On Harmonic ν-Bloch and ν-Bloch-type mappings
aa r X i v : . [ m a t h . C V ] J u l ON HARMONIC ν -BLOCH AND ν -BLOCH-TYPE MAPPINGS GANG LIU AND SAMINATHAN PONNUSAMY
Abstract.
The aim of this paper is twofold. One is to introduce the class of harmonic ν -Bloch-type mappings as a generalization of harmonic ν -Bloch mappings and thereby wegeneralize some recent results of harmonic 1-Bloch-type mappings investigated recentlyby Efraimidis et al. [12]. The other is to investigate some subordination principles forharmonic Bloch mappings and then establish Bohr’s theorem for these mappings and ina general setting, in some cases. Introduction
A significant part of function theory deals with univalent functions, function spaces suchas Bloch spaces, Bohr’s phenomenon and their various generalizations. Several authorshave contributed a lot to this development, and most importantly, in the area of planarharmonic mappings. For basic results about harmonic mappings, the reader may refer[8], the monograph of Duren [11] and the recent survey of some basic materials from [20].Concerning classical Bloch spaces, see [3, 4, 10]. In recent years, Bohr’s phenomenon, itsvarious generalizations including higher dimensional analogues and its harmonic analogueshave been studied by various authors. For more details of the importance, background,development and results, we refer to the recent survey on this topic [2] and the referencestherein. The recent results on this topic for harmonic mappings may be obtained from[15, 16]. Our primary goal here is to continue to study harmonic Bloch-type mappingsand as applications, we consider Bohr’s inequality in a general setting.Throughout we consider complex-valued harmonic mappings in the open unit disk D = { z : | z | < } : ∆ f = f zz = 0. It is well known that every harmonic mapping in D has a canonical decomposition f = h + g , where h and g are analytic functions with g (0) = 0.Thus, we may express h and g as h ( z ) = ∞ X n =0 a n z n and g ( z ) = ∞ X n =1 b n z n . (1)Moreover, the function f = h + g is locally univalent and sense-preserving in D if andonly if its Jacobian J f = | f z | − | f z | = | h ′ | − | g ′ | > D by Lewy’s theorem (see [17]),i.e., | h ′ | > | g ′ | or | ω f | < D , where ω f = g ′ /h ′ is the dilatation of f .For a given ν ∈ (0 , ∞ ), a harmonic mapping f = h + g in D is called a harmonic ν -Blochmapping if β ν ( f ) := sup z ∈ D (1 − | z | ) ν ( | h ′ ( z ) | + | g ′ ( z ) | ) < ∞ . Mathematics Subject Classification.
Primary: 30A10; 30B10; 30H30; 31A05; Secondary: 30C55.
Key words and phrases. harmonic ν -Bloch mapping, harmonic ν -Bloch-type mapping, uniformly lo-cally univalent, pre-Schwarzian, subordination, p -Bohr radius. This defines a seminorm, and the space equipped with the norm || f || B H ( ν ) := | f (0) | + β ν ( f )is called the harmonic ν -Bloch space , denoted by B H ( ν ). It is a Banach space. In particularthe space B ( ν ) defined by B ( ν ) = { f = h + g ∈ B H ( ν ) : g ≡ } forms a Banach space equipped with the norm || f || B ( ν ) := | f (0) | + β ν ( f ). Clearly, f = h + g ∈ B H ( ν ) if and only if h, g ∈ B ( ν ), sincemax { β ν ( h ) , β ν ( g ) } ≤ β ν ( f ) ≤ β ν ( h ) + β ν ( g ) . The harmonic ν -Bloch space B H ( ν ) was introduced in [6], which was a generalization of B H (1) that was studied by Colonna in [10] as a generalization of classical Bloch space B (1).One can refer to [3, 4, 5, 7, 19, 23] for information on B (1) and its extension. Motivated byresults on analytic Bloch functions, Efraimidis et at. [12] introduced harmonic Bloch-typemappings, which coincide with the following harmonic 1-Bloch-type mappings. Definition 1.
