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MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN AN HAI ON THE UNIQUENESS AND FINITENESS OF FAMILIES OF ADMISSIBLE MEROMORPHIC FUNCTIONS ON AN ANNULUS IN THE COMPLEX PLAN[.]
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN AN HAI ON THE UNIQUENESS AND FINITENESS OF FAMILIES OF ADMISSIBLE MEROMORPHIC FUNCTIONS ON AN ANNULUS IN THE COMPLEX PLANE Major: Geometry and Topology Code: 9.46.01.05 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ha Noi - 2023 This dissertation has been written at Hanoi National University of Education Supervisor: Prof Dr Si Duc Quang Referee 1: Prof Dr Sc Ha Huy Khoai - Thang Long University Referee 2: Assoc Prof Dr Sc Ta Thi Hoai An - Institute of Mathematics Referee 3: Prof Dr Tran Van Tan - Ha Noi National University of Education INTRODUCTION Motivation of the problem In 1926, R Nevanlinna showed that two nonconstant distinct meromorphic functions on the complex plane must coincide to each other if they have the same inverse images of five distinct values regardless of multiplicity, and that they are linked by a Măobius transformation if they have the same inverse images of four distinct values counted with multiplicity These results were known as Nevalinna’s five and four values theorems These two results are obtained by using Nevanlinna’s Second Fundamental Theorem for meromorphic functions with fix targets which are values in C ∪ {∞} In the past decades, many mathematicians have been interested in expanding and deepening Nevanlinna’s results by replacing the condition that has the same inverse for few values by the condition that has the same for few small functions The first results in this direction were obtained by G Gundersen, P Li, C C Yan In 2004, K Yamanoi has established the Second Fundamental Theorem for meromorphic functions on the complex plane for few small functions with counting functions truncated to level Yamanoi’s result have become a key and powerful tool in developing Nevalinna’s five and four values theorems for the case of meromorphic functions that share the inverse of small functions Nevalinna’s five and four values theorems have been largely extended by the recent publications of S D Quang and S D Quang - L N Quynh However, there are still no similar results for the for the case of meromorphic functions on doubly connected domain Here, we note that by the Doubly Connected Mapping Theorem each doubly connected domain is conformally equivalent to a annulus A = {z; ≤ r < |z| < R ≤ +∞}, which is biholomorphic to A(R0 ) = {z; R10 < |z| < R0 } for some R0 > In particular, there is no sharp second main theorem on annuli as good as the result of Yamanoi Therefore, there are almost no studies on the finiteness or uniqueness problem of meromorphic functions on the annuli with the condition of small functions are almost From the above reasons, we choose the topic "On the uniqueness and finiteness of families of admissible meromorphic functions on an annulus in the complex plane", to extend Nevanlinna’s results to the case of meromorphic functions on annuli Purpose of research The aim of the thesis is to study the the finiteness problem and uniqueness problem of admissible meromorphic functions on an annulus have the same inverse images (with multiplicity truncated by a certain level) of few values or few small functions or some pair of values with truncated multiplicities, with conditions that are more general, weaker than in previous studies, or have not been studied on these issues Moreover, in the situations that we studied, the techniques and methods of the previous authors could not be solved Object and scope of research The research object of the thesis is meromorphic functions on annuli Research scope in Value Distribution Theory Research Methods We based on the research methods, the traditional techniques of Complex Geometry and Value Distribution Theory, and