. 1 log |f (z)| = 2π. M X. y. sit. io. n. al. er. Nat.

(10)

(11)

(12)

(13) N

(14) R(z − aµ )

(15) X

(16) R(z − bν )

(17)

(18)

(19) .

(20)

(21) + log

(22) 2 − log

(23) 2

(24) R − a z R − bν z

(25) µ µ=1 ν=1. i Un. v. By taking z = 0 in Theorem 2.1, we get the Jensen’s formula.. Ch. engchi. Theorem 2.2 (Jensen’s formula) Under the assumptions of Theorem 2.1, if f (0) 6= 0, ∞, then we have 1 log |f (0)| = 2π. 2π. Z. M X. 0. N. X R R log |f (Re )|dϕ − log + . log |a | |b | µ ν µ=1 ν=1 iθ. The assumption f (0) 6= 0, ∞ in Theorem 2.1 can be eliminated. In fact, for 0 ≤ r < ∞, let n(r, f ) denote the number of poles of f in |z| ≤ r counting multiplicities. Consider the Laurent expansion of f at the origin f (z) = cλ z λ + cλ+1 z λ+1 + . . .. 3.

(26) 1 Note that λ = n(0, ) − n(0, f ). Consider the function f  λ R   f (z) if z 6= 0    z g(z) =   λ  if z = 0,   cλ R then we have the generalized Jensen’s formula.. Theorem 2.3 (Generalized Jensen’s formula) Under the assumptions of Theorem 2.1 without the condition f (0) 6= 0, ∞, then we have Z 2π M X 1 1 R log |cλ | = − n(0, ) log R log |f (Reiϕ )|dϕ − log 2π 0 |aµ | f µ=1 +. N X. 立R. log. ‧ 國. |bν |. + n(0, f ) log R,. 學. ν=1. 政治大. where cλ is the first non-zero coefficient of the Laurent expansion of f at 0.. ‧. From now on, meromorphic function means meromorphic in the whole complex. Nat. er. io. sit. y. plane. First of all, we introduce the positive logarithmic function. Definition 2.4 For x ≥ 0,. n. al. i Un. v.   log x if x ≥ 1 log+ x = max{log x, 0} =  0 if 0 ≤ x < 1.. Ch. engchi. Obviously, log+ x is a continuous non-negative increasing function on [0, ∞) 1 1 satisfying log x = log+ x − log+ and | log x| = log+ x + log+ . x x Let f be a meromorphic function, Nevanlinna [8] introduced the following notations. Definition 2.5 For 0 < r < ∞, 1 m(r, f ) = 2π. Z. 2π. log+ |f (reiθ )|dθ.. 0. 4.

(27) Definition 2.6 For 0 < r < ∞, Z r n(t, f ) − n(o, f ) N (r, f ) = dt + n(0, f ) log r, t 0 where n(t, f ) denotes the number of poles of f in the disc |z| ≤ t counting multiplicities. N (r, f ) is called the counting function of f . For 0 ≤ r < ∞, n(r, f ) denotes the number of poles of f (z) in |z| ≤ r counting multiplicities; n(r, f ) denotes the number of poles of f (z) in |z| ≤ r ignoring multiplicities; nk) (r, 1/f ) (resp. n(k (r, 1/f )) denotes the number of zeros of f (z) in |z| ≤ r with order ≤ k (resp. ≥ k) counting multiplicities; nk) (r, 1/f ) (resp. n(k (r, 1/f )) denotes the number of zeros of f (z) in |z| ≤ r with order ≤ k (resp. ≥ k) ignoring multiplicities.. 立. 政治大. ‧ 國. 學. Definition 2.7 For 0 < r < ∞, the function T (r, f ) defined by T (r, f ) = m(r, f ) + N (r, f ). ‧. is called the (Nevanlinna) characteristic function of f .. sit. y. Nat. al. er. io. It is clear that T (r, f ) is a non-negative increasing function and a convex func-. n. tion of log r. Let f be given in Theorem 2.1. It follows from the integration by. Ch. parts in Riemann-Stieltjes integral, we have M X. eZ n g c1h i R. M X. R log = |bν | ν=1. v. n(t, f ) − n(0, f1 ). R log = |aµ | µ=1. and. i Un. t. 0. R. Z 0. dt. n(t, f ) − n(0, f ) dt. t. On the other hand, the generalized Jensen’s formula can be rewritten as 1 2π. Z. 1 = 2π. Z. 2π +. iϕ. log | f (Re ) | dϕ + 0. 0. N X. log. ν=1 2π. R + n(0, f ) log R | bν |.