For a given ν ∈ (0 , ∞ ) , a harmonic mapping f on D is called a harmonic ν -Bloch-type mapping if β ∗ ν ( f ) := sup z ∈ D (1 − | z | ) ν q | J f ( z ) | < ∞ . We write B ∗ H ( ν ) for the space of all such mappings and we call || f || B ∗ H ( ν ) := | f (0) | + β ∗ ν ( f ) , the pseudo-norm of f . In Section 2, we will see that B ∗ H ( ν ) is not a linear space for any ν >
0. Because(1 − | z | ) ν q | J f ( z ) | ≤ (1 − | z | ) ν ( | f z ( z ) | + | f z ( z ) | ) , z ∈ D , it is clearly that B H ( ν ) ⊂ B ∗ H ( ν ) and thus, the space B ∗ H ( ν ) is a generalization of B H ( ν ).In addition, in the case of analytic functions f , these spaces coincide and thus, we have || f || B ∗ H ( ν ) = || f || B H ( ν ) = || f || B ( ν ) . One of the aims of this article is to generalize some of the known results of harmonic ν -Bloch mappings and ν -Bloch-type mappings (especially, the results of [12]). The paper isdivided into sections as follow: In Section 2, for the function spaces B H ( ν ) and B ∗ H ( µ ), weinvestigate its affine and linear invariance , and the inclusion relations under particularconditions. In Section 3, we find a connection between these function spaces and thespace of uniformly locally univalent harmonic mappings. Moreover, some subordination principles concerning the spaces B H (1) and B ∗ H (1) are also investigated. In Section 4,we give the growth and coefficients estimates for sense-preserving mappings in B ∗ H ( ν ).Finally, as applications of our investigations, we determine the Bohr radius for functionsin B ( ν ), and p -Bohr radius for functions in B H ( ν ) and B ∗ H ( ν ) (sense-preserving) in Section5. n Harmonic ν -Bloch and ν -Bloch-type mappings 3 Affine and linear invariance and inclusion relations
Throughout the article ν is a constant in the interval (0 , ∞ ). We first discuss the affineand linear invariance of B H ( ν ) and B ∗ H ( ν ). Let L be a family of harmonic mappingsdefined in D . Then the family L is said to be affine invariant if A ◦ f ∈ L for each f ∈ L and for all affine mappings A of the form A ( z ) = az + bz ( a, b ∈ C ). The family L iscalled linear invariant if for each f ∈ L , f ◦ ϕ α ∈ L ∀ ϕ α ( z ) = z + α αz ∈ Aut( D ) . Proposition 1. (1)
Both B H ( ν ) and B ∗ H ( ν ) are affine invariant. (2) Each of B ( ν ) , B H ( ν ) and B ∗ H ( ν ) is linear invariant. Proof. (1) Let f = h + g and A ( z ) = az + bz ( a, b ∈ C ). Then A ◦ f = ah + bg + ag + bh and thus, | ( ah + bg ) ′ | + | ( ag + bh ) ′ | ≤ ( | a | + | b | )( | h ′ | + | g ′ | )and J A ◦ f = | ( ah + bg ) ′ | − | ( ag + bh ) ′ | = ( | a | − | b | ) J f . The desired conclusion now easily follows.(2) We only need to prove that B ∗ H ( ν ) is linear invariant. For ϕ α ( z ) ∈ Aut( D ), we have J f ◦ ϕ α ( z ) = | ϕ ′ α ( z ) | J f ( ϕ α ( z )) and 1 − | ϕ α ( z ) | = | ϕ ′ α ( z ) | (1 − | z | ) , and obtain(1 − | z | ) ν q | J f ◦ ϕ α ( z ) | = (1 − | z | ) ν | ϕ ′ α ( z ) | q | J f ( ϕ α ( z )) | = (cid:18) − | z | − | ϕ α ( z ) | (cid:19) ν − (1 − | ϕ α ( z ) | ) v q | J f ( ϕ α ( z )) |≤ (cid:18) | αz | − | α | (cid:19) ν − β ∗ ν ( f ) ≤ (cid:18) | α | − | α | (cid:19) | ν − | β ∗ ν ( f ) , z ∈ D . Now it is obvious that if f ∈ B ∗ H ( ν ), then f ◦ ϕ α ∈ B ∗ H ( ν ) for each ϕ α ∈ Aut( D ).Similarly, B ( ν ) is linear invariant so that B H ( ν ) is also linear invariant. (cid:3) For each ν >
0, although both B ( ν ) and B H ( ν ) are Banach spaces, the following exampleshows that B ∗ H ( ν ) is not a linear space. It also shows that some functions in B ∗ H ( ν ) maygrow arbitrarily fast. Therefore, to study certain properties of functions in B ∗ H ( ν ) in whatfollows, we shall restrict harmonic mappings to be sense-preserving. Example 1.
Let f = h + h , where h ( z ) = ( µ − − (1 − z ) − µ for some µ > ν +1. Clearly,we have f ( z ) and the identity function z belong to B ∗ H ( ν ) whereas F ( z ) = f ( z ) + z doesnot, since (1 − x ) ν | J F ( x ) | = (1 + x ) ν − x ) µ (1 − x ) µ − ν → ∞ as (0 , ∋ x → − . G. Liu and S. Ponnusamy
Next we deal with the inclusion relations B ( ν ) ⊂ B H ( ν ) ⊆ B ∗ H ( ν ). Proposition 2.
Let µ and ν be two constants with < µ < ν . We have (1) B ( ν ) ⊂ B H ( ν ) ⊂ B ∗ H ( ν ) ; (2) B ( µ ) ⊂ B ( ν ) , B H ( µ ) ⊂ B H ( ν ) and B ∗ H ( µ ) ⊂ B ∗ H ( ν ) . Proof. (1) It only needs to find a function f ν ∈ B ∗ H ( ν ) \B H ( ν ) for each ν >
0. Forthe sake of the context later, we will prove that the one parameter family of functions f ν,t ∈ B ∗ H ( ν ) \B H ( ν ) for each ν >
0, where f ν,t ( z ) = h ν ( z ) + g ν,t ( z ) , t ∈ [0 , , (2)with h ν ( z ) = ( − log(1 − z ) for ν = 1 / , ( ν − / − (cid:2) (1 − z ) / − ν − (cid:3) for ν = 1 / , (3)and g ν,t ( z ) = − log(1 − z ) − (1 − t ) z for ν = 1 / ,z − z + (1 − t ) log(1 − z ) for ν = 3 / , (1 − z ) / − ν − ν − / − (1 − t ) (1 − z ) / − ν − ν − / ν = 1 / , / . Fix t ∈ [0 , f ν,t is sense-preserving in D with thedilatation ω f ν,t ( z ) = t + (1 − t ) z for each ν >
0. Again, by computation, we have(1 − | z | ) ν q | J f ν,t ( z ) | = (1 − | z | ) ν | − z | ν +1 / | q − | ω f ν,t ( z ) | = (1 + | z | ) ν (1 − | z | ) ν | − z | ν s − | z | − t Re( z (1 − z )) − t | − z | | − z |≤ ν +1 / √ t, which gives f ν,t ∈ B ∗ H ( ν ) for each ν >
0. Since(1 − x ) ν | h ′ ν ( x ) | = (1 + x ) ν √ − x → ∞ as (0 , ∋ x → − , we obtain that for each ν > h ν
6∈ B H ( ν ) and thus, f ν,t
6∈ B H ( ν ).(2) Let 0 < µ < ν . Clearly, B ( µ ) ⊆ B ( ν ), B H ( µ ) ⊆ B H ( ν ) and B ∗ H ( µ ) ⊆ B ∗ H ( ν ). Thenthe inclusions B H ( µ ) ⊂ B H ( ν ) and B ∗ H ( µ ) ⊂ B ∗ H ( ν ) obviously follow by (1) if we prove B ( µ ) ⊂ B ( ν ). For this, we simply consider the function f ν satisfying f ′ ν ( z ) = (1 − z ) − ν , itis easy to see that f ν ∈ B ( ν ) \B ( µ ). This completes the proof. (cid:3) It is natural to ask for the structure of the set B ∗ H ( ν ) \B H ( ν ). Proposition 3.
