ON PLANAR HARMONIC LIPSCHITZ AND PLANAR HARMONIC HARDY CLASSES

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1 Annales Academiæ Scientiarum Fennicæ Matematica Volumen 36, 11, ON PLANAR HARMONIC LIPSCHITZ AND PLANAR HARMONIC HARDY CLASSES Saolin Cen, Saminatan Ponnusamy and Xiantao Wang Hunan Normal University, Deartment of Matematics Cangsa, Hunan 4181, P. R. Cina; Indian Institute of Tecnology Madras, Deartment of Matematics Cennai-6 36, India; Hunan Normal University, Deartment of Matematics Cangsa, Hunan 4181, P. R. Cina; Abstract. In tis aer, we investigate te roerties of two classes of lanar armonic maings. First, we discuss te equivalent norms on Liscitz-tye saces of armonic K-quasiregular maings and ten we study te relationsi between te armonic area functions and te armonic Hardy classes. We also establis Landau s teorem for a class of armonic Hardy maings. 1. Introduction and reliminaries A comlex-valued function f(z) = u(z) + iv(z) defined on a simly connected domain D of C is called a armonic maing if and only if it is twice continuously differentiable and f =. Tat is te comonents u and v are real armonic in D, were reresents te comlex Lalacian oerator = 4 z z = x + y. Every armonic maing f defined in D as a canonical decomosition f = +g, were and g are analytic in D (see [3] or [4]). Since te Jacobian J f (z) of f is given by J f (z) = f z (z) f z (z) = (z) g (z), f is locally univalent and orientation-reserving if and only if g (z) < (z) in D; or equivalently (z) and te dilatation ω = g / as te roerty tat ω(z) < 1 in D. For a C, let D(a, r) = {z : z a < r}. In secial, we denote D(, r) = and D = D(, 1). Because a comosition f g wit an analytic function g remains armonic, te Riemann maing teorem allows us to assume tat D = D. Trougout tis aer, we consider armonic maings in D unless secially stated. A continuous increasing function ω : [, ) [, ) wit ω() = is called a majorant if ω(t)/t is non-increasing for t >. Given a subset Ω of C, a function f : Ω C is said to belong to te Liscitz sace Λ ω (Ω) if tere is a ositive constant C suc tat (1) f(z) f(w) Cω( z w ) for all z, w Ω. doi:1.5186/aasfm Matematics Subject Classification: Primary 3C65, 3C45; Secondary 3C. Key words: Harmonic quasiregular maing, Liscitz-tye sace, armonic Hardy class, univalent functions, Landau s teorem. Corresonding autor.

2 568 Saolin Cen, Saminatan Ponnusamy and Xiantao Wang For δ >, let () and (3) δ δ + δ ω(t) t dt C ω(δ), < δ < δ, ω(t) t dt C ω(δ), < δ < δ, were ω is a majorant and C is a ositive constant. A majorant ω is said to be regular if it satisfies te conditions () and (3) (see [5, 16]). Let G be a roer subdomain of C or R. We say tat a function f belongs to te local Liscitz sace loc Λ ω (G) if (1) olds, wit a fixed ositive constant C, wenever z G and z w < 1 d(z, G), were d(, ) denotes te Euclidean distance (cf. [6, 1]). Moreover, G is said to be a Λ ω -extension domain if Λ ω (G) = loc Λ ω (G). Te geometric caracterization of Λ ω -extension domains was first given by Gering and Martio [6]. Ten Laalainen [1] extended it to te general case and roved tat G is a Λ ω -extension domain if and only if eac air of oints z, w G can be joined by a rectifiable curve γ G satisfying (4) γ ω(d(z, G)) d(z, G) ds(z) Cω( z w ) wit some fixed ositive constant C = C(G, ω), were ds stands for te arc lengt measure on γ. Furtermore, Laalainen [1, Teorem 4.1] roved tat Λ ω -extension domains exist only for majorants ω satisfying (). Dyakonov [5] caracterized te olomoric functions of class Λ ω in terms of teir modulus. Later in [16, Teorems A], Pavlović came u wit a relatively simle roof of te results of Dyakonov. Recently, many autors considered tis toic and generalized Dyakonov s results to seudo-olomoric functions and real armonic functions of several variables for some secial majorant ω(t) = t α, were α > (see [9, 11, 1, 13, 14]). In tis aer, we first extend [16, Teorems A and B] to lanar K-quasiregular armonic maings as follows, were K 1. Teorem 1. Let ω be a majorant satisfying (), and let G be a Λ ω -extension domain. If f is a lanar K-quasiregular armonic maing of G and continuous u to te boundary G, ten f Λ ω (G) f Λ ω (G) f Λ ω (G, G), were Λ ω (G, G) denotes te class of continuous functions f on G G wic satisfy (1) wit some ositive constant C, wenever z G and w G. For any z 1, z G C, let d ω,g (z 1, z ) := inf γ ω(d(z, G)) d(z, G) ds(z), were te infimum is taken over all rectifiable curves γ G joining z 1 to z. We say tat f Λ ω,inf (G) wenever for any z 1, z G, f(z 1 ) f(z ) Cd ω,g (z 1, z ), were C is a ositive constant wic deends only on f (see [8]).