(28)

(29) M X

(30)

(31) 1 R 1

(32)

(33) dϕ + log + n(0, ) log R + log | cλ |. log

(34)

(35) iϕ f (Re ) | aµ | f +. µ=1. 5.

(36) Therefore, we obtain 1 1 m(R, f ) + N (R, f ) = m(R, ) + N (R, ) + log |cλ |, f f that is, 1 T (R, f ) = T (R, ) + log |cλ |, f which is another form of the generalized Jensen’s formula and is also known as the Nevanlinna-Jensen’s formula.. Theorem 2.8 (Nevanlinna-Jensen’s formula) Let f be a meromorphic function, then for r > 0,. 政治大. 1 T (r, f ) = T (r, ) + log |cλ |, f. 立. where cλ is the first non-zero coefficient of the Laurent expansion of f at 0.. ‧ 國. 學. By the Nevanlinna-Jensen’s formula, we can get the Nevalinna’s first funda-. ‧. mental theorem.. y. Nat. io. sit. Theorem 2.9 (Nevanlinna’s First Fundamental Theorem) Let f be a mero-. er. morphic function and a be a finite complex number. Then, for r > 0, we have. a1l. iv C ) =hT (r, f ) + log |cλ |U +nε(a, r), f −a engchi. n T (r,. where cλ is the first non-zero coefficient of the Laurent expansion of. 1 at 0, and f −a. |ε(a.r)| ≤ log+ |a| + log 2.. Usually, Nevanlinna’s first fundamental theorem is written as T (r,. 1 ) = T (r, f ) + O(1). f −a. Now, we come to the most important theorem in the theory of value distribution, namely, Nevanlinna’s second fundamental theorem.. 6.

(37) Theorem 2.10 (Nevanlinna’s Second Fundamental Theorem) Let f be a nonconstant meromorphic function and aj ∈ C, 1 ≤ j ≤ q, be q distinct finite values (q ≥ 2). Then m(r, f ) +. q X. m(r,. j=1. 1 ) ≤ 2T (r, f ) − N1 (r) + S(r, f ), f − aj 1 ) and f0. where N1 (r) = 2N (r, f ) − N (r, f 0 ) + N (r,. q X f0 f0 S(r, f ) = m(r, ) + m(r, ) + O(1). f f − aj j=1. 政治大 1. Given a ∈ C, by Nevanlinna’s first fundamental theorem,. 立. 1 ) = T (r, f ) − N (r, ) + O(1). f −a f −a. 學. ‧ 國. m(r,. Hence, Nevanlinna’s second fundamental theorem can be rewritten as follows.. ‧. Theorem 2.11 Let f be a non-constant meromorphic function and aj ∈ C∞ , 1 ≤. Nat. n. al. sit. 1 ) − N1 (r) + S(r, f ), f − aj. N (r,. er. io. (q − 2)T (r, f ) <. q X. y. j ≤ q, be q distinct values (q ≥ 3). Then. i n C U where N1 (r) and S(r, f ) are given h aseinnTheorem g c h i 2.10. j=1. v. Note that, in Theorem 2.11, if some aj = ∞, then N (r,. 1 ) should be read f − aj. as N (r, f ). Let n1 (t) = 2n(t, f ) − n(t, f 0 ) + n(t,. 1 ) and let n(t, f ) denote the number of f0. distinct poles of f in |z| ≤ t. Define Z r n(t, f ) − n(0, f ) N (r, f ) = dt + n(0, f ) log r, t 0 which is called the reduced counting function of f.. 7.

(38) Note that, if z0 is a pole of f of order k in |z| ≤ t, then z0 is counted k − 1 times by n1 (r). Similarly, for a finite value a, if z0 is a zero of f − a of order k in |z| ≤ t, then z0 is also counted k − 1 times by n1 (r). Hence, q X. q. N (r,. j=1. X 1 1 ) − N1 (r) ≤ ). N (r, f − aj f − a j j=1. Therefore, we have the third form of Nevanlinna’s second fundamental theorem.. Theorem 2.12 Let f be a non-constant meromorphic function and aj ∈ C∞ , 1 ≤ j ≤ q, be q distinct values (q ≥ 3). Then ) + S(r, f ), 政 N (r,治 f −a 大. (q − 2)T (r, f ) <. 立. q X. 1. j. j=1. where S(r, f ) is given as in Theorem 2.10.. ‧ 國. 學. In Nevanlinna’s second fundamental theorem, the remainder term S(r, f ) is. ‧. a complicated object which can be estimated by using the method of logarithmic. sit. y. Nat. derivative. It turns out that S(r, f ) is small comparing to T (r, f ). In order to make. io. al. er. it clear, we need the concept of the growth of meromorphic function.. v. n. Classically, we use the maximum modulus to measure the growth of an entire function.. Ch. engchi. i Un. Definition 2.13 Let f be a meromorphic function. The order λ of f is defined to be λ = lim sup r→∞. log+ T (r, f ) log r. and the lower order µ of f is defined to be µ = lim inf r→∞. log+ T (r, f ) . log r. Definition 2.14 Let f (z) and a(z) be meromorphic functions. If T (r, a) = S(r, f ), then a(z) is called a small function of f (z).. 8.