Let f = h + g be a harmonic mapping in D . Then f ∈ B H ( ν ) if and onlyif f ∈ B ∗ H ( ν ) and either h ∈ B ( ν ) or g ∈ B ( ν ) . We get B ∗ H ( ν ) \B H ( ν ) = { f = h + g ∈ B ∗ H ( ν ) : h
6∈ B ( ν ) and g
6∈ B ( ν ) } . n Harmonic ν -Bloch and ν -Bloch-type mappings 5 Proof.
It suffices to observe that | h ′ | ≤ p | J f | + | g ′ | and | g ′ | ≤ p | J f | + | h ′ | for a harmonicmapping f = h + g . (cid:3) The following question arises.
Problem 1.
Suppose that f ∈ B ∗ H ( ν ) . Does there exist a constant c ( ν ) depending onlyon ν such that f ∈ B H ( c ( ν )) ? In order to give an affirmative answer to this problem, we need some extra condi-tions based on the following observation for the function f = h + h , where h ( z ) =exp ((1 + z ) / (1 − z )) . Clearly, f ∈ B ∗ H ( ν ) for all ν >
0. However, f
6∈ B H ( ν ) for any ν >
0, since h
6∈ B ( ν ), which can be deduced from(1 − x ) ν | h ′ ( x ) | = 2(1 + x ) ν − "(cid:18) − x x (cid:19) ν − e x − x → ∞ as (0 , ∋ x → − . Proposition 4.
Let f be a locally univalent harmonic mapping in D . If f ∈ B ∗ H ( ν ) , then f ∈ B H ( ν + 1 / . Moreover, the constant / is sharp for each ν > . Proof.
Note that f ∈ B H ( ν ) (resp. B ∗ H ( ν )) if and only if f ∈ B H ( ν ) (resp. B ∗ H ( ν )).Without loss of generality, we may thus assume that f = h + g is sense-preserving withthe dilatation ω = ω f so that g ′ = ωh ′ and J f = | h ′ | (1 − | ω | ) or | h ′ | = s J f − | ω | . It follows (see [13, Corollary 1.3]) that | ω ( z ) | ≤ | z | + | ω (0) | | ω (0) z | , z ∈ D . Now we suppose that f ∈ B ∗ H ( ν ). Then we get(1 − | z | ) ν q | J f ( z ) | ≤ β ∗ ν ( f ) < ∞ , z ∈ D . Consequently, we have | g ′ ( z ) | < | h ′ ( z ) | in D , where | h ′ ( z ) | = s J f ( z )1 − | ω ( z ) | ≤ β ∗ ν ( f )(1 − | z | ) ν p − | ω ( z ) | (4) ≤ β ∗ ν ( f )(1 − | z | ) ν " − (cid:18) | z | + | ω (0) | | ω (0) || z | (cid:19) − / = β ∗ ν ( f )(1 − | z | ) ν " | ω (0) | | z | p (1 − | z | )(1 − | ω (0) | ) ≤ β ∗ ν ( f )(1 − | z | ) ν + s | ω (0) | − | ω (0) | , (5) G. Liu and S. Ponnusamy which shows that h (and hence g ) belongs to B H ( ν + 1 / f ∈ B H ( ν + 1 / / ν >
0, it suffices to check for the function f ν, = h ν + g ν, defined by (2). From the proof of Proposition 2, the function f ν, ( ∈ B ∗ H ( ν ))is sense-preserving in D . On the other hand, it is easy to see that h ν ∈ B ( ν + 1 / g ν, ∈ B ( ν + 1 /
2) and thus, f ν ∈ B H ( ν + 1 / < ε < ν + 1 / h ν
6∈ B ( ε ), which means f ν,
6∈ B H ( ε ). We complete the proof. (cid:3) Uniformly locally univalent and subordination principles
Connection with uniformly locally univalent harmonic mappings.
