Remarks on Bohr's Phenomenon

Transcription

1 The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics Remarks on Bohr's Phenomenon Presented By Faten S. Abu Shoga Supervised By Prof. Dr. Jasser H. Sarsour SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENE 48 ھ م

2 Contents ACKNOWLEDGMENTS ABSTRACT ii iii INTRODUCTION PRELIMINARIES 5. Topological Spaces Complex Space Holomorphic Functions Caratheodory s Inequality Bohr s Theorem 4. Bohr s Theorem For One Complex Variable A new Generalization of Bohr Radius Bohr Radius of Majorant Series Types of Bohr Radius in Multidimensional 6 3. First Type of Bohr Radius Second Type of Bohr Radius The Third Type of Bohr Radius Complex Manifold Preparation Bohr Radius of One Dimension Type One Of Bohr Radius Type Two Of Bohr Radius Relation Between the Tow Types A Generalization of Bohr Radius REFERENCES 69 i

3 ACKNOWLEDGEMENTS Firstly, I m happy to grateful to my god for every thing he gave me. Also, I am grateful to my lovely parents for their immolation, subsidization and guidance, I want to thank them for there great immolation they produced on my account and I send a lot of thanks to their defy of every problems I found in my way to complete this work. I would like to thank Prof. Dr. Jasser Sarsour, my supervisor for his guidance and his helpful suggestions during my work to complete this thesis. I wish to thank my teachers in Department of Mathematics in Islamic University for their encouragement and for supporting me in my studying. It is pleasure to express my great thanks to every mathematical teacher who teach me, specially master Fawzy El-Kaildy pity of allah for him, and my lovely teacher Aaesha Hamada, a good supporter to me. Finally I wish to thank my brothers Ahmad and Ebrahim and my sister May for their love and helpful. May, 007 Faten S. Abu Shoga ii

4 Abstract Bohr theorem [4] states that holomorphic functions bounded by in the unit disk have power series c k z k such that c k z k < in the disk of radius (the so- 3 called Bohr Radius). In particular, we studied a classical Bohr Radius in one complex variable, and Mf(r) the best upper bound on inf where Mf(z) is the majorant function of 0<r< r f(z). Also, we studied three types of Bohr Radius in multidimensional complex variables K(G),B(G),and J(G), and the exact value of Bohr Radius in the last types J(G). We proved an important facts of multidimensional complex variables which we shall use in our work. At the end, we studied Bohr Radius in a complex manifold domain, also the two types of Bohr Radius in this domain, and we shall study the conditions where these -types will be equal. iii

5 Introduction A Power series c k (λ λ o ) k is said to be nowhere convergent if it converges only for λ = λ o, it is said to be every where convergent if it is converges for all λ. If it is neither nowhere convergent nor everywhere convergent, then there exists a(finite) number r > 0, so that it converges for λ λ o < r and diverges for λ λ o > r, the number r is called the radius of convergent, and {λ C : λ λ o < r} the disk of convergence of the series. It is well known that holomorphic functions have a power series expansion. Under this fact a question asked: What are the properties of a holomorphic function that can be detected from the module of its power series coefficient? One of these properties is the radius of convergence of the power series equals. Harold Bohr studied this property in 94 [4] and state the following remarkable result on a power series in one complex variable. Theorem(Bohr phenomenon). If a power series a k z k converges in the unit disk and its sum has modulus less than, then and the constant cannot be improved. 3 a k z k < in the disk {z C : z < 3 } Actually, Bohr obtained the inequality for 0 r, Wiener M. Riez and Schuar 6 gave independent proofs and each established that is the best possible constant, 3 and other proofs are given by Sidon [5], Tomić [6] and Paulsen and singh [].

6 It is truly rare that a mathematical paper intensive research activity after lying dormant for almost a century following its publication. Yet this exactly what the short 94 paper [4] of H. Bohr has accomplished in the last decade. The interest in the Bohr phenomena was revived in the nineties due to the discovery of generalizations to domains in C n and to more abstract setting. But the only domain on which the exact value of the Bohr radius known is the open unit disc, and for holomorphic functions of several variable, there is no domain for which the Bohr radius is known exactly. Djakov and Ramanjan studied the growth of series of the type a n p r p for bounded holomorphic functions in [6]. They obtained bounds on the Bohr radii R p and extended their results to a several variables context. However, Boas and Khevinson had previously obtained bounds on the Bohr radii for any complete Reinhardt domain in []. They noticed that the radii depend on the dimension of the space being considered and tend to zero as the dimension increases. In particular, there is no Bohr radius for holomorphic functions of infinitely many variables, contrary to what Bohr himself had probably envisioned (cf, [4]). Aizenberg, Aytuna, Djakov and Tarkharov, in a series of papers (cf,[5],[3],[],[4]) have studied Bohr phenomenon in C n and also for various bases different from that of monomials in spaces of holomorphic and harmonic functions equipped with a supremum-type norm. Power series bounded by are in effect members of the unit ball of the functions space H with respect to the norm.. Bohr properties of the unit balls of other functions spaces can be analyzed by supplanting. by other norms.. This course was pursued by Batherine, Dahkvor, and Khavinson [] on unit disc with equivalent Hardy norms and yield a characterization of norms. that display a Bohr property which is turn was applied to a direct proof of the higher order and multivariable Schwarz-pick estimates.

7 The terminology Bohr Radius is somewhat whimsical, for physicists consider the Bohr Radius a 0 of the hydrogen atom to be a fundamental constant: its value is 4πɛ 0 h /m e e, or about 0.59 A. The physicists Bohr Radius is named for Niels Bohr, a founder of the quantum theory and 9 recipient of the Nobel Prize for physics. Since Niels was the elder brother of Harold Bohr, adopting the term Bohr Radius for mathematical purposes keeps the honor within the family. This thesis consist of four chapters:- In chapter, we present a brief summary for topology, complex analysis and functional analysis that we use. In section.4 we present an important theorem called Caratheodory s theorem, that we will use in proofs of many theorems. We recommend that this chapter be covered quickly, or used as back ground material, returning later as necessary. In chapter, this chapter contains 3 sections, in section we discuss a classical Bohr s Theorem, and in section we studied a new generalization of Bohr Radius, and in section 3 we studied Bohr Radius by using Majorant series Mf(z). In chapter 3, since some researchers in several complex variables feel that power series are boring, because the really interesting parts of multi-dimensional complex analysis are the parts that differ from the one variable theory, while the elementary theory of power series superficially appears the same in all dimensions. So in this chapter we studied three types of Bohr Radius in multi-dimensional complex variables: The Bohr radius K n (G), which is the largest r such that if a multiple power series c α z α is bounded by in G, then c α z α < z in the scaled α 0 domain r.g. α 0 The Bohr radius B N (G) which is somewhat easier to work within, where it is the largest r such that if c α z α is bounded by in G, then sup c α z α. rg α 0 α 0 The Bohr Radius J(Bp n ) which is the largest r such that if c α z α is bounded 3 α 0

