Chapter 1. Introduction and Preliminaries

Transcription

1 Chapter Introduction and Preliminaries

2 . Introduction Nevanlinna theory ( also called Value Distribution Theory ) is a branch of complex analysis developed by Finnish mathematician Rolf Nevanlinna. Nevanlinna theory studies the distribution of the roots of the equation f(z) = a, where f is an entire or meromorphic function and a is any complex number. One of the early path breaking results in the theory of value distribution is a theorem by the famous mathematician Picard who proved in 897 that any transcendental entire function f(z) must take every finite complex value infinitely many times, with at most one exception. Later, E. Borel by introducing the concept of exponent of convergence of an entire function, proved his well-known and classical result that the exponent of convergence of the a -points of an entire function of finite order equals its order with at most one exception. This result, generally known as the Picard- Borel theorem, led the foundation for the theory of value distribution. It was R. Nevanlinna who made the decisive contribution to the development of the theory of value distribution. Before him, the principle object and tool of the theory were the class of entire functions and the maximum modulus, respectively. It was Nevanlinna who elevated the theory of meromorphic functions to a new level by introducing the characteristic function T (r, f) for the meromorphic function f, as an efficient tool. In 929, Nevanlinna proved the first and second fundamental theorems, which had been significant breakthroughs in the development of Nevanlinna theory. Picard s and Borel s theorems are immediate consequences of Nevanlinna s second fundamental theorem. Nevanlinna theory has emerged as a powerful tool in the study of uniqueness of meromorphic functions, normal families, complex differential and difference equations etc. 2

3 The uniqueness theory of meromorphic functions mainly studies conditions under which there exists essentially only one function satisfying these conditions. It is well-known that any polynomial is determined by its zero points ( the set on which the polynomial take zeros ) except for a non-constant factor, but it is not true for transcendental entire or meromorphic functions. For example, functions e z and e z have the same ±, 0 and points. Therefore, how to uniquely determine a meromorphic function is interesting and complex. Early, Nevanlinna himself proved that any non-constant meromorphic function can be uniquely determined by five values. In other words, if two non-constant meromorphic functions f and g take same five values at the same points, then f g. Many mathematicians like J.H.Zhu, S.Z.Ye, N.Toda, Q.D.Zhang, Y.H.Yi, Li-Qiao and H.X.Yi etc made significant contribution to the introduction and development of the theory of uniqueness. Five value Theorem is a very important result of Nevanlinna on the uniqueness of meromorphic functions. The multiple values play a very cardinal role in uniqueness of meromorphic functions. The problems of the multiple values and uniqueness of meromorphic functions mainly study the effect of the multiple value on the uniqueness. Many mathematicians such as Q.L.Xiong, L.Yang, Gopalakrishna-Bhoosnurmath, H.C.Xie, W.C.Lu, Ueda, Jiang-Lin, H.X.Yi etc studied uniqueness theory by adopting multiple values and made remarkable works and contribution to the development of the uniqueness theory of meromorphic functions. The foundation of the normal family theory was laid by P.Montel in the early twentieth century. Later on, with in-depth study of Nevanlinna theory, normal family theory developed rapidly. Many mathematicians like Q.L.Xiong, C.T.Chuang, L.Yang, G.H.Zhang and others made contribution to the introduction and development of the theory. The research 3