Moti-vated by the characterization of Bloch space B (1) and the recent work of the authors [18]concerning equivalent conditions of uniformly locally univalent (briefly, ULU) harmonicmappings, we will show the connections among harmonic ν -Bloch, ν -Bloch-type mappingsand ULU harmonic mappings.We first introduce the notion and some properties of ULU harmonic mappings. Aharmonic mapping f = h + g in D is called ULU if there exists a constant ρ > f is univalent on the hyperbolic disk D h ( a, ρ ) = (cid:26) z ∈ D : (cid:12)(cid:12)(cid:12)(cid:12) z − a − az (cid:12)(cid:12)(cid:12)(cid:12) < tanh ρ (cid:27) , of radius ρ , for every a ∈ D . One of equivalent conditions of ULU is stated in terms ofthe pre-Schwarzian derivative or norm . Let f be a locally univalent harmonic mappingin D . The pre-Schwarzian derivative and the norm of f are defined as [14] (see also [9]) P f = (log J f ) z , z ∈ D , and || P f || = sup z ∈ D (1 − | z | ) | P f ( z ) | , respectively. Clearly, the two definitions coincide with the corresponding definitions inthe analytic case. Similar to the proof of [14, Theorem 7], the function f = h + g in D isULU if and only if || P f || < ∞ (see also [18, Theorem 4.1]). Several equivalent conditionsof ULU mappings can be found in these three papers and the references therein.Now let’s restrict f to be analytic in D . It is well-known that f ∈ B (1) if and only ifthere exists a constant c > F such that f = c log F ′ (see [19]). On the other hand, f is ULU if and only if there exists a constant c > F such that f ′ = ( F ′ ) c (see [22, Theorem 2]). Thus, f ∈ B (1)if and only if there exists a ULU analytic function F such that f = log F ′ . Furthermore, aharmonic mapping f = h + g belongs to B H (1) if and only if there exist two ULU analyticfunctions H and G such that f = log H ′ +log G ′ . A natural question is to ask: What aboutthe characterization of B ∗ H (1) ? The following theorem and example show some extraneouscomplexities of the structure of the space B ∗ H (1), which are different from Example 1. Theorem 1.
Let F = H + G be sense-preserving and ULU in D . Then for each ε ∈ D ,the function f ε = h ε + g ε belongs to B ∗ H (1) , where h ε = log( H ′ + εG ′ ) and ω = g ′ ε /h ′ ε isbounded in D . n Harmonic ν -Bloch and ν -Bloch-type mappings 7 Proof.
Suppose that F = H + G is a sense-preserving and ULU in D . It follows from [18,Theorem 4.1] that || P H + εG || < ∞ for all ε ∈ D . By assumption, for each ε ∈ D , we have(1 − | z | ) q | J f ε ( z ) | ≤ (1 − | z | ) | h ′ ε ( z ) | (1 + sup z ∈ D | ω ( z ) | )= (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) H ′′ + εG ′′ H ′ + εG ′ (cid:12)(cid:12)(cid:12)(cid:12) (1 + sup z ∈ D | ω ( z ) | ) ≤ || P H + εG || (1 + sup z ∈ D | ω ( z ) | ) < ∞ and the assertion follows. (cid:3) Example 2.
Consider the function f = h + g in D with the dilatation ω f ( z ) = e iθ z , where h = log H ′ and H ( z ) = exp r z − z ! =: exp( q ( z )) , z ∈ D , and the principal branch of the square root is chosen such that q (0) = 1. We claim that f ∈ B ∗ H (1) \B H (1) and H is locally univalent but not ULU in D . To do this, straightforwardcomputations give that H ′ ( z ) = 1(1 − z ) r − z z exp r z − z ! = 0 , z ∈ D , and || P H || = sup z ∈ D (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) (1 + 2 z ) √ − z + √ z (1 − z ) √ − z (cid:12)(cid:12)(cid:12)(cid:12) = ∞ , showing that H is locally univalent but not ULU in D . Again, elementary computationsshow that(1 − | z | ) q | J f ( z ) | = (1 − | z | ) | h ′ ( z ) | p − | z | = (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) (1 + 2 z ) √ − z + √ z (1 − z ) √ − z (cid:12)(cid:12)(cid:12)(cid:12) p − | z | ≤ (cid:12)(cid:12)(cid:12) (1 + 2 z ) √ − z + √ z (cid:12)(cid:12)(cid:12) p | z | < ∞ , z ∈ D , which implies f ∈ B ∗ H (1). Moreover, because H is not ULU, we find that h
6∈ B (1) andthus, f
6∈ B H (1). Hence we conclude that, f ∈ B ∗ H (1) \B H (1).Although Theorem 1 is a generalization of [12, Theorem 2], we can’t give a complementcharacterization of B ∗ H (1), let alone to B ∗ H ( ν ) ( ν > ν -Bloch-type mapping for some ν >
0. Recall that f is ULU ifand only if || P f || < ∞ . In view of this, to describe our result more precisely, we definethe set B H ( ν ) = { f : f is a locally univalent harmonic mapping in D with || P f || ≤ ν } and its subset B ( ν ) of all analytic functions in B H ( ν ). Theorem 2.
For any ν > , we have B H ( ν ) ⊂ B ∗ H ( ν/ . In particular, B ( ν ) ⊂ B ( ν/ .Moreover, these two inclusions are best possible. G. Liu and S. Ponnusamy
Proof.