(39) Let f be an entire function. Define, for r ≥ 0, M (r, f ) = max |f (z)|. |z|≤r. Then the relation between M (r, f ) and T (r, f ) is given as follows.. Theorem 2.15 Let 0 ≤ r < R < ∞ and f be an entire function, we have T (r, f ) ≤ log+ M (r, f ) ≤. R+r T (R, f ). R−r. In particular, T (r, f ) ≤ log+ M (r, f ) ≤ 3T (2r, f ).. 政治大 By Theorem 2.15, the order and lower order of an entire function are unam立. biguous. Now, we can state the properties of S(r, f ).. ‧ 國. 學. Lemma 2.16 Let f be a non-constant meromorphic function. If f is of finite. (r → ∞).. al. n. f0 ) = O(log(rT (r, f ))), f. Ch. where E is a set of finite measure.. er. io. If f is of infinite order, then m(r,. y. f0 ) = O(log r), f. sit. Nat. m(r,. ‧. order, then. i Un. v. (r → ∞, r ∈ / E),. engchi. Theorem 2.17 Let f be a non-constant meromorphic function and S(r, f ) be defined in Theorem 2.10. If f is of finite order, then S(r, f ) = O(log r),. (r → ∞).. If f is of infinite order, then S(r, f ) = O(log (rT (r, f ))), where E is a set of finite measure.. 9. (r → ∞, r ∈ / E),.

(40) In the thesis, we will denote by S(r, f ) any quantity satisfy S(r, f ) = o(T (r, f )) as r → ∞ if f is of finite order, and S(r, f ) = o(T (r, f )) as r → ∞, r ∈ / E if f is of infinite order, where E is a set of finite measure. By Lemma 2.16, m(r,. f0 ) = S(r, f ). Moreover, Milloux [7] proved the following. f. Theorem 2.18 Let f be a non-constant meromorphic function and k be a positive integer and let Ψ(z) =. k X. ai (z)f (i) (z),. i=1. where a1 (z), a2 (z), . . . , ak (z) are small functions of f . Then. 治政 Ψ m(r, ) = S(r, f ). 大 f. 立. ‧ 國. 學. For three small functions, we still have the generalization of second fundamental theorem.. ‧ sit. y. Nat. Theorem 2.19 [9] Let f be a non-constant meromorphic function and a1 (z), a2 (z). io. n. al. T (r, f ) <. 3 X. Cj=1 h. N (r,. er. and a3 (z) are three distinct small functions. Then. 1 ) + S(r, f ). f − aj. engchi. i Un. v. In 1929, Nevanlinna [9] introduced the quantity δ(a, f ) to measure the degree of a meromorphic function misses a value a.. Definition 2.20 Let f be a non-constant meromorphic function and a ∈ C∞ . The deficiency of a with respect to f is defined by δ(a, f ) = lim inf r→∞. 1 m(r, f −a ). T (r, f ). 10. = 1 − lim sup r→∞. 1 N (r, f −a ). T (r, f ). ..