8 α 0 by in Bp n = {z C n : z p z n p < }, then c α z α < in the hyper cone rb n = {rz C n : z z n < } where z = (z, z,..., z n ) C n, α = (α, α,..., α n ), α i Z + {0}. In section 3.3 we find the exact Bohr Radius for type 3 defined above, and we find that it is equal as Bohr radius for one dimension. 3 In chapter 4, We studied Bohr Radius in complex manifold domain and we studied two types of Bohr Radius in this domain. In section 4.5 of this chapter we find the conditions where the two types are equals, also we studied a new generalization of Bohr Radius in this domain. 4

9 Chapter Preliminaries In this chapter we set up our basic theoretical framework and introduce the reader to those parts of complex analysis and topology that are used frequently in the main theme of this thesis.. Topological Spaces In this section we introduce definitions of some kinds of spaces and definitions of open and closed sets. Definition... [7](Topological Space) A topology on a set X is a collection τ of subsets of X, called the open sets, satisfying:-. Any union of elements of τ belong to τ,. Any finite intersection of elements of τ belongs to τ, 3. φ and X belongs to τ. We say (X, τ) is a topological space. Definition... [9](Metric Space) A metric space is a pair (X, d), where X is a set and d is a metric on X (or distance function on X), that is, a function defined on X X such that for all x, y, z X we have:-. d is real-valued, finite and non-negative.. d(x, y) = 0 if and only if x = y. 5

10 3. d(x, y) = d(y, x) (Symmetry) 4. d(x, y) d(x, z) + d(z, y) (Triangle inequality). Definition..3. [9] A sequence (x n ) in a metric space (X, d) is said to be converges to x, if for every ɛ > 0 there is an N = N(ɛ) such that d(x n, x) < ɛ for every n > N Definition..4. [9] A sequence (x n ) in a metric space (X, d) is said to be cauchy (or fundamental) if for every ɛ > 0 there is an N = N(ɛ) such that d(x m, x n ) < ɛ for every m, n > N The space X is said to be complete if every cauchy sequence in X converges (that is, has a limit which is an element of X). Definition..5. [9](Vector Space) A vector space ( or linear space) over a field K is a nonempty set X of elements x,y,...(called vectors) together with two algebraic operations. These operations are called vector addition and multiplication of vectors by scalars, that is, by elements of K. Definition..6. [9](Normed Space, Banach Space) A normed space X is a vector space with a norm defined on it. A Banach space is a complete normed space. A norm on a complex (or real) vector space X is a real-valued function on X whose value at an x X is defined by x, and which has the properties:- (N) x 0. (N) x = 0 if and only if x = 0 (N3) αx = α x (N4) x + y x + y (Triangle Inequality) where x and y are arbitrary vectors in X and α is any scalar. Definition..7. [9](Seminorm) A seminorm on a vector space X is a mapping P : X R satisfying (N), (N3), and (N4) in definition (..6). 6

11 Definition..8. [7] A subset S of a topological space (X, τ) is said to be compact if for every open cover of S there exists a finite subcover of S. Definition..9. [7] A topological space (X, τ) is said to be sequentially compact if and only if every sequence in X has a convergent subsequence. Theorem..0. [7] A metric space is compact if and only if it is sequentially compact.. Complex Space In this section we gave some facts about complex space which will be used in our work. Remark... [] Any complex number z can be written as z = re iφ where r = z and φ = tan ( y x) where z = x + iy. Facts... [] Let z, z C. () z = 0 if and only if z = 0. () z = z, Re z z, Im z z. (3) z + z = z + z, z z = z z. (4) z z = z z, z = z z z z 0. (5) z + z + z z = ( z + z ), (Parallelogram identity). (6) Triangle inequalities (i) z + z z + z. (ii) z z z z. (7) e iθ = 0 θ π. The following definitions were taken from []. Definition..3. A subset S C is said to be closed if its complement S c = C\S is open. 7

12 Definition..4. A boundary point of a set S C is a point of which every neighborhood contains at least one point of S and at least one point not in S. S, the boundary of S, is defined as the set of all the boundary points. Remark..5. The boundary S is always closed in C. Definition..6. S, the closure of S C, is defined by S = S S. Definition..7. In the Euclidean metric space (C, d) the open ball B(z 0 ; ɛ) = {z C : z z 0 < ɛ} is called an open disc of radius ɛ centered at z 0 C or ɛ neighborhood or a neighborhood of z 0. Definition..8. A subset S C is called open set ( in C) if for every z 0 S there is a δ > 0 such that B(z 0, δ) S. [This means that some disc a round z 0 lies entirely in S]. Definition..9. B(z 0 ; ɛ) = {z C : z z 0 = ɛ} is the circle of radius ɛ > 0 centered at z 0. Definition..0. A point z 0 is called an interior point of S C if there exist a δ > 0 such that B(z 0 ; δ) S. The interior of S, denoted by Int S, is the set of all interior points of S. Remark... It is clear from the definition that A set is open if and only if each of its points is an interior point. Definition... An open subset S C is disconnected if A, B S such that: (i) S = A B, A φ, B φ. (ii) Ā B = φ and A B = φ. If S is not disconnected, it is called connected. Definitions..3.. A domain is an open connected set.. A domain together with some, none or all of its boundary points is referred to as region. 8

13 Definition..4. [] A function f : D C C is continuous at z 0 D if and only if lim z z0 f(z) exists and equals the function value f(z 0 ). We say that f is continuous on D when f is continuous at all points of D. Theorem..5. [7] Let X and Y be topological spaces and f : X Y is a function, then the following are equivalent:- (a) f is continuous. (b) For each open set H in Y, f (H) is open in X. Definition..6. [8] A set S C n is called circular (or circled set) if z S implies ( e iθ z,..., e iθ z n ) S for all 0 θ π. The set S C n is called a Reinhardt domain if z S implies ( e iθ z,..., e iθn z n ) S for all 0 θ j π, j =,..., n. The set S C n is called a complete circular domain if z S implies (ζ z,..., ζ n z n ) S for all ζ j C with ζ j, j =,..., n. Now we consider some definitions and remarks in real case. Theorem..7. [0] Let a, b, n, m, c R then:-. lim n n n =,. If c > 0 then lim n c n =, 3. If 0 < a, 0 < b then a < b if and only if a n < b n, 4. If c >, then c m > c n m > n. Definition..8. [8] Let f be defined on an open set S in C n. We say that f is homogenous of degree P over S if f(λ z) = λ p f(z) for every real λ and for every z in S for which λ z S. Remark..9. [0] Let X be a nonempty set, and let φ, φ 0 be defined on X and have bounded ranges in R. Then sup X {φ + φ 0 } sup{φ} + sup{φ 0 }. X X 9