Assume f ∈ B H ( ν ) for some ν >
0. Note that || P f || = || P f || . Without loss ofgenerality, we may assume that f is sense-preserving. Then, because f z (0) = 0, we mayconsider F ( z ) = f ( z ) − f (0) f z (0) . Then F is sense-preserving in D with the normalization F (0) = F z (0) − || P F || = || P f || and thus, F ∈ B H ( ν ). It follows from [18, Theorem 6.1] that J F ( z ) ≤ (1 − | F z (0) | ) (cid:18) | z | − | z | (cid:19) ν , z ∈ D , which implies F ∈ B ∗ H ( ν/ B ∗ H ( ν ) preserves affine invariance for each ν >
0, weget f ∈ B ∗ H ( ν/ B H ( ν ) ⊂ B ∗ H ( ν/
2) from Example 1. The sharpness follows ifwe choose f ( z ) = f ν ( z ) = Z z (cid:18) t − t (cid:19) ν/ dt + b Z z (cid:18) t − t (cid:19) ν/ dt, z ∈ D , where | b | <
1. Indeed, it is easy to see that || P f ν || = ν and f ν ∈ B ∗ H ( ν/
2) but f ν
6∈ B ∗ H ( ε )for any 0 < ε < ν/ f is restricted to be analytic, then a similar proof shows that B ( ν ) ⊂ B ( ν/ f ν with b = 0. (cid:3) Subordination principles.
Every bounded analytic function in D belongs to the(analytic) Bloch space B (1). This fact also holds for harmonic mappings (see [10, The-orem 3]) and for a simpler proof of it (using subordination), we refer to [6, Theorem A].That is, if a harmonic mapping f is bounded in D , then f belongs to the (harmonic) Blochspace B H (1). Next we will investigate some subordination principles for some harmonicBloch mappings.Let A D denotes the class of analytic functions φ : D → D and A D denotes the subclassof A D with the normalization φ (0) = 0. In 2000, Schaubroeck [21] generalized the notionof subordination from analytic functions to harmonic mappings. Let f and F be twoharmonic mappings in D . Then f is subordinate to F , denoted by f ≺ F , if there is afunction φ ∈ A D such that f = F ◦ φ . We denote f (cid:22) F if there exists a function φ ∈ A D such that f = F ◦ φ . Clearly, if f ≺ F then f (cid:22) F . Theorem 3. (Subordination principle)
Let f and F be two harmonic mappings in D . If f (cid:22) F and F ∈ B H (1) (resp. B ∗ H (1) ), then f ∈ B H (1) (resp. B ∗ H (1) ). In particular, if f (cid:22) F and F ∈ B (1) , then f ∈ B (1) . Proof.
We just need to prove the case of ν -Bloch-type mappings since the proof of theremaining cases are similar. Assume that f (cid:22) F and F ∈ B ∗ H (1). Then there exists afunction φ ∈ A D such that f = F ◦ φ . We find that J f ( z ) = J F ( φ ( z )) | φ ′ ( z ) | n Harmonic ν -Bloch and ν -Bloch-type mappings 9 and by the Schwarz-Pick lemma, we get (1 − | z | ) | φ ′ ( z ) | ≤ − | φ ( z ) | . Consequently,(1 − | z | ) q | J f ( z ) | = (1 − | z | ) | φ ′ ( z ) | p | J F ( φ ( z )) |≤ (1 − | φ ( z ) | ) p | J F ( φ ( z )) | ≤ β ∗ ( F ) < ∞ , z ∈ D , which clearly shows that f ∈ B ∗ H (1). (cid:3) Remark 1.
We remind that f = h + g ∈ B H (1) does not mean that either h , g or f isbounded even if f is sense-preserving in D . For instance, consider f ( z ) = h ( z ) + g ( z ) = log(1 − z ) + z + log(1 − z ) = z + 2 log | − z | . and f ( z ) = h ( z ) − g ( z ) = log(1 − z ) − z + log(1 − z ) = − z + 2 i arg(1 − z ) . Then it is easy to verify that f , f ∈ B H (1), and both f and f are sense-preserving in D . However, except f , neither h , nor g nor f is bounded in D .4. Growth and coefficients estimates
In this section, we investigate some growth and coefficients estimates for functions in B ∗ H ( ν ). For corresponding results in the case of B H ( ν ), the reader can refer to [6, 23]. Theorem 4.
Suppose that f = h + g ∈ B ∗ H ( ν ) is sense-preserving in D with the dilatation ω f , where h and g are given by (1) . Then max {| h ( z ) − a | , | g ( z ) |} ≤ β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | h ν ( r ) , | z | = r < , where h ν is defined by (3) . The estimate is sharp in order of magnitude for each ν > / .If ν < / , then each of h, g, f is bounded in D . Proof.
Let | z | = r <
1. Following the proof of Proposition 4 and (5), because f issense-preserving, we havemax {| h ( z ) − a | , | g ( z ) |} = max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) z Z h ′ ( tz ) dt (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) z Z g ′ ( tz ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≤ r Z | h ′ ( tz ) | dt ≤ β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | Z r (1 − r t ) ν +1 / dt ≤ β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | Z r (1 − rt ) ν +1 / dt = β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | h ν ( r ) . For each ν > /
2, the sharpness of the order of magnitude can be seen from thefunctions f ν,t = h ν + g ν,t defined by (2) for t ∈ [0 , h ν from its formulation. Fix t ∈ [0 , g ν,t , since for x ∈ (0 ,
1) andany ε > − x ) ν − / − ε | g ν,t ( x ) | = (1 + x ) ν − / − ε (1 − x ) ε (cid:12)(cid:12)(cid:12)(cid:12) − (1 − x ) ν − / ν − / − − tν − / (cid:2) (1 − x ) − (1 − x ) ν − / (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) → ∞ as x → − when ν > / ν = 3 /
2, and(1 − x ) − ε | g / ,t ( x ) | = (1 + x ) − ε / (1 − x ) ε | x + (1 − t )(1 − x ) log(1 − x ) | → ∞ as x → − .If ν < /
2, then for | z | = r < {| h ( z ) − a | , | g ( z ) |} ≤ β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | (1 / − ν ) − (1 − (1 − r ) / − ν ) ≤ β ∗ ν ( f ) s | ω f (0) | − | ω f (0) | (1 / − ν ) − . Obviously, both h and g are bounded in D and thus, f is also bounded in D . (cid:3) If f ∈ B ∗ H (1 /
2) is sense-preserving in D , then the boundedness of f is uncertain, whichmay be verified easily by considering the two functions f and f in Remark 1. Indeed, f , f ∈ B ∗ H (1 / Theorem 5.