(41) Definition 2.21 Let f be a non-constant meromorphic function and a ∈ C∞ . We define Θ(a, f ) = 1 − lim sup r→∞. and θ(a, f ) = lim inf. 1 ) N (r, f −a. T (r, f ). ,. 1 1 N (r, f −a ) − N (r, f −a ). T (r, f ). r→∞. .. Clearly, 0 ≤ δ(a, f ) ≤ 1, 0 ≤ Θ(a, f ) ≤ 1 and 0 ≤ θ(a, f ) ≤ 1. Also, 0 ≤ δ(a, f ) + θ(a, f ) ≤ Θ(a, f ). By Theorem 2.12, we have. 政治 X 大 δ(a, f ) + θ(a, f ) ≤ Θ(a, f ) ≤ 2. 立. Theorem 2.22 Let f be a non-constant meromorphic function. Then X a. a. ‧ 國. 學. In order to study uniqueness theorems of meromorphic functions, we state a. ‧. Nevanlinna theorem which plays an important role.. y. Nat. sit. Theorem 2.23 [9] Suppose f1 , . . . , fn are linearly independent meromorphic func-. n. al. er. io. tions satisfying the following identity n X. C h j=1 fj ≡ 1. U n i engchi. v. Then for 1 ≤ j ≤ n, we have T (r, fj ) ≤. n X. N (r,. k=1. −. n X. 1 ) + N (r, fj ) + N (r, D) fk. N (r, fk ) − N (r,. k=1. 1 ) + o(T (r)), D. where D is the Wronskian of f1 , . . . , fn , and T (r) = max {T (r, fk )}, 1≤k≤n. E is a set with finite linear measure.. 11.

(42) Moreover, we can get a useful result in the uniqueness theorem of meromorphic functions.. Theorem 2.24 [9] Let fj (j = 1, 2, 3) be meromorphic functions where f1 be not a constant function. If 3 X. fj (z) ≡ 1,. j=1. and. 3 X. 3 X 1 N (r, ) + 2 N (r, fj ) < (λ + o(1))(T (r)) fj j=1 j=1. where λ < 1,. (r ∈ I),. 政治大. T (r) = max {T (r, fj )},. 立. 1≤j≤3. and I is a set of r ∈ (0, ∞) with infinite measure, then f2 (z) ≡ 1 or f3 (z) ≡ 1.. ‧ 國. 學. Finally, we review some theorems which will be needed in the following sections.. ‧. er. io. sit. y. Nat. Theorem 2.25 [9] Suppose that f is a meromorphic function in |z| < R and af (z) + b g(z) = , where a, b, c, and d are constant satisfying ad − bc 6= 0. Then for cf (z) + d 0 < r < R, we have. n. a lT (r, g) = T (r, f ) + O(1). i v n Ch U engchi. Theorem 2.26 [9] If f is a transcendental meromorphic function in the complex plane, then T (r, f ) = ∞. r→∞ log r lim. 12.

(43) 3. Meromorphic Functions of Class A Let A denote the collection of all non-constant meromorphic functions f sat-. isfying 1 N (r, f ) + N (r, ) = S(r, f ). f Such functions are called meromorphic functions of class A. Clearly, ez , zez , . . . are functions of class A.. Proposition 3.1 If f is a non-constant rational function, write. 政治大. ap z p + ap−1 z p−1 + · · · + a0 f (z) = , bq z q + bq−1 z q−1 + · · · + b0. 立. ‧ 國. 學. where ap (6= 0), ap−1 , . . . , a0 and bq (6= 0), bq−1 , . . . , b0 are complex numbers, p, q are non-negative integers satisfying p + q ≥ 1, and ap z p + ap−1 z p−1 + · · · + a0 and. ‧. bq z q + bq−1 z q−1 + · · · + b0 have no common factors. Then   (p − q) log r + O(1) if p > q, m(r, f ) =  O(1) if p ≤ q. n. al. er. io. sit. y. Nat. and. Ch. i Un. N (r, f ) = q log r + O(1). engchi. v. holds for a sufficiently large r. Thus,. T (r, f ) = max {p, q} log r + O(1).   (p − q) log r + O(1) if p > q, P roof . First, we prove that m(r, f ) =  O(1) if p ≤ q, P (z) = ap z p + ap−1 z p−1 + · · · + a0 and Q(z) = bq z q + bq−1 z q−1 + · · · + b0 . Let A(r) =. |ap−1 | 1 |a0 | 1 |bq−1 | 1 |b0 | 1 + ··· + and B(r) = + ··· + . p |ap | r |ap | r |bq | r |bq | rq. 13.