14 Remark..0. Let X be a nonempty set, and let φ, φ 0 be defined on X and have bounded ranges in R, then sup X {φ} sup{φ 0 } sup{φ φ 0 }.3 Holomorphic Functions X This section contains some definitions and theorems about holomorphic functions and a power Series as a Holomorphic Function. The following Definitions and Theorems are taken from []. Definition.3.. A function f defined on an open set D C is differentiable at an interior point z 0 of D if the limit exists. X f (z 0 ) = lim z z0 f(z) f(z 0 ) z z 0 Definition.3.. A function f of the complex variable z is holomorphic on a set S C if it has a derivative at each point of some open set containing S. Definition.3.3. f is holomorphic at a point z 0 if it is holomorphic in a neighborhood of z 0. Fact.3.4. Any holomorphic function has a power series expansion f(z) = c k z k Theorem.3.5. (Taylor s Theorem) Let f(z) be a holomorphic function, we can write f(z) as f(z) = n 0 f n (z 0 ) (z z 0 ) n, z z 0 < R n! This is often called Taylor series expansion for f in z z 0 < R. In special case when z 0 = 0, it is called Maclaurin series expansion. 0

15 Remark.3.6. Let G be a domain, The Maclaurin series of f(w) = ( w) k is ( w) = ( ) n w n k for w G k+ n k k which is the k-fold differentiation of the geometric series w n. n 0 Note.3.7. We know that the series series called Laurant series. z = n 0 z n, converge for z <, this Theorem.3.8. [] Suppose that f is holomorphic in a bounded domain D and continuous on D. Then f attains its maximum on the boundary D of D. Theorem.3.9. [][Maximum Modules Theorem] Let D be a domain and f : D C be holomorphic, suppose f has a local maximum, then f is constant..4 Caratheodory s Inequality [5] Let D C n be a circular domain, that is a Cartan domain, characterized by the fact that if z D then ze iφ D, where z = (z,..., z n ) and 0 φ π. It is well known fact that in this type of domains every holomorphic function can be expanded into a series of homogenous polynomial. Firstly we have the following definitions. Definition.4.. A set S in a space X is called convex if S contains the line segment x x whenever x and x are distinct points of S. This is equivalent to the requirement that, if x, x S, then also α x + α x S whenever α and α are positive number such that α + α =. Definition.4.. If S is any set in X, the convex hull of S is defined to be the intersection of all convex sets which contains S. Definition.4.3. [9] The distance dist(x, B) from a point x to a nonempty subset B of (X, d) is defined to be dist(x, B) = inf d(x, b). b B

16 Now we are ready to formulate our theorem. Theorem.4.4. Let D be a Cartan domain and f be a function holomorphic in D. If in the domain D the expansion f(z) = P k (z), z D. for the function f is valid and f(d) G C, then for every z D the following inequality holds for every k where G is the convex hull of G. P k (z) dist(p 0 (z), G), Theorem (.4.4) has the following interesting corollaries. Corollary.4.5. Let D be a Cartan domain and f be a function holomorphic in D. If f(z) < for every z D, then for every k the following holds P k (z) ( P 0 (z) ), z D Corollary.4.6. Let D be a complete, bounded, Reinhardt domain. Let f be a function holomorphic in D with the corresponding multidimensional power series f(z) = c α z α, z D α 0 where α = (α,..., α n ), α = α α n, z α = z α... z αn n and all α i are nonnegative integers. If f(d) G C, then for every α such that α the following holds where d α (D) = max z α. D c α dist(c 0, G) d α, In one complex variable, Theorem (.4.4) leads to, Corollary.4.7. If in the unit disk K = {z C : z < } the function f is a power series, that is f(z) = c k z k and f(k) G, then for every k we have c k dist(c 0, G).

17 The following result is the known Caratheodory s inequality, (see [5] and [3]). Lemma.4.8. If in the unit disk K = {z C : z < } the holomorphic function f(z) written as f(z) = c n z n, and if n=0 Ref(z) > 0 z K, then c k Re c 0 k. Corollary.4.9. If in the unit disk K = {z C : z < } the equality f(z ) = c k z k holds and if f(z ) < for every z K then c k ( c 0 ), k. 3

18 Chapter Bohr s Theorem. Bohr s Theorem For One Complex Variable In this section we will study a classical Bohr s Theorem with proof which due to Edmund Landau [0] (see also [3]), where his proof based on a classical inequality of Caratheodory (.4.8) Definition... [] The classical Bohr Radius K(G) in a domain G is the largest r such that if a power series c k z k is bounded by in G, then c k z k < for z in the scaled domain rg. Theorem... [] If a power series sum has modules less than, then the constant 3 cannot be improved. c k z k converges in the unit disk and its c k z k < in the disk {z C : z < }, and 3 Proof. Let f(z) be a holomorphic function, that represented by the series satisfying that c k z k < for all z <. c k z k Let φ be an arbitrary real number, and set the function g = e iφ f, since f(z) < for all z < then e iφ f(z) < [ because e iφ =, φ R] So [Re(e iφ f(z))] + [Im(e iφ f(z))] <, then [Re(e iφ f(z))] < so Re(e iφ f(z)) <, therefore < Re(e iφ f(z)) <, and 0 < 4

19 Re(e iφ f(z)) <. But Reg(z) = Re(e iφ f(z)), so Reg(z) > 0, then by Corollary (.4.9), c k Re( e iφ c 0 ) when k. Since φ is arbitrary, let φ be such that c 0 = re iφ, where r = c 0 [ by Remark..]. Then r = e iφ c 0 = c 0, and since ( c 0 ) R it follows that c k Re( c 0 ) = ( c 0 ) when k. (.) Now, if f is a constant function i,e f(z) = c 0, then c k z k = c 0 = c k z k < when z < 3. For nonconstant f(z), the inequality (.) shows that if z < 3, then c k z k = c 0 + < c 0 + c 0 + c k z k c k ( 3 )k ( c 0 )( 3 )k [by (.)] = c 0 + ( c 0 ) ( 3 )k = c 0 + ( c 0 )( ) = To see that the radius in bohr s theorem is the best possible, consider the linear 3 fractional transformation f a defined by f a (z) = z a where 0 < a <, and z < az, then az < and since > then z so az 0 then f a a a(z)is holomorphic function on z < so we can write it as f a (z) = c k (a)z k. 5

20 Now, z a az = z az a az = z (az) k a (az) k [by Laurant series since az < ] = z + az + a z 3 + a 3 z 4 + a a z a 3 z = a + ( a )z + a( a )z + a ( a )z 3 + Since 0 < a < and 0 < a <, then f a ( z ) = a + ( a ) z + a( a ) z + = a + ( a )z + a( a )z + So, a + f a ( z ) = a + ( a )z + a( a )z + = c k (a)z k We claim that a + f a ( z ) > when z >, to prove it, note that if + a z > + a, then a z > a a, so a z <, and a z < + a + a a + a = + a a = + a + a + a So, Then, a z > + a + a (.) a + f a ( z ) = a + z a a z > a + ( z a) + a + a [from.] 6