Suppose that f = h + g ∈ B ∗ H ( ν ) is sense-preserving in D with the dilatation ω f , where h and g are given by (1) . Then | b | < | a | ≤ β ∗ ν ( f ) p − | ω f (0) | , and max {| a n | , | b n |} ≤ β ∗ ν ( f ) (cid:18) e ν + 1 (cid:19) ν +1 / s | ω f (0) | − | ω f (0) | ( n + 2 ν ) ν − / , n ≥ . Proof.
The first inequality follows if we set z = 0 in (4). For the second inequality, werecall from (5) that | g ′ ( z ) | < | h ′ ( z ) | ≤ | ω f (0) | − | ω f (0) | β ∗ ν ( f ) (1 − | z | ) ν +1 , z ∈ D . We integrate this inequality over the circle | z | = r and get ∞ X n =1 n | b n | r n − < ∞ X n =1 n | a n | r n − ≤ | ω f (0) | − | ω f (0) | β ∗ ν ( f ) (1 − r ) ν +1 . Thus, for n ≥
2, we obtainmax {| a n | , | b n |} ≤ β ∗ ν ( f ) n s | ω f (0) | − | ω f (0) | r n − (1 − r ) ν +1 / . n Harmonic ν -Bloch and ν -Bloch-type mappings 11 It is a simple exercise to see that r − n (1 − r ) − ( ν +1 / is maximized in r ∈ (0 ,
1) for r = q n − n +2 ν . Consequently,max {| a n | , | b n |} ≤ β ∗ ν ( f ) n s | ω f (0) | − | ω f (0) | (cid:18) n + 2 νn − (cid:19) n/ − / (cid:18) n + 2 ν ν + 1 (cid:19) ν +1 / = β ∗ ν ( f ) φ ν ( n )(2 ν + 1) ν +1 / s | ω f (0) | − | ω f (0) | ( n + 2 ν ) ν − / , where φ ν ( x ) = "(cid:18) ν + 1 x − (cid:19) x − ν +1 ν +1 / (cid:18) νx (cid:19) , x ≥ . Next we prove that φ ν is an increasing function of x to its limit e ν +1 / in [2 , ∞ ). Clearly, φ ν ( x ) > x ≥
2. For convenience, we letΦ ν ( x ) = (log φ ν ( x )) ′ = φ ′ ν ( x ) φ ν ( x ) = 12 log (cid:18) x + 2 νx − (cid:19) − (2 ν + 1) x + 4 ν x ( x + 2 ν ) . Differentiating with respect to x yieldsΦ ′ ν ( x ) = − ψ ν ( x )2 x ( x − x + 2 ν ) , ψ ν ( x ) = (2 ν − x + 8( ν − ν ) x + 8 ν . If ν = 1 /
2, then ψ ν ( x ) = 2 x + 2 ≥ ψ ν (2) = 6 > x ≥
2. If ν = 1 /
2, then we obtain ψ ′ ν ( x ) = 2(2 ν − x + 8( ν − ν ) ≥ ψ ′ ν (2) = 8( ν − / + 2 > x ≥ ψ ν ( x ) ≥ ψ ν (2) = 4(2 v + 1) > x ≥ ′ ν ( x ) < , ∞ ) so that Φ ν ( x ) > lim x →∞ Φ ν ( x ) = 0 for all x ∈ [2 , ∞ ).Therefore, we obtain φ ′ ν ( x ) > , ∞ ) and the proof is complete. (cid:3) Bohr’s inequalities
One of the classical problems in the theory of analytic functions which inspire manyresearchers is to determine r = sup ( r ∈ (0 ,
1) : M f ( r ) := ∞ X n =0 | a n | r n ≤ ) , where the supremum is taken over the class which consists of all functions of the form f ( z ) = P ∞ n =0 a n z n that converges in D and | f ( z ) | ≤ D . It is well-known that r = 1 / / D . Many authors have discussed the Bohr radius and extendedthis notion to various settings which led to the introduction of Bohr’s phenomenon. Asremarked in the introduction, we refer to [2, 15, 16] and the references therein for resultson this topic. Moreover, in [15] the authors introduced the notion of p -Bohr radius forharmonic mappings which is defined as follows: Let f = h + g be a harmonic mapping in D , where h and g have the form (1) . For p ≥ , the p -Bohr radius for f is defined to bethe largest value r p such that | a | + ∞ X n =1 ( | a n | p + | b n | p ) /p r n ≤ for | z | = r ≤ r p . Clearly, all these radii coincide in the analytic case. The classical case p = 1 is consideredfirst time in [1].In this section, we determine the Bohr radius for analytic functions in B ( ν ) and p -Bohrradius for harmonic mappings in B H ( ν ) and B ∗ H ( ν ). The following results are generaliza-tions of that of the results of Kayumov et al. [16, Section 4]. Theorem 6.