4 on the normal family theory of meromorphic functions is a very active international subject in recent decades, especially with the foundation of Zalcman-Pang s Lemma. In this period, many Chinese mathematicians such as Y.X.Gu, H.H.Chen, X.C.Pang, M.L.Fang and others have made remarkable works and contribution in the development of the normal family theory. In the theory of differential equations, in many cases, it is impossible to find an explicit solution for a given differential equation. But, Nevanlinna theory offers an efficient way for this problem. The only requirement is that the solution must be meromorphic either in the whole complex plane or in small domain where the growth of the solution is sufficiently large near the boundary of the domain. In 942, H.Wittich, who was the first to make the systematic study in the application of Nevanlinna theory in to the complex differential equations. Later due to the efforts of S.Bank, I.Laine and others, Nevanlinna theory became the leading tool in analyzing solutions of complex differential equations. Logarithmic derivative lemma has numerous applications in complex differential equations and it also plays a crucial role in proving the Nevanlinna s second fundamental theorem. It has many useful results in Nevanlinna theory and vast number of applications in the theory of ordinary differential equations. For instance, Yosida s generalization of Mamquist theorem,depends heavily on the lemma of logarithmic derivative. In contrast to differential equations, non-linear differential equations often admit meromorphic solutions and hence Nevanlinna s value distribution theory is applicable. The foundation of the theory of complex difference equations was laid by Nolund, Julia, Brikhoff, Batchelder and others in the early part of the twentieth century. Later on, Shimomura, Yanagihara and Laine studied non-linear complex difference equations from the view point of Nevanlinna theory. 4

5 We now present some basic results of Nevanlinna theory : Let f(z) be a meromorphic function (i.e analytic except for poles ) and not constant in the in the disc z R (0 < R < ). Firstly, we define positive logarithmic function, for x 0, logx if x log + x = 0 if x < 0 < Clearly logx = log + x log + x The term m(r, f) is called Proximity function of f, which is the average of the positive logarithm of f(z) on the circle z = r and is defined as m(r, f) = m(r,, f) = 2π log + f(re iθ ) dθ. 2π 0 Next, we define the counting function N(r, f) as follows N(r, f) = r 0 n(t, f) n(0, f) dt + n(0, f)logr, t where n(t, f) denotes the number of poles of f(z) in the disc z t, multiple poles are counted according to their multiplicities. n(0, f) denotes the multiplicity of poles of f(z) at the origin ( if f(0) =, then n(0, f) = 0 ). Here, N(r, f) is called the counting function of poles of f(z). Similarly, we can define the counting function N r, for the zeros of f(z). f The characteristic function T (r, f) of f(z) is defined as follows T (r, f) = m(r, f) + N(r, f) Clearly T (r, f) is a non-negative function. It plays a cardinal role in the whole theory of meromorphic functions. 5

6 Let a be a complex number. Clearly, /(f(z) a) is meromorphic in the disc z R. Similarly, we can define the above definitions as Here m r, And N m r, = 2π log + f a 2π 0 f(re iθ ) dθ. can also be written as m(r, f = a) or m(r, a). r n(t, r, = ) n(0, f a 0 t ) dt + n 0, logr, f a where n(t, /(f a)) denotes the number of zeros of f(z) a in the disc z t counting multiplicities and n(0, /(f a)) the multiplicity of zeros of f(z) a at the origin. N r, can be expressed as N(r, f = a) or N(r, a). T ( r, ) ( = m r, f a ) ( + N r, f a ). f a T (r, /(f a)) is said to be the characteristic function of /(f a). We now define the order of a meromorphic function f(z) as and the lower order of f(z) is defined as log + T (r, f) ρ(f) = lim r logr λ(f) = lim r log + T (r, f). logr The characteristic function of products and sums of functions Let a, a 2,..., a p be p complex numbers. By the definition of positive logarithms, we have p log + a j log + a j log + a j log + a j + logp. 6

7 For p meromorphic functions f j (z) (j =, 2,..., p), we have p m r, f j m(r, f j ) (..) ( m r, ) f j m(r, f j ) + logp. (..2) Clearly, the following two inequalities p N r, f j ( N r, hold. ) f j N(r, f j ) (..3) N(r, f j ). (..4) Hence it follows from (..), (..2), (..3) and (..4) that the following inequalities p T r, f j T (r, f j ) ( T r, hold. ) f j T (r, f j ) + logp. We now state some of the basic fundamental results of Naevanlinna theory. Theorem.. [38]. ( First fundamental theorem of Nevanlinna) Suppose that f(z) is meromorphic function in z < R( ) and a is any complex number. Then for 0 < r < R we have T r, = T (r, f) + log c λ + ε(a, r), f a where c λ is the first non-zero co-efficient of the Laurent expansion of f(z) a and ε(a, r) log + a + log2. at the origin, 7