21 ( ) ( ) + a > a + + a a + a ( ) ( ) a a + a = a + + a + a = a + a a + a = a + a + a a + a = + a + a = Therefore, a + f a ( z ) >. So, a+ z a a z >, i,e a + f a( z ) = c k z k > when but 0 < a < implies that < + a < 3, so It follows that the radius 3 + a > 3. in Bohr s theorem cannot be increased. + a < z,. A new Generalization of Bohr Radius In this section we will study a new generalization of Bohr Radius in theorem... Definition... [8] A transformation of Euclidean space with respect to a certain point O that brings each point M in a one-to-one correspondence with a point Ḿ on the straight line OM in accordance with the rule OḾ = kom where k is a constant number, not equal to zero, which is known as the homothety ratio. Definition... [6] Let G C be any domain. A point p G is called a point of convexity if p G, where G is the convex hull of G. Definition..3. [6] A point of convexity p is called regular if there exist a disk U G so that p U. Recall definition.4. of convex hull 7

22 Notation..4. U = {z C : z < } denotes the unit disk. Remark..5. Under the definition of convex hull.4. we have the convex hull of any disk U is itself, i.e Ũ = U. Lemma..6. Under the hypotheses of Corollary (.4.7), if z < 3, then c kz k < dist(c 0, U). Proof. By the above Remark (..5) and Corollary (.4.7) we have So, if z < 3, then c k z k < c k dist(c 0, U) ( ) k c k 3 < dist(c 0, U) = dist(c 0, U) = dist(c 0, U). ( 3 ( ) ) k Theorem..7. [6] In a domain G, if the function f(z) = c k z k (.3) is such that f(u ) G, with G C, then for z < 3 the inequality is valid. The constant 3 point of convexity. c k z k < dist(c 0, G) (.4) cannot be improved if G contains at least one regular 8

23 Proof.. If z < 3 and f(u ) G then corollary [.4.7] yields:- c k z k = c k z k ( ) k < c k 3 ( ) k < dist(c 0, 3 G) [by corollary.4.7] = dist(c 0, G) ( ) k 3 = dist(c 0, G). = dist(c 0, G).. We will prove the exactness of the constant contains at least one regular point of convexity. ( ) 3 in the case the boundary In the classical case of Bohr s Theorem this is obtained by considering the family of functions Here f(z) = a z az c k z k = if and only if z = 0 < a < (.5) + a, because c k z k = a f( z ) = a a +a a ( ) +a = a a+a +a +a a +a = a = a (a ) =. (a )(a + ) + a See proof of Theorem (..) 9

24 Furthermore, taking a, we obtain the desired result. In equation (.5), we will have c 0 = a, note that instead of the family (.5) one can use the family e iφ f(z), where f(z) is taken from (.5), i,e. e iφ f(z) = aeiφ e iφ z az In this case it follows that c 0 = ae iφ and when a we get that c 0 = e iφ tends to U along the radius of argument a [because if a then c 0 = e iφ = the unit circle]. If G is an arbitrary disk U, then by Lemma..6 we have c k z k dist(c 0, G). dose not change under homotheties and translations, we deduce the exactness of 3 in the case of any disk. For any domain G, let ζ be a regular point of convexity, then there exist a disk U G such that And by hypothesis ζ U (.6) ζ G (.7) and since ζ is a point of convexity then It follows from (.6),(.7) and (.8) that ζ G (.8) ζ U G G Consider the function f(z) = c k z k such that f(u ) U, let c 0 = αe iφ by using definition.4.3 of distance we can choose c 0 to given. dist(c 0, U) = d(c 0, ζ). 0

25 Since ζ U G G and U G G then dist(c 0, U) = dist(c 0, G) = dist(c 0, G) = d(c 0, ζ). Therefore, in the inequality (.4) one cannot take z < r, where r > 3..3 Bohr Radius of Majorant Series In 956, Aurel Wintner raised the following question related to Bohr s Theorem on Majorant series: For the class of holomorphic functions f in the unit disk with modulus bounded by Mf(r), what is the best upper bound on inf or inf Mf(z) 0<r< r 0< z < z? It follows by taking r = that the value 3 is an upper bound, Wintner claimed 3 -incorrectly- that the bound 3 cannot be improved. After time, Günther Schlenstedt see from the maximum principal that the best bound on inf Mf(z) 0< z < z is actually. Mf(r) But in fact the best bound of inf is neither 3 nor, but, as Harold P. 0<r< r Boas demonstrate in [3]. And now we will study it. Definition.3.. [8] Let U C be the unit disk. For 0 < p < let ( π ) H p (U ) := {f holomorphic on U : sup f(re iθ ) p p dθ 0<r< 0 f H p < } Remark.3.. If p= in definition.3. we have a Hardy Space H (U ) which defined as, ( π ) H (U ) := {f holomorphic on U : sup f(re iθ ) dθ 0<r< π 0 f H < } Definition.3.3. [3] Let f be a holomorphic function then the majorant function

26 Mf(z) is defined by, if f(z) = c k z k then Mf(z) = c k z k. We write theorems (.3.4) and (.3.5) without proofs. Theorem.3.4. [9](Cauchy Schwarz Inequality) Let (ζ) i=, and (η) i= be sequences of complex numbers then, ζ i η i ζ k η m i= Theorem.3.5. [4] (Parseval s Theorem). Suppose f is Riemman-integrable function with period π, and f(x) c n e inx, wherer this notation means that if m= c n e inx converges then it converges to f(x), then π f(x) dx = π π c n. Theorem.3.6. [3] If f is a holomorphic function such that f(z) < when z <, then the majorant function Mf satisfies the inequality: Mf(r) inf 0<r< r, and the bound cannot be replaced by any smaller number. Proof. Here, we shall prove some what more than is stated in the Theorem. Namely, I do not need the function f to belong to the unit disk of H, all we will use is that f belong to the disk of H, because H H. Let f H (U ), where U is unit disk of the Hardy space H. By Remark (.3.), we have ( f H = sup 0<r< π ( then, π So, π π 0 π 0 π 0 ) f(re iθ ) dθ ) f(re iθ ) dθ f(re iθ ) dθ (.9)

27 Since f(z) = c k z k, then by Parseval s Theorem (.3.5), By using (.0) and (.9) we have c k = π f(re iθ ) dθ. (.0) π 0 c k, so c 0 + c k, then c k c 0 (.) Since, Mf(r) = c 0 + c k r k, then Mf(r) r = c 0 r + c k r k (.) Now, apply Cauchy Schwarz inequality (.3.4) to the term c k r k, we have ( c k r k n= By using (.3), the equation (.) becomes, Mf(r) r c 0 r + ( c n ) ( n= Since 0 < r < then 0 < r <, so series, then So, ( ) (r ) m = m= r. m= c n ) ( ( r m ) m= ) ( r m ) (r ) m = r m= ) (.3) (.4) is a geometric Mf(r) r ( c 0 r + c 0 r + n= c n ) c0 r r (.5) [by using (.)] (.6) 3