Assume that f ( z ) = P ∞ n =0 a n z n belongs to B ( ν ) and || f || B ( ν ) ≤ . Then ∞ X n =0 | a n | r n ≤ for | z | = r ≤ r ( ν ) = max { r ( ν ) , r ( k ) } when ν ∈ ( k/ , ( k + 1) / for some k ∈ N := N ∪ { } . Here r ( ν ) is the unique solution in (0 , to the equation − r ) ν − π r = 0 (6) and r ( k ) is the unique solution in (0 , to the equation rF k ( r ) − r = 0 , (7) where F k ( r ) = ∞ X n =1 r n n for k = 0 , log 11 − r for k = 1 , k log 11 − r + 1 k k − X n =1 n (cid:18) − r ) n − (cid:19) for k ≥ . (8) Moreover, r ( ν ) can not be replaced by r ( ν ) when ν ≥ , where r (1) = 0 . , and r ( ν ) = min (cid:26) . , q − / ν − √ ν − (cid:27) for ν > . Proof.
By hypothesis, we have || f || B ( ν ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 na n z n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | f ′ ( z ) | ≤ (1 − | a | ) (1 − | z | ) ν , z ∈ D . Integrating the inequality over the circle | z | = r yields ∞ X n =1 n | a n | r n − ≤ (1 − | a | ) (1 − r ) ν . (9) n Harmonic ν -Bloch and ν -Bloch-type mappings 13 By the classical Cauchy–Schwarz inequality, we obtain | a | + ∞ X n =1 | a n | r n ≤| a | + vuut ∞ X n =1 n | a n | r n vuut ∞ X n =1 n ≤| a | + (1 − | a | ) r (1 − r ) ν r π ≤ r ≤ r ( ν ) , where r ( ν ) is the unique solution in (0,1) to the equation of (6). In fact, for each ν ∈ (0 , ∞ ), the function r/ (1 − r ) ν increases from 0 to ∞ in [0 , ν ∈ ( k , k +12 ] for some k ∈ N , then it follows from (9) that ∞ X n =1 n | a n | r n − ≤ (1 − | a | ) (1 − r ) k +1 . Integrating the above inequality twice (with respect to r ) yields ∞ X n =1 | a n | r n ≤ (1 − | a | ) F k ( r ) , where F k ( r ) is defined by (8). Applying the Cauchy–Schwarz inequality again, we have | a | + ∞ X n =1 | a n | r n ≤| a | + vuut ∞ X n =1 | a n | r n vuut ∞ X n =1 r n ≤| a | + (1 − | a | ) r F k ( r ) r − r ≤ r ≤ r ( k ) , where r ( k ) is the unique solution to the equation of (7). Note that both F k ( r ) and r/ (1 − r ) are strictly increasing in [0 , r ( ν ), since B ( ν ) ⊇ B (1) for any ν ≥
1, it follows from [16,Theorem 9] that r ( ν ) can not be replaced by 0 . ν ≥
1. In addition, let’sconsider the function f ν ( z ) = (1 − z ) − ν − ν −
1) = ∞ X n =1 a ν,n z n , z ∈ D . A basic computation shows that f ν ∈ B ( ν ) and || f ν || B ( ν ) = 1 when ν >
1. It is easy tosee that all coefficient a ν,n are non-negative real number for each ν > a ν,n = 0 forodd integer values of n ≥
1. For ν >
1, we consider the following inequality ∞ X n =1 a ν,n r n = (1 − r ) − ν − ν − ≤ , provided r ≤ q − / ν − √ ν − (cid:3) It is easy to see that the function r ( ν ) is monotonically decreasing to 0 in (0 , + ∞ ).In the following table, the notation ( r ( ν ) ց r ( ν )] means that the value of r ( ν ) ismonotonically decreasing from lim ν → ν +1 r ( ν ) = r ( ν ) to r ( ν ) when ν < ν ≤ ν . Sodoes ( r ( ν ) ց r ( ν )]. Note that the function r ( ν ) is monotonically decreasing from0 . , + ∞ ) and r (5 . . ν r ( ν ) r ( k ) r ( ν )(0,1/2] (0 . ց . . . ց . . ց . . . ց . . ց . . . ց . . ց . . . ց . . ց . . . ց . . ց . . . ց . Theorem 7.
Suppose that f = h + g ∈ B H ( ν ) , where h and g are given by (1) . If || f || B H ( ν ) ≤ and p ≥ , then we have | a | + ∞ X n =1 ( | a n | p + | b n | p ) /p r n ≤ for | z | = r ≤ max { r ( ν, p ) , r ( k, p ) } when ν ∈ ( k/ , ( k + 1) / for some k ∈ N . Here r ( ν, p ) is the unique solution in (0 , to the equation − r ) ν − M p π r = 0 (10) and r ( k, p ) is the unique solution in (0 , to the equation M p rF k ( r ) − r = 0 , where M p = max { (2 /p ) − , } and F k ( r ) is defined by (8) . Proof.