3 On lanar armonic Liscitz and lanar armonic Hardy classes 569 Teorem. Let ω be a majorant satisfying (). If f is a lanar K-quasiregular armonic maing in G, ten f Λ ω,inf (G) f Λ ω,inf (G). Many autors discussed te relationsis between Hardy classes of olomoric functions and integral means (see [7, 15]). In order to derive an analogous result of [7, Teorem 1] for te setting of armonic maings, we need to introduce some notation. For a armonic function f in D, for > and r < 1, we define (5) I (r, f) = 1 f(re iθ ) dθ and say tat f belongs to te armonic Hardy class H f = su (I (r, f)) 1/ < +. <r<1 For a armonic maing f in D, te generalized armonic area function A (r, f) is defined by A (r, f) = f(z) da(z), were da denotes te normalized Lebesgue measure on D and f = ( f z + f z ) 1/. Te following teorem is an analogous result of [7, Teorem 1]. Teorem 3. Let f be armonic in D and δ >. Ten, if 1 <, 1 (6) f H (D) wile if >, (7) 1 Teorem 4. Let f H A (r, f)(1 r) δ( ) dr < +, A (r, f)(1 r) δ( ) dr < + f H (D). (D) and δ >. If 1 <, ten δ( )+ lim (1 r) A (r, f) =. r 1 Finally, we rove a Landau s teorem for a class of armonic Hardy maings. Teorem 5. Let f be a armonic in D wit f M and f() = λ f () 1 =, were M is a ositive constant, λ f (z) = f z (z) f z (z) and 1. Ten f is univalent in D ρ, were ρ = ϕ(r ) = max <r<1 ϕ(r), ϕ(r) = r ( 1 if t 1 + t ), wit t = 4 1 M. r(1 r) 1

4 57 Saolin Cen, Saminatan Ponnusamy and Xiantao Wang Moreover, f(d ρ ) contains a univalent disk D R R = r ϕ(r ) r ϕ(r ). wit In Teorem 5, we remark tat max <r<1 ϕ(r) does exist, since lim ϕ(r) = lim ϕ(r) =. r + r 1 Te roofs of tese teorems are resented in te following sections. We end tis section wit te following roblem wic is suggested by te referee: Does Teorem 1 still old if te yotesis maings being armonic is droed?. Harmonic Liscitz classes In order to rove our main results, we need te following result. Teorem A. [1, Teorem 7] If f is a K-quasiregular armonic maing of D into itself, ten f z (z) + f z (z) 4K cos( f(z)/ ) 1 z olds for z D. Proof of Teorem 1. Te imlications f Λ ω (G) f Λ ω (G) f Λ ω (G, G) are obvious. We only need to rove f Λ ω (G, G) f Λ ω (G). For a fixed oint z G, let F (η) = f(z + d(z)η)/m z, η D, were d(z) := d(z, G) and M z := su{ f(ζ) : ζ z < d(z)}. By a simle calculation, we obtain tat F η (η) + F η (η) F η (η) F η (η) = f ξ(ξ) + f ξ (ξ) f ξ (ξ) f ξ (ξ) K, were ξ = z + d(z)η. Ten F is a K-quasiregular armonic maing of D into itself. By Teorem A, we ave wic in turn gives F η () + F η () 4K(1 F () ) (8) d(z)( f ξ (z) + f ξ (z) ) 8K (M z f(z) ). Witout loss of generality, we let ζ G wit ζ z = d(z), and let w D(z, d(z)). Ten f(w) f(z) f(w) f(ζ) + f(ζ) f(z) and so, wic imlies tat Cω(d(z)) + Cω(d(z)) 3Cω(d(z)) su ( f(w) f(z) ) 3Cω(d(z)), w D(z,d(z)) (9) M z f(z) 3Cω(d(z)).