we added new techniques to solve the problems posed in the thesis Scientific and practical significances The thesis contributes to enriching and deepening the results on the uniqueness and finiteness of meromorphic functions on annuli The thesis is also one of the references for bachelor, master and PhD students in this direction of research Structures of thesis Together with the Introduction; Conclusion and recommendations; Works related to the thesis; References, the thesis includes four chapters: Chapter Overview Chapter Two meromorphic functions on an annulus having the same inverse images of few small functions Chapter Finiteness problem of families of meromorphic functions on an annulus having the same inverse images of the four values Chapter Two meromorphic functions on an annulus sharing some pairs of values The thesis is based on four articles, published in journals:Complex Analysis and Operator Theory (SCIE); Mathematica Bohemica (ESCI/Scopus); Bulletin of the Iranian Mathematical Society (SCIE); Indagationes Mathematicae (SCIE) The place where the thesis is carried out This dissertation has been written at Hanoi National University of Education Chapter OVERVIEW In this chapter, we summarize the results of previous authors on the finiteness problem and the uniqueness problem of meromorphic functions on the complex plane Next, we state the new results we have obtained in the study the finiteness problem and uniqueness problem of meromorphic functions on an annulus have the same inverse images (with multiplicity truncated by a certain level) of few values or few small functions or some pairs of values I Two meromorphic functions on an annulus having the same inverse images of few small functions Nevalinna’s four and five values theorems have attracted the interest of many mathematicians and they have expanded and developed it in several directions The first direction is to ignore inverse images with multiples greater than a certain value The second direction is to replace the values with small functions In 1999, I Yuhua and Q Jianyong generalized the result of Nevanlinna to the case where the five distinct values are replaced by five small functions a1 , , a5 In 2002, W Yao gave an improvement of this result He showed that if k ≥ 22 and min{νf −ai ,≤k , 1} = min{νg−ai ,≤k , 1} (1 ≤ i ≤ 5), then f = g H X Yi showed that the result of Yao holds for k ≥ 14, and Đ Đ Thai and T V Tan showed that this result still holds for k ≥ by using the Yamanoi’s Second Main Theorem for meromorphic functions and small functions S Đ Quang improved the result of Đ Đ Thai and Tan by proving that two functions f and g must coincide if min{νf −ai ,≤3 , 1} = min{νg−ai ,≤3 , 1} (1 ≤ i ≤ 3) and min{νf −ai ,≤2 , 1} = min{νg−ai ,≤2 , 1} (4 ≤ i ≤ 5) However, the above results are proven only for meromorphic functions on the complex plane The proofs of the above results is based on the Cartan auxiliary function and Yamanoi’s Second Main Theorem for meromorphic functions and small functions on C with counting functions truncated to level We note that Yamanoi’s Second Main Theorem is in this case the best possible theorem Recently, Khrystiyanyn and Kondratyuk proposed the Nevanlinna theory for meromorphic functions on annuli A(R0 ), then completed by M Lund and Z Ye Using the Second Main Theorem for meromorphic functions on annuli, T B Cao, H X Yi and H Y Xu proved a uniqueness theorem for meromorphic functions on an annulus sharing at least five values As far as we know, this is the only result for meromorphic functions on an annulus having the same inverse images of the few values In particular, the problem of meromorphic functions on an annulus having the same inverse images of several small functions has not been studied In this thesis, we refer to the uniqueness problem of meromorphic functions on an annulus having the same inverse images of the few small functions However, as far as we know, in