8 Lemma.. [38]. (Lemma of the logarithmic derivative ) Let f(z) be a non-constant meromorphic function in the complex plane. If order of f(z) is finite, then m (r, f ) = O(logr), (r ), f If the order of f(z) is infinite, then m (r, f ) = O (log(rt (r, f))), (r, r E 0 ), f where E 0 is a set whose linear measure is not greater than 2. Theorem..2 [38].( Second fundamental theorem of Nevanlinna) Suppose that f(z) is meromorphic function in z < R( ) and a j (j =, 2,..., q) are q( 2) distinct finite complex numbers. Then for 0 < r < R, we have where m(r, f) + q ( m r, ) 2T (r, f) N (r) + S(r, f), f a j N (r) = 2N(r, f) N(r, f ) + N (r, f ), and S(r, f) = m (r, f ) ( + m r, f q f f a j ) + O(). (..5) Theorem..3 [38]. Let f(z) be a non-constant meromorphic function in the complex plane and S(r, f) be defined as (..5) in Theorem..2. If the order of f(z) is finite, then S(r, f) = O(logr), (r ). If the order of f(z) is infinite, then S(r, f) = O (log(rt (r, f))), (r, r E), where E is a set with finite linear measure. 8

9 We now give a precise form of the Second fundamental theorem of Nevanlinna. Theorem..4 [38]. Suppose that f(z) is meromorphic function in the complex plane and a, a 2,..., a q are q( 3) distinct values in the extended complex plane. Then (q 2)T (r, f) < q N ( r, ) + S(r, f), f a j where S(r, f) is the remainder term with the same properties as in Theorem..3. In this thesis, we investigate uniqueness problems of entire or meromorphic functions concerning differential polynomials sharing fixed points, sharing one value with multiplicity using weighted sharing method, sharing three entire small functions and one small meromorphic function and uniqueness in class A. Finally we study some results of Nevanlinna theory in complex Banach spaces. Our thesis organized into six chapters. Chapter. Introduction and Preliminaries In this chapter, we give brief introduction of Nevanlinna theory of meromorphic functions and few fundamental results to be required in the whole thesis. Chapter 2. Uniqueness of Meromorphic Functions Sharing Fixed Points In this chapter, we study the uniqueness problems of entire or meromorphic functions concerning differential polynomials sharing fixed point and obtain some significant results, which improves the result due to Lin and Yi [2]. Chapter 3. Weighted Sharing and Uniqueness of Entire or Meromorphic Functions In this chapter, we present a unified approach of investigating uniqueness problem of 9

10 entire or meromorphic functions concerning differential polynomials that share one value with multiplicity using weighted sharing method. We prove two main theorems which generalize and improve the results of Fang and Fang [5], Dyavanal [6] and others. This method also yields some new results. In this chapter, we also solve the open problem posed by Dyavanal. Chapter 4. On the Uniqueness Problems in the Class A of Meromorphic Functions In this chapter, we obtain some new results in the class A of meromorphic functions f in C which satisfy the condition N(r, f) + N r, = S(r, f). We study the value f distribution and uniqueness of meromorphic functions of the type f n f (k) in A. We obtain significant results which improve the result of Yang and Hua [42] in class A. It is interesting to note that for functions in class A, the condition on n is greatly improved. In this chapter, we also pose open problem. Chapter 5. Uniqueness of Meromorphic Functions This chapter deals with the various uniqueness problems of meromorphic functions concerning differential polynomials. Section. Uniqueness of Meromorphic Functions Concerning Differential Polynomials In this section, we study the uniqueness problems of meromorphic functions concerning differential polynomials that share three entire small functions as an application of the result due to Anupama Patil [33]. Here our techniques employed are much different and relatively simple and lead to several other results. 0