5 On lanar armonic Liscitz and lanar armonic Hardy classes 571 Tus, by (8) and (9), we ave (1) f ξ (z) + f ξ (z) 4CK ω(d(z)), z G. d(z) Finally, given any two oints z 1, z G, let γ G be a curve wic joins z 1, z satisfying (4). Integrating (1) along γ, we obtain tat (11) f(z 1 ) f(z ) ( f z (z) + f z (z) ) ds(z) 4CK ω(d(z)) ds(z). d(z) Terefore, (4) and (11) yield γ f(z 1 ) f(z ) C 1 ω( z 1 z ), were C 1 is a ositive constant. Tis comletes te roof. Proof of Teorem. Te imlication f Λ ω,inf (G) f Λ ω,inf (G) is obvious. We need only to rove tat f Λ ω,inf (G) f Λ ω,inf (G). For a fixed oint z G, let F (η) = f(z + d(z)η)/m z, η D, were d(z) := d(z, G) and M z := su{ f(ζ) : ζ z < d(z)}. From te roof of Teorem 1, it follows tat (1) d(z)( f ξ (z) + f ξ (z) ) 8K (M z f(z) ), were ξ = z + d(z)η. For w D(z, d(z)), tere exists a ositive constant C suc tat ω(d(ζ, G)) (13) f(w) f(z) Cd ω,g (w, z) C ds(ζ), d(ζ, G) were [w, z] denotes te straigt segment wit endoints w and z. We observe tat if ζ [w, z], ten one as [w, z] D(z, d(z)) G and terefore wic gives tat (14) [w,z] d(ζ, G) d(ζ, D(z, d(z))), ω(d(ζ, G)) d(ζ, G) ω(d(ζ, D(z, d(z)))). d(ζ, D(z, d(z))) For any w D(z, d(z)), (13) and (14) imly tat ω(d(ζ, G)) ω(d(ζ, D(z, d(z)))) f(w) f(z) C ds(ζ) C ds(ζ) [w,z] d(ζ, G) [w,z] d(ζ, D(z, d(z))) ω(d(z) ζ z ) d(z) ω(t) = C ds(ζ) C dt Cω(d(z)). d(z) ζ z t From tis we obtain tat [w,z] (15) M z f(z) Cω(d(z)). Again, for any z 1, z G, by (1) and (15), tere exists a ositive constant C 1 suc tat f(z 1 ) f(z ) C 1 d ω,g (z 1, z ) and te roof of tis teorem is comleted. γ

6 57 Saolin Cen, Saminatan Ponnusamy and Xiantao Wang 3. Harmonic Hardy classes It is wort to remark tat te standard tecnologies of analytic functions are not useful to rove Teorem 3 and terefore, we use Green s Teorem in its roof. Green s teorem states tat if g C (D), ten (16) 1 g(re iθ ) dθ = g() + 1 ( ) r g(z) log da(z), r < 1. z Proof of Teorem 3. First we rove te imlication (7). Let f H (D) and δ >. For r < 1, by te Poisson integral formula, we ave Using Jensen s inequality, we ave and so f(z) = 1 r z z re iθ f(reiθ ) dθ, z. f(z) 1 r z z re iθ f(reiθ ) dθ ri (r, f), r z (17) f(z) (r z ) ri (r, f), were I (r, f) is defined by (5). It follows tat (18) r r (r ρ) δ M (ρ, f) dρ ri (r, f) dρ (r ρ) r1+δ I (r, f), 1 δ δ were M(r, f) = su{ f(z) : z = r}. By (16), we ave I (r, f) = f() + 1 ( f(z) ) log r z da(z) and terefore r d dr I (r, f) = 1 ( f(z) ) da(z) D r [ ( = 1) f(z) 4 f z (z)f(z) + f(z)f z (z) ] + f(z) f(z) da(z) ( 1) f(z) f(z) da(z) ( 1)M (r, f) f(z) da(z) = ( 1)A (r, f)m (r, f).

7 On lanar armonic Liscitz and lanar armonic Hardy classes 573 By integration, Hölder s inequality and (18), we ave I (r, f) f() + ( 1) r [ r f() + ( 1) [ r ( A (ρ, f) ρ f() + ( 1) [ r ( A (ρ, f) ρ A (ρ, f) M (ρ, f) dρ ρ (r ρ) δ M (ρ, f) dρ ) (r ρ) δ( ) dρ ( δ ) I (r, f) ] ) ] (r ρ) δ( ) dρ. ] Witout loss of generality, we may now assume tat f(). Ten we ave ( I (r, f) f() + ) [ r ( A (ρ, f) ) ] + ( 1) (r ρ) δ( ) dρ δ ρ wic sows tat 1 A (r, f)(1 r) δ( ) dr < + f H (D). Next, we rove te imlication (6). By a simle calculation, we get r d [( ) dr I (r, f) = 1 f(z) 4 f z (z)f(z) + f(z)f z (z) ] + f(z) f(z) da(z) ( 1) f(z) f(z) da(z) and A (r, f) = f(z) da(z) M (r, f) f(z) f(z) da(z), wic imlies tat ( 1)A (r, f) r d dr I (r, f)m (r, f). By (18) and Hölder s inequality, we see tat [ ] r [ A (ρ, f) ( 1) ρ r [ r (r ρ) δ( ) (r ρ) δ( ) M ( ) (r, f) ] dρ ( ) d dr I (r, f) dρ ] (r ρ) δ M (r, f)dρ (I (r, f) I (, f)) ( δ ) I (r, f),