28 Now, let r c 0 < then and, 0 < c 0 r, then So, c 0 r Now, by using (.5),(.7) and (.8) < r (.7) 0 < r c 0 c0 (.8) r Mf(r) r c 0 r + < r + c0 r Take inf of both sides, 0<r< inf 0<r< Because 0 < r < and r Therefor, we have ( ) Mf(r) r >, so inf Mf(r) inf 0<r< r ( inf 0<r< 0<r<. ( r ) r + = + = ) =. To see that the bound cannot be replaced by any smaller number, let f(z) = z + then we have, f(z) = z z + By Laurant series, f(z) becomes, f(z) = z z,where z < ( ) k z + ( z ) k = z + z + z z + z +... z, 4

29 = + = z 0 + So, Mf(r) = r 0 + Then, Since Mf(r) r = r + [ ( ) k ( ) ] k+ + z k r <, then r <, so r >. [ ( ) k ( ) ] k+ + z k [ ( ) k ( ) ] k+ + r k [ ( ) k ( ) ] k+ + r k Therefore, Mf(r) inf 0<r< r = inf 0<r< = inf 0<r< = + k= ( ( r + r + [ ( ) k ( ) ] ) k+ + r k ( ) 0 + ( ) [ ( ) k ( ) ] ) k+ + r k ( ) ( ) = + 3 = 4 =. 5

30 Chapter 3 Types of Bohr Radius in Multidimensional This chapter contains three types of Bohr Radius in multidimensional variables and important definitions, in section we present a first type of Bohr Radius, in section and 3 we present the other types which defined by Aizenberg in [5] and [5]. Notation For the power series c α z α :-. α denotes an n-tuple (α, α,..., α n ) of nonnegative integers.. α denotes the sum α + α α n of its components. α 3. α! denotes the product α!α!... α n! of the factorials of its components. 4. z denotes an n-tuple (z, z,..., z n ) of complex numbers. 5. z α denotes the product z α zα... zα n n. 6. c α z α = c α z α = c α k α coefficient). α >0 c α z α. = k!, which called Multidimensional coefficient (or multinomial α! 6

31 3. First Type of Bohr Radius In this section we will study a first type of Bohr Radius. firstly we will present some theorems without prove that will be used later, also we will prove some theorem and remarks that important in the multidimensional space C n. Theorem 3... [9](Holder Inequality) Let p, q R such that p + q = then ( ζ j γ j j= ζ k p ) p ( Definition 3... [3] Define B n p = {z C n : m= γ m q ) q. n z j p < } which is called the unit ball of the complex Banach Space l n p whose norm is defined by z l n := p ( n ) p z j p. j= Definition [3] Bohr Radius K(G) is the largest r such that whenever α c α z α for z in a domain G, it follows that c α z α for z in the scaled domain r.g. j= α Definition The surface of the torus with as radius vector, in the Cartesian coordinates of the Euclidean space C 3, r = a sin uk + l( + ɛ cos u)(i cos v + j sin v) (here (u, v) are the intrinsic coordinates, a is the radius of the rotating circle, l is the radius of the axial circle, and ɛ = a l is the eccentricity), is often also called a torus. Its line element is ds = a du + l ( + ɛ cos u) dv. In the writings of Bohr [4] and Landau [0], one finds a useful trick employed by F. Wiener in the context of coefficient bounds for the one-dimensional Bohr theorem. 7

32 Lemma [3](Winner Method) Let G be a complete Reinhardt domain, and let F be the set of holomorphic functions on G with modulus bounded by. Fix a multi- index α other than (0, 0,..., 0), and suppose that the positive real number b is an upper bound for the modulus of the derivative f (α) (0) for every function f in F, then f (α) (0) ( f(0) )b for every function f in F. Proof. See reference [3]. Note In our work we denote f α (0) = c α and f(0) = c 0. Theorem [5](Generalization of Cauchy Inequality) Let f : D C n C be a function given by f(z) = α c α z α be a series, if α c α z α < in the domain D, then for each α = (α, α,..., α n ), α i Z + 0 i N we have, where d α (D) = max z α. D c α d α (D). Remark < d α (B n p ) = sup{ z α : z B n p } (3.) exists. Proof. Since B n p = {z C n : then n z j p < }, So z j <, for all j {,,..., n}, j= Therefore, z α < set. Hence sup{ z α : z B n p } exist. z α = z α... zα n n = z α... z n α n <.... = α, and z B n p. So { zα : z B n p } is a bounded 8

33 Theorem [8] (Lagrange Multipliers). Let f(x, x,..., x n ), g(x, x,..., x n ) C in a domain D, and g p + gp gp n > 0 in D. The set of points (x, x,..., x n ) on the curve g(x, x,..., x n ), where f(x, x,..., x n ) has maxima or minima, is included in the set of simultaneous solutions (x, x,..., x n, λ) of the equations, f (x, x,..., x n ) + λg (x, x,..., x n ) = 0 f (x, x,..., x n ) + λg (x, x,..., x n ) = 0. f n (x, x,..., x n ) + λg n (x, x,..., x n ) = 0 g(x, x,..., x n ) = 0 ( α Proposition d α (Bp n ) = sup{ z α : z Bp n α } = (z,..., z n ), α = (α,..., α n ), p and α 0. Proof. We will proof it by using Lagrange Multipliers. α α Let f(z) = z α zα... zα n n, and g(z) = zp + zp zp n Now, we have ) p where z = U(z) = z α zα... zα n n + λ(z p + zp zp n ) (3.) By differential with respect to z we have, U z (z) = α z α z α... zα n n + p λ z p = 0 (3.3) Again, by differential with respect to z we have, U z (z) = α z α zα... z α n n + p λ z p = 0 (3.4) In general, for any fixed i {,,..., n}, and U zi (z) = α i z α zα... zα i i... z α n n + p λ z p i = 0 (3.5) g(z) = z p + zp zp n = 0 So, z p + zp zp n = (3.6) 9

34 Now from equation (3.3) we have Now, by (3.7) and (3.4) we have, λ = α z α z α... zα n n p z p α z α zα... z α n n + p z p ( α z α z α, z 0 (3.7) p z p... ) zα n n = 0, α z α zα... z α n n α z α p z α +p... z α n n = 0, (z α p z α z α zα n n )(α z p α z p ) = 0 So we have, z α p z α... z α n n = 0 (3.8) or α z p α z p = 0 (3.9) However, the equation (3.8) is impossible because z α 0. Hence we have, α z p α z p = 0. (3.0) Similarly, using equation (3.5) for any i {,..., n} with equation (3.7) we have α i z p α z p i = 0 (3.) So we have n equation [equations (3.) for any i {,..., n} and equation (3.0)], then by solving this system as a linear system we have z p i = α i α + α α n = α i α (3.) then, z i = z α i i = ( αi ) p α ( α α i i α α i ) p 30