(44) Given ε > 0 , there exists r0 > 0, such that |A(r)| < ε and |B(r)| < ε for r ≥ r0 . So for all r ≥ r0 and |z| = r, (1 − ε)|ap |rp ≤ |P (z)| ≤ (1 + ε)|ap |rp and (1 − ε)|bq |rq ≤ |Q(z)| ≤ (1 + ε)|bq |rq . Let α =. (1 − ε)|ap | (1 + ε)|ap | and β = , then for all r ≥ r0 , (1 + ε)|bq | (1 − ε)|bq | αrp−q ≤ |f (z)| ≤ βrp−q .. 政治大 |m(r, f ) − (p − q) log r| ≤ M, 立. 學. ‧ 國. If p > q and r ≥ 0, then there exists M > 0, such that. which means that m(r, f ) = (p − q) log r + O(1). If p = q, then for all r ≥ r0 and |z| = r,. ‧. α ≤ |f (z)| ≤ β.. y. Nat. sit. So we get log α+ ≤ m(r, f ) ≤ log β + , which means that m(r, f ) = O(1).. al. n. we get m(r, f ) = O(1).. er. io. If p < q, then by choosing r > 0, such that 0 < αrp−q < 1 and rp−q < 1,. Ch. engchi. i Un. v. Now, we prove N (r, f ) = q log r + O(1) for a sufficiently large r. Choose r0 > 0 such that Q(z) has q zeros in |z| < r0 . We may assume that Q has a zero at z = 0 of multiple m ≥ 0. Then for all r ≥ r0 , Z r n(t, f ) − n(o, f ) N (r, f ) = dt + n(0, f ) log r t 0 Z r n(t, f ) − n(o, f ) = dt + n(0, f ) log r t δ = (q − m)(log r − log δ) + m log r = q log r − (q − m) log δ = q log r + O(1),. 14.

(45) where δ > 0 is small. Therefore, we have T (r, f ) = max {p, q} log r + O(1) for a sufficiently large r. q In general, the following class of meromorphic function is of class A.. Proposition 3.2 Let α be a non-constant entire function and h be a non-zero rational function. Then f (z) = h(z)eα(z). 政治大. is a meromorphic function of class A.. 立. ‧ 國. 學. P (z) be a nonzero rational function, deg P (z) = p, deg Q(z) = Q(z) q, (P (z), Q(z)) = 1. Choose r0 > 0, so that all zeros of P (z) and Q(z) lie in |z| < r0 . P roof . Let h(z) =. Then by Theorem 3.1, for all r ≥ r0 , we have. n. al. 1 N (r, ) = p log r. h. Ch. er. io By assumption, e. α(z). sit. y. Nat. and. ‧. N (r, h) = q log r. i Un. v. is an entire function without zeros, so we have. engchi. N (r, f ) = N (r, h) ≤ N (r, h) = q log r and 1 1 1 N (r, ) = N (r, ) ≤ N (r, ) = p log r, f h h which imples that 1 N (r, f ) + N (r, ) ≤ q log r + p log r. f Therefore, we get 1 N (r, f ) + N (r, ) = S(r, f ) f by Theorem 2.26. So we conclude that f ∈ A.. 15. q.

(46) Proposition 3.3 If f is a meromorphic function of class A, then so is. 1 . f. 1 P roof . Since f ∈ A, we have N (r, f ) + N (r, ) = S(r, f ). f 1 By the Theorem 2.25 , T (r, f ) = T (r, ) + O(1), we have f 1 1 N (r, f ) + N (r, ) = S(r, ), f f and 1/f is a meromorphic function of class A.. q. However, if f and g are of functions of class A, f + g and f g may not be of. 政治大 functions of class A, but f + g and f h are not meromorphic functions of class A. 立. class A; for example, f (z) = ez , g(z) = −ez and h(z) = e−z are of meromorphic. ‧ 國. 學. Proposition 3.4 All functions in A are transcendental meromorphic functions.. ‧. P (z) be a non-constant rational function, deg P (z) = p, Q(z) deg Q(z) = q, (P (z), Q(z)) = 1 and p + q ≥ 1. Choose r0 > 0, so that all zeros of P roof . Let f (z) =. sit. y. Nat. n. al. er. io. P (z) and Q(z) lie in |z| ≤ r0 . Then by Theorem 3.1, for all r ≥ r0 , N (r, f ) = q log r,. Ch. engchi. 1 N (r, ) = p log r, f. i Un. v. and T (r, f ) = max {p, q} log r + O(1). Now, assume that P (z) has s distinct zeros and Q(z) has t distinct zeros, then, s ≤ p, t ≤ q, and for all r ≥ r0 , we have N (r, f ) = t log r and 1 N (r, ) = s log r. f 16.