By assumption, we see that | h ′ ( z ) | + | g ′ ( z ) | ≤ ( | h ′ ( z ) | + | g ′ ( z ) | ) ≤ (1 − | a | ) (1 − | z | ) ν , z ∈ D . Integrating the inequality over the circle | z | = r so we get ∞ X n =1 n ( | a n | + | b n | ) r n − ≤ (1 − | a | ) (1 − r ) ν . Using the Cauchy-Schwarz inequality, we obtain | a | + ∞ X n =1 ( | a n | p + | b n | p ) /p r n ≤| a | + vuut ∞ X n =1 n ( | a n | p + | b n | p ) /p r n vuut ∞ X n =1 n ≤| a | + vuut M p ∞ X n =1 n ( | a n | + | b n | ) r n r π ≤| a | + p M p (1 − | a | ) r (1 − r ) ν (cid:18) π √ (cid:19) n Harmonic ν -Bloch and ν -Bloch-type mappings 15 which is less than or equal to 1 provided r ≤ r ( ν, p ), where r ( ν, p ) is defined by (10).If ν ∈ ( k/ , ( k + 1) /
2] for some k ∈ N , then we can combine the above proof with thecorresponding proof of Theorem 6. The resulting discussion completes the proof. (cid:3) Next we will study p -Bohr radius for functions in B ∗ H ( ν ). Consider f ( z ) = h ( z ) + g ( z ) = 11 − z + (cid:18) z − z (cid:19) so that a = 1 and a n = b n = 1 for n ≥
1. Clearly, f ∈ B ∗ H ( ν ) and || f || B ∗ H ( ν ) = | a | = 1for any ν >
0. However, we have | a | + ∞ X n =1 ( | a n | p + | b n | p ) /p r n > | a | = || f || B ∗ H ( ν ) for any r >
0. In this case, the p -Bohr radius for f is 0. This is the reason why we addthe condition of sense-preserving in the following result. Theorem 8.
Suppose that f = h + g ∈ B ∗ H ( ν ) is a sense-preserving harmonic mapping,where h and g are given by (1) . If || f || B ∗ H ( ν ) ≤ and p ≥ , then | a | + ∞ X n =1 ( | a n | p + | b n | p ) /p r n ≤ for | z | = r ≤ max { r ( ν, p, | ω f (0) | ) , r ( k, p, | ω f (0) | ) } when ν ∈ ( k/ , ( k + 1) / for some k ∈ N . Here r ( ν, p, | ω f (0) | ) is the unique solution in (0 , to the equation − | ω f (0) | )(1 − r ) ν +1 − M p π (1 + | ω f (0) | ) r = 0 and r ( k, p, | ω f (0) | ) is the unique solution in (0 , to the equation M p (1 + | ω f (0) | ) rF k +1 ( r ) − (1 − | ω f (0) | )(1 − r ) = 0 , where M p = max { /p − , } and F k ( r ) is defined by (8) . Proof.
By hypothesis | g ′ ( z ) | < | h ′ ( z ) | and || f || B ∗ H ( ν ) ≤ | h ′ ( z ) | ≤ − | a | (1 − | z | ) ν + s | ω f (0) | − | ω f (0) | , z ∈ D . As in the proof of the previous theorem, we obtain that ∞ X n =1 n ( | a n | + | b n | ) r n − ≤ | ω f (0) | − | ω f (0) | (1 − | a | ) (1 − r ) ν +1 . The remaining part of the proof is identical to Theorem 7 and thus, we omit the details.The proof is complete. (cid:3)
The dependence of | ω f (0) | about p -Bohr radius in Theorem 8 can be seen from thefollowing example. Example 3.
For t ∈ [1 / , f t on D given by f t ( z ) = h t ( z ) + g t ( z ) = ∞ X n =0 a ( t ) n z n + ∞ X n =1 b ( t ) n z n , z ∈ D , where h t ( z ) = 1 − √ t − t + 12 log 1 + z − z and g t ( z ) = t −
12 log(1 − z ) + t z − z . It is easy to see that each f t is sense-preserving in D with the dilatation ω f t ( z ) = (1 − t ) z + t .We find that h ′ t ( z ) = 11 − z , | g ′ t ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12) (1 − t ) z + t − z (cid:12)(cid:12)(cid:12)(cid:12) ≥ t − (1 − t ) | z || − z | ≥ t − | − z | , z ∈ D , and thus,(1 − | z | ) q J f t ( z ) ≤ (1 − | z | ) s | − z | − (2 t − | − z | ≤ √ t − t , z ∈ D , which implies that f t ∈ B ∗ H (1). Also, we observe that(1 − | x | ) q J f t ( x ) → √ t − t as ( − , ∋ x → − + , which infers β ∗ ( f t ) = 2 √ t − t and || f t || B ∗ H (1) = 1. Clearly, | a ( t )0 | + ∞ X n =1 ( | a ( t ) n | p + | b ( t ) n | p ) /p r n > | a ( t )0 | = 1 − √ t − t for any r >
0. Note that | ω f t (0) | = | g ′ t (0) | / | h ′ t (0) | = t and 1 − √ t − t → || f t || B ∗ H (1) as t → − . This means that if t approaches to 1 − , then the p -Bohr radius for f t approachesto 0. Acknowledgments.
The work was completed during the visit of the first author to theIndian Statistical Institute, Chennai Centre and this author thanks the institute for thesupport and the hospitality. The research of the first author was supported by the NSFsof China (No. 11571049), the Construct Program of the Key Discipline in Hunan Provinceand the Science and Technology Plan Project of Hunan Province (No. 2016TP1020). Thesecond author is on leave from IIT Madras, Chennai.
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E-mail address : [email protected] S. Ponnusamy, Stat-Math Unit, Indian Statistical Institute (ISI), Chennai Centre, 110,Nelson Manickam Road, Aminjikarai, Chennai, 600 029, India.
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