35 Now, z α = Π n i=z α i i = Π n i= ( α α i ) p i α α i = (Πn i= αα i i ) p (Π n i= α α i ) = ( α α α α ) p p = (αα ) p ( α α ) p ( α Therefore, sup{ z α : z Bp n α } = α α Proposition 3... Let z = (z, z,..., z n ) and α = (α, α,..., α n ) then, k ( n ) k z α = z k l α = z n j j= where k α = k! α!. ) p Proof. Case : If : α α = z α = α =! α! zα =!! z +!! z !! z k n = z i. i= Case : If k=: α α = z α = α =! α! zα 3

36 If α = such that α i = for some i and α j = 0 i j then α! =! and so! α! zα =!! z i = z i If α = such that α i =, α j = for some i j and α k = 0 k i, k j, then α! = (!)(!) =.! So, α! zα =!! z i zj =! z i z j. Therefore, z α = α α = = = α =! α! zα n!! z i + i= ( n ) z i i= n i,j=,i j! z i z j Case 3: For any k: = = k α n i= z α = k! k! z i k + + n j,..,j k = n z i k + k i= + k! ( n = z i i= n i,j=,i j k! α! zα k! (k )! z i k z j +... k!! z j... z jk. j i j m n z i k z j +... i,j=,i j n j,..,j k = ) k z j... z jk. j i j m 3

37 Therefore, k α And by using definition (3..) ( n z k l = n j= ( n k z α = z i ). z j ) k i= ( n = z j j= ) k Corollary 3... Proof. Since z k l n k = n k α = k ( n z α = z j α j= ) k Let z = (,,..., ), then z α = α α... α n = k = So, k ( n k = ) = n k. α j= Corollary n k Proof. Since k α, then by using corollary (3..) we have k α = n k Therefore, nk. 33

38 The lower bound in the next theorem is due to Lev Aizenberg in [5] when p =, the lower bound when p is due jointly to Dmitry Khavinson and Harold P. Boas in []. Theorem When n >, the Bohr Radius K(B n p ) of l n p admits the following bounds:- unit ball in Cn If p, then If p, then 3 3 e ( ) p K(B n n p ) 3 n K(Bn p ) Proof. [Proof of The Lower Bound when p ][3] By (3..0) we have: ( α d α (Bp n ) = sup{ z α : z Bp n α } = Assume α c αz α < when z B n p then by generalization of Cauchy inequality (3..7) when α 0 we have, α α ) p c α d α (B n p ) = So, by Winner Method (3..5) ( α α α α ) p α α α α, since p Hence, we have when k > 0 that c α z α = c α z α c α α α α α ( c0 ) (3.3) ( c0 ) α α = ( c 0 ) 34 α α zα by using(3.3) k k α α zα (3.4)

39 Then multiplying and dividing by the multinomial coefficient k α = k! α! then the inequality (3.4) become, c α z α ( c 0 ) = ( c 0 ) k k α α k! α! k k α! α α k! α! k! zα k α z α since α α < α! then then we have, c α z α ( c 0 ) = ( c 0 ) k k α! k k k! α! k z α k! α k α z α (3.5). c α z α = c 0 + α c 0 + c α z α ( c 0 ) k k k! k α z α [by (3.5)] = c 0 + ( c 0 ) = c 0 + ( c 0 ) k k k! k k k! z k l n k α z α [by proposition (3..)] (3.6) 35

40 If we apply Holder inequality for z k l n z k l n = ( (( n z j ) k j= m q ) q ) k (( i= = ( n j= z j ) k, we have s z t p ) p ) k, where p + q = t= = (n p z l n p ) k (3.7) From inequality (3.7), the inequality (3.6) become, c α z α c 0 + ( c 0 ) α Note that the real quadratic function t t + ( t ) t + ( t ) = (t t) k k k! (n p z l n p ) k (3.8) = (t t + ) = [(t ) ] = (t ) + Now, replace t by c 0 so we have c 0 + c 0. So, for the inequality (3.8) we will try to find a value that make We know that the series k k k! ( n p z l n p ) k = follows that if x is the unique positive number such that, never exceeds, since k k k! xk converges if xe < by using ratio test, It k k k! xk = 36 (3.9)

41 then the Bohr Radius K(B n p ) is at least as big as n p In combinators, one encounters the tree function T, which satisfies the functional equation T (x)e T (x) k k = x and has the series expansion T (x) = x k, then T (x) = = So x T (x) = k k k! x k k k! xk = z l n p. k x k k k k! xk = k! (3.0) If we differentiate the functional equation T (x)e T (x) = x with respect to x we have, then x T (x) = T (x)e T (x) T (x)e T (x) T (x) = T (x)(e T (x) T (x)e T (x) ) = ] x T (x) [e T (x) T (x)e T (x) T (x) = x = T (x)e T (x) T (x) T (x), so equations (3.9) and(3.0) yield that T (x) = then T (x) = T (x) so T (x) = 3. Since, T (x)e T (x) = x then x = e 3 3 = 3 3 e This completes the proof of the Lower Bound in theorem when p. [Proof of The Lower Bound when ( p )] [3] If α c α z α when z Bp n, define a one-variable function f(ζ) as 37

42 f(ζ) = α c α z α ζ α and α 0, then, f(ζ) = c 0 + c α z α ζ α = c 0 + c α z α ζ k So by Winner Method (The one dimensional version)(3..5), c α z α c 0 when k Integrating the square of the left-hand side of this inequality over a torus (3..4) we have π π c α (re iθ ) α dθ π π ( c 0 ) = π( c 0 ) (3.) then, π sup c α (re iθ ) α dθ π( c 0 ) 0<r< π but, π π c α (`re iθ ) α dθ π sup c α (re iθ ) α dθ 0<r< π (3.) for all w = `re iθ in the unit ball of l n p, then π π c α (`re iθ ) α dθ π( c 0 ) (3.3) Since f(w) = c α(`re iθ ) α, then by using Parseval s theorem (.3.5) we have, c α = π c α (`re iθ ) α dθ (3.4) π π We integrate over a torus because the Riemann surface in complex plane is a torus or sphere 38

43 But w α <, then c α w α c α, so equation (3.4) becomes where w = `re iθ. c α w α π π π By using (3.5) the inequality (3.3) becomes Therefore, c α w α ( c 0 ) c α w α c α w α c α, then c α (`re iθ ) α dθ (3.5) c 0 (3.6) for every point w in the unit ball of l n p. Now if z lies in the l n p ball of radius 3 n then z ln p ball, z 3 n then 3 nz, then apply the Cauchy-Schwarz inequality (.3.4)with w = 3 nz shows that c α z α = ( c 0 ) = ( c 0 ) ( c α (3 ) ( ) α n) α z α 3 n c α w α ( ) k 9n ( ) α 3 n ( ) k 9n [by (.3.4)] [by inequality(3.6)] 39