11 Section 2. Uniqueness of Meromorphic Functions that Share One Small Function with its Differential Polynomials In this section, we study the uniqueness theorems that share one small function with its differential polynomials. We improve the result due to Anupama Patil [33] by considering one small meromorphic function instead of three non-zero distinct entire small functions. As consequence of this result, we obtain several other significant results in this direction. Chapter 6. Some Results of Nevanlinna Theory in Complex Banach Space E In this chapter, we study some results of Nevanlinna theory in complex Banach space E. Section. E -Valued Borel Exceptional Values of Meromorphic Functions In this section, we deal with the Borel exceptional values for meromorphic functions in complex Banach space E. We extend a result due to Gopalakrishna and Bhoosnurmath [29] to meromorphic functions in E. Section 2. On a Result of Wittich and Milloux in Banach Spaces In this section, we obtain results analogous to Wittich and Milloux in complex Banach spaces.

12 .2 Preliminaries Throughout the thesis, we denote by C, the se of all complex numbers and by a meromorphic function f(z), we always mean a non-constant meromorphic function defined in the whole complex plane C. Following are some of the basic definitions and lemmas that are needed further in our thesis..3 Definitions Definition.3. For any non-constant meromorphic function f(z) we denote by S(r, f) any function satisfying S(r, f) = o(t (r, f)) as r, except possibly outside a set of r of finite measure. Definition.3.2 A meromorphic function a is said to be a small function of f provided that T (r, a) = S(r, f) i.e., T (r, a) = o(t (r, f)), as r outside of a possible exceptional of finite linear measure. For any non-constant meromorphic function f, we denote by S(f) the family of all meromorphic functions g ( including constants and the infinite constant ) satisfying T (r, g) = o(t (r, f)). Definition.3.3 Let f(z) be a meromorphic function. A complex number a is said to be a Picard s exceptional value or exceptional value in the sense of Picard or evp of f if n(r, a, f) = O(), i.e, f(z) a has finitely many zeros. We now state the famous Picard theorem which is a simple application of the second fundamental theorem of Nevanlinna. 2

13 Theorem.3. (Picard s theorem). Any transcendental meromorphic function in the complex plane has at most two Picard exceptional values. Definition.3.4 Let f be a transcendental meromorphic function in C with order ρ(f). A complex number a is said to be a Borel exceptional value in the sense of Borel or evb of f if log + N(r, a, f) lim r logr < ρ(f). Clearly, every Picard exceptional value is a Borel exceptional value. We state Borel s theorem on Borel exceptional value. Theorem.3.2 (Borel s theorem). Let f be a transcendental meromorphic function in C with finite positive order λ. Then f has at most two Borel exceptional values. Definition.3.5 [38]. Let f(z) be a non-constant meromorphic function in the complex plane and a be any complex number. The deficiency of a with respect to f(z) is defined by m r, δ(a, f) = lim r T (r, f) = lim r N r, T (r, f) It is obvious that 0 δ(a, f). Definition.3.6 [38]. If δ(a, f) > 0, then the complex number a is named as deficient value of f(z). The deficient value is also called exceptional value in the sense of Nevanlinna. Definition.3.7 [38]. Let f(z) be a non-constant meromorphic function in the complex plane and a be any complex number. We define r, Θ(a, f) = lim r N T (r, f) N, θ(a, f) = lim r r, N r, T (r, f), 3

14 Clearly 0 Θ(a, f), 0 θ(a, f) and δ(a, f) + θ(a, f) = Θ(a, f). Theorem.3.3 [38]. Suppose that f(z) is a non-constant meromorphic function in the complex plane. Then the set of values a for which Θ(a, f) > 0 is countable, and on summing over all such values a {δ(a, f) + θ(a, f)} a a Θ(a, f) 2. An immediate consequence of the above theorem is the following result. Corollary.3. [38]. Suppose that f(z) is a non-constant meromorphic function in the complex plane. Then there are at most countable many deficient values of f(z) and δ(a, f) 2. a 4