the case of meromorphic functions on an annulus there is still no Second Main Theorem as good as the result of K Yamanoi This is the main difficulty in studying this problem Therefore, the study of meromorphic functions on an annulus based on Value Distribution Theory has many limitations In Chapter 2, write based on the article [1] (in Works related to the thesis), by using new techniques and in combination with those of P Li, without Yamanoi’s theorem we have established the uniqueness theorem for admissible meromorphic functions on an annulus sharing at least five small functions without counting multiplicities Moreover, in our result, all intersection points of such meromorphic functions with small functions not need to be counted if their multiplicities are bigger than a certain number Our results are stated as follows Theorem Let f, g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , , aq (q ≥ 5) be distinct small (with respect to f and g ) functions on A(R0 ) Let k1 , , kq be q positive integers or +∞ with q X 2q(q − 4) 2q < + , k 5(q + 4) 5k i i=1 where k0 = max1≤i≤q ki Assume that min{νf0−ai ,≤ki , 1} = min{νg−a , 1}, ∀1 ≤ i ≤ q i ,≤ki Then f = g Our above results not only generalize, but also improve most of the results on transcendental meromorphic functions that have the same inverse images of at least five small functions on the complex plane In addition, in the case of admissible meromorphic functions f and g sharing four distinct small functions counted with multiplicity, we will show that they are linked by a quasi-Măobius transformation Namely, we have proved the following theorem Theorem Let f and g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , a2 , a3 , a4 be four distinct small (with respect to f and g ) functions P Assume on A(R0 ) Let ki (1 ≤ i ≤ 4) be positive integers with 4i=1 ki1+1 < 219 that νf0−ai ,≤ki = νg−a , ∀1 ≤ i ≤ q i ,≤ki Then f is a quasi-Măobius transformation of g Our above theorem is a generalization and improvement of Nevalinna’s four values theorem to the case of admissible meromorphic functions having the same inverse images of four distinct small on an annulus II Finiteness problem of families of meromorphic functions on an annulus having the same inverse images of the four values In 1998, H Fujimoto improved the Four Values Theorem of Nevanlinna by proving that there are at most two meromorphic functions on C which share four distinct values with multiplicities truncated by level This kind of such results are called finiteness theorems for meromorphic function sharing values For the case of meromorphic functions on C, there are many extension of the Four Values Theorem However, as far as we known there is still no finiteness theorems for the case of meromorphic functions on doubly connected domain sharing four values, for intance on annuli In this thesis, we refer to the finiteness problem of meromorphic functions on an annulus having the same inverse images of the four values In Chapter 3, write based on the article [2] (in Works related to the thesis), we prove that if three admissible meromorphic functions f1 , f2 , f3 on an annulus share four distinct values regardless of multiplicity and have the complete identity set of positive counting function then f1 = f2 or f2 = f3 or f3 = f1 Namely, we have proved the following result Theorem Let f1 , f2 , f3 be three meromorphic functions on an annulus A(R0 ) and let a1 , a2 , a3 , a4 be four distinct values in C ∪{∞} Assume that f1 , f2 , f3 share a1 , a2 , a3 , a4 regardless of multiplicities If f1 is admissible and f1 , f2 , f3 have the identity complete set of positive counting function then f1 = f2 or f2 = f3 or f3 = f1 From Theorem it follows that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicities truncated by level and sharing three other values regardless of multiplicities In Chapter 3, write based on the article [3] (in Works related to the thesis), we