44 ( c 0 ) ( ( ) ) k n k 9n [Corollary(3..3)] = c 0 3 k (3.7) Consequently, we have for each point z where z < 3 that, c α z α = c 0 + α = c 0 + c α z α ( c0 ) 3 k c 0 + ( c 0 ) = c 0 + ( c 0 ) = ( c 0 ) + < This complete the proof of theorem when p ( ) k by (3.7) 3 3. Second Type of Bohr Radius In this section we will introduce a second type of Bohr Radius B(G), Lev Aizenberg define this type in [5], he formulate and proof theorem (3..5) to find lower bound of second Bohr Radius. Definition 3... [5] The Bohr Radius B n (D) is the largest r such that if the series c α z α converges in a complete n-circular bounded domain D and α α holds in it, then α sup z rd c α z α, where rd is called the homothetic transformation of D. c α z α <, Now we define a negative Binomial series which we will use in the next theorem. 40

45 Definition 3... [8](Negative Binomial Series) Negative Binomial Series is the series which arises in the binomial theorem for negative integer n, (x + a) n = n x k a n k k = ( ) k n + k k x k a n k Remark Take x = r and a = in definition (3..), then ( r) = ( ) k n + k ( r) k n k = n + k r k k Remark Let d α (D) = max z α for a domain D, then z D d α (rd) d α (D) = = = = max (rz) α rd max z α D max rd max rd max rd (rz ) α... (rz n ) α n max z α... D zα n n (r) α (z ) α... (r) α n (z n ) α n max z α... D zα n n ( ) r α +α +...+α n (z ) α... (z n ) α n max z α... D zα n n = ( ) max (z ) α... (z n ) α n r α +α +...+α n D max z α... D zα n n = r α +α +...+α n = r α 4

46 Theorem [5] The inequality n 3 B n(d). is true for any complete, bounded n-circular domain D. Proof. Let f(z) = α then by theorem (3..7) c α z α be a holomorphic function such that, α c α z α <, c α d α (D), where d α(d) = max z α D Since f α (0) = c α, then by using Winner method (3..5), take b = d α (D) then c α c 0 d α (D) for α (3.8) Now, by (3.8) we get, α sup z rd c α z α = α c α sup z α z rd c α d α (rd) α = c 0 + c α d α (rd) α by (3.8) c 0 + = c 0 + ( c 0 ) α ( c0 ) d α (rd) d α (D) = c 0 + ( c 0 ) α = c 0 + ( c 0 ) max D (rz) α max D z α α d α (rd) d α (D) r k by remark (3..4) 4

47 since = n + k [see [5]], then k α sup z rd c α z α c 0 + ( c 0 ) n + k k r k = c 0 + ( c 0 ) [ ] ( r) n by Remark(3..3) (3.9) Now, if r ( ) n 3 then ( r) ( ) n 3 ( r) n and ( r) n 3 hence (3.30) then inequality (3.9) becomes, α sup z rd c α z α c 0 + ( c0 ) = c 0 + c 0 = ( c0 c 0 + ) = ( c 0 ) < So, α sup c α z α < if r n rd 3. The following theorem states a relation between first type and second type of Bohr Radius. Theorem If K(G) is the first type of Bohr Radius and B(G) is the second type of Bohr Radius, then we have K(G) B(G). 43

48 Proof. Since Hence So if z α sup w α then w rg c α z α c α sup w α and so c α z α sup c α w α w rg w rg c α z α sup c α w α. α α w rg c α w α < then c α z α <. α sup w rg Therefor K(G) B(G). α 3.3 The Third Type of Bohr Radius As it point out already in [5], it seems more natural to consider not a single number in the Bohr problem in C n, but the largest subdomain D B of D, such that c α z α < holds. α 0 At this stage, Aizenberg state and prove an interesting result in [5] for the domain B n = {z C n : z z n < }, But in this section we will state and prove a generalization of this result for any domain Bp n for any p. Proposition Let α = (α,..., α n ), α 0 and p > then n N Proof. By induction on n. ( α α α α... αα n n ) p α! α!... α n!. Let n =, then α = α and α! = α α!!, then α! = α! α! ( ) α α ( p α α ) p α α = α α =, So (3.3) is true for n =.. Let n =, then α = (α, α ), α, α Z {0}. (3.3) =, and Since, α!α! (α + α )! (3.3) 44

49 and since, If p > then p α α αα (α + α ) α +α (3.33) <, so the inequality (3.33) becomes ( (α + α ) α +α ) p (α α αα ) p (3.34) It follows from inequalities (3.3) and (3.34) that ( (α + α ) α +α ) p α!α! (α α αα ) p (α + α )! (3.35) then, Therefore, is true when n =. ( (α + α ) α +α α α αα ( α α α α ) p ) p (α + α )! α!α! α! α! (3.36) (3.37) 3. Assume that (3.3) is true for any m < n +, i,e, if α = (α,..., α m ), then m < n +, ( α α α α... αα m m ) p α! α!... α m! (3.38) 4. We want to show that (3.3) is true for m = n +. Let ά = (α,..., α n, α n+ ), since n < n +, then ( α α α α... αα n n ) p α! α!... α n! and so ( α α... αα n n ) ( ) p α! p α!... α n! α α (3.39) 45

50 and if α 0 = ( α, α n+ ) then α 0 = α + α n+ = ά and m = < n, then Now, ( ά ά ( ά ά α α α α n+ n+ α α... αα n n α α n+ n+ ) p ) p = ά! α! α n+! ( ά ά ) p (α α... αα n n ) p ( α α n+ ) p n+ (3.40) by using (3.39) = ( α α n+ n+ ) ( ά ά α α n+ n+ α α ( ά ά ) p α! ( p α α ) p (α!... α n!) ) ( p α! ) α!... α n! by using (3.40) = ά! α n+! α!. ά! α!... α n! α n+!. α! α!... α n! Hence, inequality (3.3) is true for n + and the induction is complete. Lemma If P k (z) is the homogenous polynomial P k (z) = c α z α, and P k (z) < in the ball Bp n, then c α z α < for every z in the hypercone B n. Proof. Since P k (z) < in the ball Bp n, then by Theorem (3..7) and Proposition (3..0), c α ( ) α α p d α (Bp n ) = α α... αα n n 46

51 Then for every z B n we have that c α z α ( ) α α p α α... z α αα n n ( ) α! z α [ by proposition (3.3.)] α!... α n! = α z α [ by proposition (3..)] α ( n k = z j ) ] [ by proposition (3..)] j= < because z B n. Remark If z B n and z = (z, z,..., z n ), let then we have z = ( 3 z = z 3,..., z ) n 3 3 Bn, z α = = = ( z ) α ( zn ) αn ( ) α (z α 3... zα n ( ) k z α 3 n ) Since z B n then zα <, so z α = < ( ) k z α 3 ( ) k. 3 47