(47) Therefore, lim. N (r, f ) + N (r, f1 ) T (r, f ). r→∞. =. r+s max {p, q}. which is not zero. So f is not a meromorphic function of class A.. q. Proposition 3.5 Let f be a meromorphic function with Θ(0, f ) = Θ(∞, f ) = 1, then f ∈ A.. P roof . Since Θ(0, f ) = Θ(∞, f ) = 1, we have Θ(0, f ) = 1 − lim sup. 立. Θ(∞, f ) = 1 − lim sup r→∞. T (r, f ). = 1,. ‧ 國. N (r, f ) + N (r, f1 ). N (r, f1 ) N (r, f ) ≤ lim sup + lim sup r→∞ T (r, f ) r→∞ T (r, f ). ‧. T (r, f ). Nat. y. = 1 − Θ(0, f ) + 1 − Θ(∞, f ) = 0.. al. er. io. sit. lim. r→∞. =1. 學. which imply. T (r, f ). 政治大 N (r, f ) r→∞. and. N (r, f1 ). v. n. 1 Therefore, N (r, f ) + N (r, ) = o(T (r, f )) = S(r, f ) and f ∈ A. f Remark.. Ch. engchi. i Un. q. In the literature, a non-constant meromorphic function f satisfying. Θ(0, f ) = Θ(∞, f ) = 1 is called meromorphic functions of class K.. Proposition 3.6 Let f be a meromorphic function with δ(0, f ) = δ(∞, f ) = 1, then f ∈ A.. P roof . Since δ(0, f ) = δ(∞, f ) = 1, we have Θ(0, f ) = Θ(∞, f ) = 1. By Proposition 3.5, we get f ∈ A.. q. 17.

(48) Remark.. In the literature, a non-constant meromorphic function f satisfying. δ(0, f ) = δ(∞, f ) = 1 is called meromorphic functions of class F. For further properties of meromorphic functions of class A, we recall the following proposition. Proposition 3.7 [9] If f ∈ A and k is a positive integer, then (i) T (r,. f (k) ) = S(r, f ); f. (ii) T (r, f (k) ) = T (r, f ) + S(r, f );. 政治大. (iii) f (k) (z) ∈ A.. 立. ‧. ‧ 國. 學. 1 P roof . Since f ∈ A, we have N (r, f )+N (r, ) = S(r, f ). In particular, N (r, f ) = f 1 S(r, f ) and N (r, ) = S(r, f ). f f (k) ) = S(r, f ). Therefore, By Lemma 2.16 , m(r, f f (k) f (k) f (k) ) = N (r, ) + m(r, ) f f f 1 ≤ k{N (r, f ) + N (r, )} + S(r, f ) f = S(r, f ),. n. al. er. io. sit. y. Nat. T (r,. which implies (i).. Ch. engchi. i Un. v. By the basic property of characteristic function & (i), we have. T (r, f (k) ) ≤ T (r,. f (k) ) + T (r, f ) ≤ T (r, f ) + S(r, f ). f. Similarly, we have T (r, f ) ≤ T (r, f (k) ) + T (r,. f. ) f (k) f (k) = T (r, f (k) ) + T (r, ) + O(1) f = T (r, f (k) ) + S(r, f ).. 18.

(49) We obtain T (r, f (k) ) = T (r, f ) + S(r, f ). So, (ii) holds.. Finally, by (i) & (ii), we have N (r, f (k) ) = N (r, f ) = S(r, f ) = S(r, f (k) ), and N (r,. 1 f. ) ≤ N (r, (k) ≤ T (r,. f. 1 N (r, ) ) + f (k) f f f (k). ) + S(r, f ). 政治大. f (k) ) + S(r, f ) f = S(r, f ). ≤ T (r,. 立. ‧ 國. 1 f (k). ) = S(r, f (k) ) and (iii) holds.. 學. Therefore, N (r, f (k) ) + N (r,. q. Now, we can prove the main result in this section.. ‧. Nat. y. sit. n. al. er. io. Theorem 3.8 Let a and b be distinct complex numbers and f be a non-constant 1 1 meromorphic function satisfies N (r, ) + N (r, ) = S(r, f ). Then f is a f −a f −b Möbius transformation of a function in class A. Ch. engchi. i Un. v. P roof . Consider the meromorphic function g defined by g(z) =. f (z) − b . f (z) − a. Then, by Theorem 2.25, T (r, f ) = T (r, g) + O(1). Obviously, we have N (r, g) = N (r,. 1 ) f −a. and 1 1 N (r, ) = N (r, ). g f −b. 19.

(50) Therefore, by assumption, we have 1 1 1 ) + N (r, ) N (r, g) + N (r, ) = N (r, g f −a f −b = S(r, f ) = S(r, g), which says that g is a function of class A. By a simple calculation, we get f=. ag − b g−1 q. which says that f is a Möbius transformation of g.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 20. i Un. v.