have also extent and improve the Four Values Theorems of Nevanlinna and Fujimoto to the case of admissible meromorphic functions on an annulus We have proven that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicities truncated by and other three values regardless of multiplicities Furthermore, in these results, the inverse images of these values by the functions with multiplicities more than a certain number not need to be counted Namely, we have proved the following result Theorem Let f1 , f2 , f3 be three meromorphic functions on the annulus A(R0 ) Let a1 , , a4 be for distinct values in C ∪ {∞} Let k1 , , k4 be four positive integers or +∞ with 11 11 11 14 + + + < k1 + k2 + k3 + k4 + Assume that (i) min{νf01 −a1 ,≤k1 , 2} = min{νf02 −a1 ,≤k1 , 2} = min{νf03 −a1 ,≤k1 , 2}, (ii) min{νf01 −ai ,≤ki , 1} = min{νf02 −ai ,≤ki , 1} = min{νf03 −ai ,≤ki , 1}, ∀ ≤ i ≤ If f1 is admissible then f1 = f2 or f2 = f3 or f3 = f1 We also show that that there are no more than three nonconstant meromorphic functions on an annulus having the same inverse images of four values regardless of multiplicity Moreover, in our result, the inverse images of these values by the functions with multiplicities more than a certain number not need to be counted Namely, we have proved the following result Theorem Let f be a nonconstant meromorphic function on A(R0 ) Let a1 , , a4 be for distinct values in C ∪ {∞} Let k1 , , k4 be positive integers or maybe +∞ with 4 17 X 1 25 25 X + < + , 64 i=1 ki 16 i=1 ki + 32 32k0 where k0 = max1≤i≤4 ki Then ]V(f, {ai , ki }4i=1 , 1) ≤ III Two meromorphic functions on an annulus having the same inverse images of some pairs of values In 1997, T Czubiak and G Gundersen proved the following Theorem Let f and g be two non constant meromorphic functions on C that share six pairs of values (ai , bi ), ≤ i ≤ 6, IM (ignoring multiplicities), where , bi ∈ C and 6= aj , bi 6= bj whenever i 6= j , i.e., min{νf0−ai , 1} = min{νg−b , 1} (1 ≤ i 6) i Then f is a Măobius transformation of g After that, the sharing pairs of values problem of meromorphic functions has been studied by many authors For the case of meromorphic functions on C, there are some extensions of the above results of T Czubiak and G Gundersen, where the pairs of values are replaced the by pairs of small functions For example, in 2014 S Đ Quang and L N Quynh prove that two meromorphic functions on C must be linked by a quasi-Măobius transformation if they share a pair of small functions ignoring multiplicities and share other four pairs of small functions with multiplicities truncated by At the same time, Quang and Quynh also proved that the two meromorphic functions on C must be linked by a quasi-Măobius transformation if they share the same inverse, regardless of multiples, of q (q ≥ 6) pairs of small functions However, as far as we known there is still no such theorem for the case of meromorphic functions on doubly connected domains sharing pairs of values Therefore, in this thesis, we pose the problem of studying the case where admissible meromorphic functions on an annulus having the same inverse images of pairs of values Theorem 2.2.2 Let f : A(R0 ) → PN (C) be a linearly nondegenerate holomorphic mapping Let {Hi }qi=1 (q ≥ N + 2) be a set of q hyperplanes in PN (C) in general position Then (q − N − 1)T0 (r, f ) ≤ q X [N ] N0 (r, f ∗ Hi ) + Sf (r), i=1 where f ∗ Hi denotes the pull back divisor of Hi by f and f ∗ H = ν(f,H) 2.3 Two meromorphic functions on an annulus having the same inverse images of at least five small functions Our result is stated as follows Theorem 2.3.1 Let f, g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , , aq (q ≥ 5) be distinct small (with respect to f and g ) functions on A(R0 ) Let k1 , , kq be q positive integers or +∞ with q X 2q(q − 4) 2q < + , k 5(q + 4) 5k i i=1 where k0 = max1≤i≤q ki Assume that min{νf0−ai ,≤ki , 1} = min{νg−a , 1}, ∀1 ≤ i ≤ q i ,≤ki Then f = g Corollary 