52 Theorem If the power series f(z) = c α z α, z B converge in the unit ball α 0 Bp n = {z C n : z p z n p < } (3.4) c α z α < holds in and the modulus of its sum in there is less than, then the hypercone α 0 3 Bn = {z C n : z z n < } and the constant 3 cannot be improved. Proof. By corollary (.4.7) and the hypothesis of theorem we get that for every k the estimate then, c α z α < ( c 0 ) z B n p (3.4) because c 0 0, then by Lemma (3.3.), z B n c α ( c 0 ) zα < then Now, if z 3 Bn, so z = 3 z, z Bn, then c α ( c 0 ) zα < (3.43) we have, c α z α < ( c 0 ) (3.44) c α z α = c 0 + α 0 = c 0 + c α z α ( k c α z 3) α 48

53 ( ) k = c 0 + c α z α 3 ( ) < c 0 + ( c 0 ) = c 0 + ( c 0 ) = c 0 + c 0 ( ) 3 k (3.44) and Remark (3.3.3) Therefore, = c α z α <, z 3 Bn. α 0 The fact that constant cannot be improved can be seen if we consider the function 3 of the type f(z, 0, 0,..., 0), this function plays as a function of single variable which proved in theorem (..). 49

54 Chapter 4 Complex Manifold Our aim in this chapter is to obtain multidimensional generalization of theorem (..) and theorem (4..) in a more general setting and in the spirit of functional analysis and by using complex manifold domain with Holomorphic function with positive real part. 4. Preparation In this section some notation and facts will be listed, which are of fundamental importance for every thing that follows. Definition 4... [7] A space X is Hausdorff if and only if whenever x and y are distinct points of X, there are disjoint open sets U and V in X with x U, and y V. Definition 4... [7] A one-to-one correspondence between two topological spaces such that the two mutually-inverse mappings defined by this correspondence are continuous. These mappings are said to be homeomorphic, or topological, mappings, and also homeomorphisms, while the spaces are said to belong to the same topological type or are said to be homeomorphic or topologically equivalent. Definition [3] By an n-dimensional complex manifold we mean a connected Hausdorff space M such that each point has a neighborhood which is homeomorphic to a ball in C n. We sometimes express this by saying that a complex manifold is a connected Hausdorff space which is locally Euclidean. 50

55 Remark If a complex manifold M is not open, then Int(M). Proof. By definition (4..3) a complex manifold M is a Hausdorff space, then for all x, y M such that x y, there exist open sets U, V M such that U V = and x U, y V. So, if Int(M) =, there is no open sets in M that contains x, y which implies a contradiction. Therefore, Int(M). Definition [7] C (X) is the set of all bounded real valued continuous functions on a space X. Definition [7] On C (X), we define the topology of uniform convergence to be the topology defined by the metric where f, g C (X). d(f, g) = sup{ f(x) g(x) : x X} Definition [8] A Fréchet space is a complete topological vector space (real or complex) whose topology is induced by a countable family of a seminorms. To be more precise, there exist seminorm functions. n : U R n N such that the collection of all balls B ɛ (n) (x) = {y U : x y n < ɛ}, x U, ɛ > 0, n N is a base for the topology of U. We suppose M is a complex manifold. We denote by H(M) the space of holomorphic functions on M and for any compact subset K M we set f K = sup f(z), K f H(M). 5

56 The system of seminorms f K, K M, defines the topology of uniform convergence on compact subsets of M. Equipped with this norm system H(M) is a Fréchet space. Let. r, r (0, ), be one-parametric family seminorms in H(M), that are continuous with respect to the topology of uniform convergence on compact subsets on M. We always assume in the following that:. r. r if r r. 4. Bohr Radius of One Dimension Theorem 4... [] If the function f(z) = part, then cannot be improved. c k z k, z < has a positive real c k z k < f(0) in the disk {z : z < 3 } and the constant 3 Proof. Firstly, note that f(0) = c 0, since the holomorphic function f(z) = c k z k holomorphic function has a positive real part in z <, then by lemma.4.8 we have c k c 0 k, so for z < 3 c k z k = c 0 + c k z k c 0 + c 0 ( 3 )k = c 0 + c 0 ( 3 )k it follow that, = c 0 + c 0 ( ) = c 0. K M:-this symbol mean that K is a compact subset of M. 5

57 So, c k z k f(0). Now, consider the function f(z) = + z z = + + z z = z k, since ( ) ( ) + z z = z + iimz. z z z Then,Re(f(z)) = Re( + z z ) = z > 0 since z <. So f(z) has a z positive real part and the sum of the moduli of the terms of its expansion equals for z = 3. Note that if z > 3,then + z k > + ( 3 )k = = f(0) So the constant 3 is the best possible. 4.3 Type one of Bohr Radius In this section we will consider the following problem, Is there an r (0, ) and a K M such that f r f K f H(M)?. Aizenberg in [] see that proof of theorem (4.3.4) depends on Lemma (4.3.3), But in this section we will proof theorem (4.3.4) without using Lemma (4.3.3). Firstly, we present a theorem without proof, this Theorem will be useful in our work. Definition [8] A sequence {f n } is said to be uniformly bounded on S if there exists a constant M > 0 such that f n (x) M for all x in S and all n. Theorem [9] (Montel Theorem). Let G C be open, F a family of holomorphic functions on G. Suppose that the functions f, f,... are uniformly bounded on each compact subset of G. Then there exists a subsequence of {f n } which converges uniformly on compact subsets. Proof. See reference [9]. 53

58 Lemma [] If G is an open domain on a complex manifold then for any z 0 G and K G, there exist a constant C > 0 such that whenever f H(G) and f(z 0 ) = 0, we have,. Proof. See reference []. f K = sup K f(z) C sup Ref(z) G Theorem Let M be a complex manifold, z 0 M, f(z 0 ) is real, and. r, r (0, ), be a one parametric family of continuous seminorms in H(M), such that: (i). r. r if r r. (ii) f r f(z 0 ) as r 0. Then there is r (0, ) and K M such that f r f K f H(M), Re f > 0 Proof. We will proof this theorem in 3 steps; Step. Let K M, and consider the set Φ = {φ H(M) : sup φ(z) =, φ(z 0 ) = ɛ 0 K, where ɛ 0 < }, Claim : Φ is a compact set. To see that, let {φ n } be a sequence in Φ and define g n = φ n + φ n n N, then g n (z 0 ) = ɛ 0. + ɛ 0 Now, since Reφ n < Re(φ n + ) and Imφ n = Im(φ n + ) so, φ n φ n < + φ n n N, then + φ n <, then g n = φ n + φ n <, so g n is uniformly bounded sequence in H(M), since M is a complex manifold subset of C n, then we have -cases: 54