(51) 4. Multiple Values of Meromorphic Functions of Class A. Definition 4.1 Let f be a non-constant meromorphic function, and a ∈ C∞ . We say that a is a multiple value of f if all the zeros of f (z) − a are multiple. Example 4.2 0 is a multiple value of f (z) = (z − 1)2 (z + 1)4 . Example 4.3 0, ∞ are multiple values of ez. For general meromorphic functions, we have the following well-known result about multiple values.. 立. 政治大. Theorem 4.4 Let f be a non-constant meromorphic function, then f has at most. ‧ 國. 學. four distinct multiple values.. ‧. P roof . Suppose that f has five distinct multiple values, say a1 , a2 , a3 , a4 , a5 ∈ C∞ .. y. Nat. n. al. 5 X. Ch. N (r,. j=1. 1 ) + S(r, f ) f − aj. er. io. (5 − 2)T (r, f ) <. sit. By Theorem 2.12,. i Un. v. 5 1 X 1 ≤ N (r, ) + S(r, f ) 2 j=1 f − aj. engchi. 5 1 X T (r, f ) + S(r, f ) 2 j=1 5 = T (r, f ) + S(r, f ), 2. ≤. which is a contradiction. So f has at most four multiple values.. q. In fact, there exists a meromorphic function which has exact four multiple values, namely, the well-known Weierstrass ℘-function ℘(z) which satisfies the differential equation ℘0 (z)2 = 4(℘(z) − e1 )(℘(z) − e2 )(℘(z) − e3 ),. 21.

(52) where e1 , e2 , e3 are distinct constants. It is obvious that e1 , e2 , e3 and ∞ are multiple values of ℘(z). Therefore, Theorem 4.4 is sharp. Now, we consider the case of meromorphic function of class A and prove our main result in this section as follows.. Theorem 4.5 Let f be a meromorphic function of class A, then f has at most two multiple values.. P roof . Suppose that f has three distinct multiple values, say a1 , a2 , a3 ∈ C∞ .. 政治大 1 N (r, f ) + N (r, ) = S(r, f ). 立 f. Since f ∈ A, we have. ‧ 國. 學. Case 1. a1 , a2 , a3 are different from 0 and ∞. Then, by Theorem 2.12,. io. n. al. er. Nat. 3 1 X 1 ) + S(r, f ) ≤ N (r, 2 j=1 f − aj 3 ≤ T (r, f ) + S(r, f ), 2. which is impossible.. y. j=1. 1 1 ) + N (r, f ) + N (r, ) + S(r, f ) f − aj f. sit. N (r,. ‧. (5 − 2)T (r, f ) ≤. 3 X. Ch. engchi. i Un. v. Case 2. One of a1 , a2 , a3 is 0 or ∞. (4 − 2)T (r, f ) ≤. 2 X j=1. N (r,. 1 1 ) + N (r, f ) + N (r, ) + S(r, f ) f − aj f. 2 1 X 1 ≤ N (r, ) + S(r, f ) 2 j=1 f − aj. ≤ T (r, f ) + S(r, f ), which is also impossible.. 22.

(53) Case 3. Two of a1 , a2 , a3 are 0 and ∞. 1 1 ) + S(r, f ) (3 − 2)T (r, f ) ≤ N (r, f ) + N (r, ) + N (r, f f −a 1 = N (r, ) + S(r, f ) f −a 1 1 N (r, ) + S(r, f ) ≤ 2 f −a 1 ≤ T (r, f ) + S(r, f ), 2 which is a contradiction. q. Therefore, f has at most two multiple values.. 政治大 multiple values, namely, 0 and ∞. Therefore, Theorem 4.5 is sharp. 立. The function f (z) = z 2 ez is of class A by Proposition 3.2, and it has exact two. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 23. i Un. v.

(54) 5. The Unicity of Meromorphic Functions of Class A In this section, we will discuss the sharing value problem of meromorphic func-. tion of class A and obtain some results. In order to state and prove the theorems, we need some preliminaries.. Lemma 5.1 [9] If f ∈ A and a is a finite non-zero number, then N 1) (r, where N 1) (r,. 1 ) = T (r, f ) + S(r, f ), f −a. 政治大. 1 ) denotes the counting function of simple zeros of f − a. f −a. 立. ‧. ‧ 國. a proof.. 學. The following result is stated without proof in [9]. For completeness, we give. Theorem 5.2 [9] Let f and g be meromorphic functions of class A and a be a. y. Nat. er. io. g f and if necessary, we may assume that a = 1. a a. al. n. P roof . By considering. sit. non-zero complex number. If f and g share a IM, then either f ≡ g or f g ≡ a2 .. Ch. By Lemma 5.1, we have. engchi. i Un. v. N 1) (r,. 1 ) = T (r, f ) + S(r, f ) f −1. N 1) (r,. 1 ) = T (r, g) + S(r, g). g−1. and. Hence, N(2 (r,. 1 ) = S(r, f ) f −1. N(2 (r,. 1 ) = S(r, g), g−1. and. 24.