2.3.2 Let f, g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , , aq (q ≥ 5) be distinct small (with respect to f and g ) functions 3q+28 Assume that on A(R0 ) Let k be a positive integer or +∞ with k > 2(q−4) νf0−ai ,≤k , = νg−a , , ∀1 ≤ i ≤ q i ,≤k Then f = g When R0 = +∞ then from Corollary 2.3.2 we get the result of W Yao for class of transcendental meromorphic functions: If k ≥ 22 and min{νf0−ai ,≤k , 1} = min{νg−a , 1} (1 ≤ i ≤ 5) then f = g So, our above results not only generalize, i ,≤k but also improve most of the results on meromorphic functions that have the same inverse images of at least five small functions on the complex plane In order to prove Theorem 2.3.1 we need to prepare the following lemma 13 Lemma 2.3.3 Let f and g be two distinct admissible meromorphic functions on an annulus A(R0 ) and let a1 , , aq (q ≥ 5) be distinct small functions with respect to f Suppose that min{νf0−ai ,≤ki (z), 1} = min{νg−a (z), 1} (1 ≤ i ≤ q) i ,≤ki for all z outside a analytic subset A of counting function equal to Sf (r) Then we have q X X [1] [1] [1] N0 (r, νf0−ai ) ≤ N0 (r, νf0−ai ,>ki ) + N0 (r, νg−a ) +Sf (r) + Sg (r) i ,>ki i=5 2.4 i=1 Two meromorphic functions on an annulus having the same inverse images of four small functions Our result is stated as follows Theorem 2.4.1 Let f and g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , a2 , a3 , a4 be four distinct small (with respect to f and g ) P functions on A(R0 ) Let ki (1 ≤ i ≤ 4) be positive integers with 4i=1 ki1+1 < 219 Assume that νf0−ai ,≤ki = νg−a ∀1 ≤ i ≤ q i ,≤ki Then f is a quasi-Măobius transformation of g Corollary 2.4.2 Let f and g be two admissible meromorphic functions on an annulus A(R0 ) Let a1 , a2 , a3 , a4 be four distinct small (with respect to f and g ) functions on A(R0 ) Let k be a positive integer with k > 218 Assume that νf0−ai ,≤k = νg−a i ,≤k Then f is a quasi-Măobius transformation of g 14 Chapter FINITENESS PROBLEM OF FAMILIES OF ADMISSIBLE MEROMORPHIC FUNCTIONS ON AN ANNULUS HAVING THE SAME INVERSE IMAGES OF THE FOUR VALUES The main goal of Chapter is to prove the finitenes theorems for the family of meromorphic functions on an annulus which share four distinct values Chapter is written based on articles [2] and [3] (in Works related to the thesis) 3.1 Some auxiliary results In this section, we prove some new results of Nevanlinna theory for the proof of the main theorems in this chapter Lemma 3.1.1 Let f be an admissible meromorphic function on A(R0 ) and let a1 , a2 , a3 be three distinct values in C ∪ {∞} Let g be a meromorphic function on A(R0 ) such that f and g share all a1 , a2 , a3 regardless of multiplicities Then we have T0 (r, f ) = O(T0 (r, g)) + Sf (r) and T0 (r, g) = O(T0 (r, f )) + Sg (r) as r −→ R0 In particular g is admissible 15 Lemma 3.1.2 Let f be a nonconstant meromorphic function on A(R0 ) and let a ∈ C then for every positive integer k (maybe k = +∞) We have T0 (r, f ) + Sf (r) k+1 k + [1] [1] and N0 (r, νf −ai ,≤k ) ≥ N0 (r, νf0−a ) − T0 (r, f ) + Sf (r) k k Lemma 3.1.3 Let f be an admissible meromorphic function on A(R0 ) and let a1 , , a4 be four distinct values in C∪{∞} Let k1 , , k4 be four positive integers or maybe +∞ such that X < k + i i=1 [1] N0 (r, νf0−a,>k ) ≤ Let g be a meromorphic function on A(R0 ) such that min{1, νf0−ai ,≤ki } = min{1, νg−a } (1 ≤ i ≤ 4) i ,≤ki Then we have T0 (r, f ) = O(T0 (r, g)) + Sf (r) and T0 (r, g) = O(T0 (r, f )) + Sg (r) as r −→ R0 In particular g is admissible 3.2 Finiteness of meromorphic functions on an annulus having the same inverse images of four values regardless of multiplicity In the following theorem, we show that there are at most two admissible meromorphic functions on an annulus having the same inverse images of four values regardless of multiplicity Theorem 3.2.1 Let f1 , f2 , f3 be three meromorphic functions on an annulus A(R0 ) and let a1 , a2 , a3 , a4 be four distinct values in C ∪ {∞} Assume that f1 , f2 , f3 share a1 , a2 , a3 , a4 regardless of multiplicities If f1 is admissible and f1 , f2 , f3 have the identity complete set of positive counting function then f1 = f2 or f2 = f3 or f3 = f1 In order to prove Theorem 3.2.1 we need to prepare the following lemma In all three lemmas, let f1 , f2 , f3 be three meromorphic functions on A(R0 ) and let a1 , a2 , a3 , a4 be four distinct values in C \ {0} satisfying the following two conditions: (1) f1 , f2 , f3 share four values a1 , , a4 regardless of multiplicities, 16 (2) f1 is an admissible meromorphic function The quantities Sf1 (r), Sf2 (r), Sf3 (r) denoted them by the same notation S(r) We set T0 (r) = T0 (r, f1 ) + T0 (r, f2 ) + T0 (r, f3 ) For i ∈ {1, , 4}, we put Fik = (fk − )/fk Lemma 3.2.2 If f1 , f2 , f3 are distinct then the following assertions hold: (1) 2T0 (r, fk ) = X N0 (r, νi ) + S(r), ≤ k ≤ 3, i=1 (2) N0 (r, C ) = S(r), (3) N0 (r, νi,s ) = S(r) ∀1 ≤ i ≤ 4, ≤ s ≤ Lemma 3.2.3 If f1 , f2 , f3 are distinct then Cartan’s auxiliary function Φ(Fi1 , Fi2 , Fi3 ) 6≡ for every ≤ i ≤ Lemma 3.2.4 Let i be an index in {1, , 4} and let Cartan’s auxiliary function Φ := Φ(Fi1 , Fi2 , Fi3 ) If f1 , f2 , f3 are distinct then N0 (r, νi,0 ) + X [1] N0 (r, νj ) ≤ N0 (r, νΦ0 ) ≤ T0 (r) + S(r) j=1 j6=i From Theorem 3.2.1, we will show that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicities truncated by level and three other values regardless of multiplicities For detail, we have the following corollary Corollary 3.2.5 Let f1 , f2 , f3 be three meromorphic functions on an annulus A(R0 ) and let a1 , a2 , a3 , a4 be four distinct values in C ∪ {∞} Assume that f1 , f2 , f3 share a1 with multiplicities counted to level and share a2 , a3 , a4 regardless of multiplicities If f1 is admissible then f1 = f2 or f2 = f3 or f3 = f1 With weaker assumption that the meromorphic functions on an annulus share all four values regardless of multiplicities, Theorem 3.2.1 also implies the following corollary Corollary 3.2.6 Let f1 , f2 , f3 , f4 be four meromorphic functions on an annulus A(R0 )) and a1 , a2 , a3 , a4 be four distinct values in C ∪ {∞} Assume that f1 , f2 , f3 , f4 share all a1 , a2 , a3 , a4 regardless of multiplicities If f1 is admissible then there are two functions among {f1 , f2 , f3 , f4 } identify to each other 17 3.3 Finiteness of meromorphic functions on an annulus having the same inverse images of four values, and all inverse images of those values with multiplicities more than a curtain number not need to be counted In the following theorem, we will show that there are at most two meromorphic functions sharing a value with multiplicities truncated by level and sharing three other values regardless of multiplicities Theorem 3.3.1 Let f1 , f2 , f3 be three meromorphic functions on the annulus A(R0 ) (1 ≤ R0 ≤ +∞) Let a1 , , a4 be for distinct values in C ∪ {∞} Let k1 , , k4 be four positive integers or +∞ with 11 11 11 14 + + + < k1 + k2 + k3 + k4 + Assume that (i) min{νf01 −a1 ,≤k1 , 2} = min{νf02 −a1 ,≤k1 , 2} = min{νf03 −a1 ,≤k1 , 2}, (ii) min{νf01 −ai ,≤ki , 1} = min{νf02 −ai ,≤ki , 1} = min{νf03 −ai ,≤ki , 1}, ∀2 ≤ i ≤ If f1 is admissible then f1 = f2 or f2 = f3 or f3 = f1 Corollary 3.3.2 Let f1 , f2 , f3 be three nonconstant meromorphic functions on the annulus A(R0 ) Let a1 , , a4 be for distinct values in C ∪ {∞} Let k be a positive integer or +∞ with k > 46 Assume that (i) min{νf01 −a1 ,≤k , 2} = min{νf02 −a1 ,≤k , 2} = min{νf03 −a1 ,≤k , 2}, (ii) min{νf01 −ai ,≤k , 1} = min{νf02 −ai ,≤k , 1} = min{νf03 −ai ,≤k , 1} ∀2 ≤ i ≤ If f1 is admissible then f1 = f2 or f2 = f3 or f3 = f1 When R0 = +∞ then from Corollary 3.3.2 we get an extension and improvement of Fujimoto’s result In order to prove Theorem 3.3.1 we need to prepare the following lemmas Lemma 3.3.3 With the assumption of Theorem 3.3.1, if f1 , f2 , f3 are distinct then 4 X X [1] N0 (r, Ti ) ≤ 2T0 (r) − N0 (r, νf01 −ai ,≤ki ) i=1 i=1 18