8 574 Saolin Cen, Saminatan Ponnusamy and Xiantao Wang wic yields 1 f H (D) Te roof of te teorem is comleted. A (r, f)(1 r) δ( ) dr < +. Proof of Teorem 4. It is not difficult to see tat A (r, f) wic imlies tat 1 r (1 ρ) δ( ) dρ 1 +δ( ) (1 r) A + δ( ) (r, f) By Teorem 3, we conclude tat wic gives 1 r (1 ρ) δ( ) A (ρ, f) dρ 1 (1 ρ) δ( ) A (ρ, f) dρ < + δ( )+ lim (1 r) A (r, f) = r 1 and te roof of te teorem is comleted. r (1 ρ) δ( ) A (ρ, f) dρ. 4. Landau s teorem Proof of Teorem 5. By assumtion and te inequaltiy (17), we ave f(z) 1 M, z D. (1 z ) 1 Set f(z) = n=1 a nz n + n=1 b nz n, and, for ζ D, let F (ζ) = f(rζ)/r so tat F (ζ) = A n ζ n + B n ζ n, were A n = a n r n 1 and B n = b n r n 1. Ten F () = λ F () 1 = and n=1 F (ζ) 1 M r(1 r) 1 By [, Lemma 1], for n {, 3, }, we ave n=1 = M(r) for ζ D. (19) A n + B n 4M(r). To rove te univalence of F, we coose two distinct oints ζ 1, ζ D ρ1 (r) and let ζ 1 ζ = ζ 1 ζ e iθ, were t ρ 1 (r) = t,

9 On lanar armonic Liscitz and lanar armonic Hardy classes 575 wit t = 4M(r). Ten (19) yields tat F (ζ ) F (ζ 1 ) = F ζ (ζ) dζ + F ζ (ζ) dζ [ζ 1,ζ ] F ζ () dζ + F ζ () dζ [ζ 1,ζ ] (F ζ (ζ) F ζ ()) dζ + (F ζ (ζ) F ζ ()) dζ [ζ 1,ζ ] > ζ 1 ζ [ λ F () Here in te last ste we use te fact tat 1 4M(r) n= ( A n + B n )nρ n 1 1 (r) ] [ ζ 1 ζ 1 4M(r) ρ1(r)( ] ρ 1 (r)) (1 ρ 1 (r)) ρ1(r)( ρ 1 (r)) (1 ρ 1 (r)) =, i.e. ρ 1 (r) = ϕ(r) r. Tus, F (ζ ) F (ζ 1 ). Te univalence of F follows from te arbitrariness of ζ 1 and ζ. Tis imlies tat f is univalent in ρ1 (r). Now, for any ζ = ρ 1 (r)e iθ D ρ1 (r), we easily obtain tat F (ζ ) ρ 1 (r) ( A n + B n )ρ n 1(r) ρ 1 (r) n= [ = ρ 1 (r) 1 4M(r) ] ρ 1 (r) 1 ρ 1 (r) = R r, n=. 4M(r) ρ n 1(r) were te last ste is a consequence of te exression for ρ 1 given by ρ 1 (r) = r 1 ϕ(r). Terefore, f(ρ1 (r)) contains a univalent disk of radius R. Acknowledgements. Te autors would like to tank te referee for is (or er) careful reading of tis aer and useful suggestions. Te researc was artly suorted by NSF of Cina (No ) and Hunan Provincial Innovation Foundation for Postgraduate (No ). Te work was carried out wile te first autor was visiting IIT Madras, under RTFDCS Fellowsi. Tis autor tanks Centre for Cooeration in Science & Tecnology among Develoing Societies (CCSTDS) for its suort and cooeration. Te researc was also artly suorted by te Program for Science and Tecnology Innovative Researc Team in Higer Educational Institutions of Hunan Province. References [1] Cen, H.: Te Scwarz Pick lemma for lanar armonic maings. - Sci. Cina Mat. 54, 11, doi:1.17/s x. [] Cen, S., S. Ponnusamy, and X. Wang: Bloc and Landau s teorems for lanar - armonic maings. - J. Mat. Anal. Al. 373, 11, [3] Clunie, J. G., and T. Seil-Small: Harmonic univalent functions. - Ann. Acad. Sci. Fenn. Ser. A I Mat. 9, 1984, 3 5. [4] Duren, P.: Harmonic maings in te lane. - Cambridge Univ. Press, 4.

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