(55) 1 ) is the counting function of f − 1 with multiplicities greater or f −1 1 equal to 2, similarly for N(2 (r, ). g−1. where N(2 (r,. Since f and g are meromorphic functions of class A and they share 1 IM, by Theorem 2.12, we have T (r, g) = T (r, f ) + S(r, f ). Set h(z) =. f (z) − 1 . g(z) − 1. Obviously, we have = S(r, f ), 政治g −1 1 )大. N (r, h) ≤ N (r, f ) + N (2 (r,. 立. ‧ 國. 學. 1 1 N (r, ) ≤ N (r, g) + N (2 (r, ) = S(r, f ), h f −1. and. sit. j=1. N (r,. al. n. j=1. 3 X 1 )+2 N (r, fj ) = S(r, f ). fj j=1. er. io. 3 X. fj ≡ 1 and. y. Nat. Let f1 = f , f2 = h, f3 = −hg. Then. 3 X. ‧. T (r, h) ≤ T (r, f ) + T (r, g) + O(1) ≤ 2T (r, f ) + S(r, f ).. Ch. engchi. i Un. v. By Theorem 2.24, we conclude that either f2 ≡ 1 or f3 ≡ 1 which imply that either f ≡ g or f g ≡ 1 and the proof is completed.. q. Finally, we consider the sharing value problem of a meromorphic function with its derivative. The following well-known result has been proved by Frank-Weissenborn [2] in 1986.. Theorem 5.3 Let f be a non-constant meromorphic function and k ≥ 1. If f and f (k) share distinct finite value a and b CM, then f ≡ f (k) .. 25.

(56) For meromorphic functions of class A, we can use Theorem 5.2 to obtain the following result.. Theorem 5.4 Let f be a non-constant meromorphic function of class A, a be a non-zero complex number and k ≥ 1. If f and f (k) share a IM, then f ≡ f (k) .. P roof . Since f ∈ A, by Proposition 3.7, f (k) ∈ A. Thus, we can apply Theorem 5.2 to conclude that either f ≡ f (k) or f f (k) ≡ a2 . If f f (k) ≡ a2 holds, then T (r,. 政治大. f (k) a2 ) = T (r, 2 ) f f = 2 T (r, f ) + O(1).. Which contradicts to Proposition 3.7, namely, T (r,. 學. ‧ 國. 立. f (k) ) = S(r, f ). f. ‧. Therefore, we must have f ≡ f (k) .. n. er. io. sit. y. Nat. al. Ch. engchi. 26. i Un. v. q.

(57) References [1] C.-T. Chuang and C.-C. Yang, Fix-points and factorization of meromorphic functions, World Scientific Publishing Co. Inc., Teaneck, NJ, 1990. Translated from the Chinese. [2] G. Frank and G. Weißenborn, Meromorphe Funktionen, die mit einer ihrer Ableitungen Werte teilen, Complex Variables Theory Appl., 7 (1986), pp. 33–43. [3] F. Gross, Factorization of meromorphic functions, Mathematics Research. 政治大. Center, Naval Research Laboratory, Washington, D. C., 1972.. 立. [4] G. G. Gundersen, Meromorphic functions that share three or four values, J.. ‧ 國. 學. London Math. Soc. (2), 20 (1979), pp. 457–466.. , Meromorphic functions that share four values, Trans. Amer. Math. Soc.,. ‧. [5]. 277 (1983), pp. 545–567.. y. Nat. sit. [6] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs,. n. al. er. io. Clarendon Press, Oxford, 1964.. Ch. i Un. v. [7] H. Milloux, Les fonctions méromorphes et leurs dérivées. Extensions d’un. engchi. théorème de M. R. Nevanlinna. Applications, Actualités Sci. Ind., no. 888, Hermann et Cie., Paris, 1940. [8] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Chelsea Publishing Co., New York, 1974. Reprinting of the 1929 original. [9] C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, vol. 557 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 2003.. 27.

(58) [10] L. Yang, Value distribution theory, Springer-Verlag, Berlin, 1993. Translated and revised from the 1982 Chinese original.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 28. i Un. v.

(59)