INTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen INTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS Shaolin Chen Saminathan Ponnusamy and Xiantao Wang Hunan Normal University Department of Mathematics Changsha Hunan 48 P. R. China; Indian Institute of Technology Madras Department of Mathematics Chennai-6 36 India; Hunan Normal University Department of Mathematics Changsha Hunan 48 P. R. China; Abstract. In this paper we investigate some properties of planar harmonic mappings in Hardy spaces. First we discuss the integral means of harmonic mappings and those of their derivatives and as a consequence we solve the open problem of Girela and Peláez in the setting of harmonic mappings. In addition we establish coefficient estimates and a distortion theorem for harmonic mappings in Hardy spaces.. Introduction and preliminaries For each r ( ] we denote by D r the open disk {z C: z < r} and by D the open unit disk D. A complex-valued function f defined on D is called a harmonic mapping in D if and only if it is twice continuously differentiable and f = i.e. the real and imaginary parts are real harmonic in D where represents the usual complex Laplacian operator = 4 z z = x + y. An obvious fact is that every harmonic mapping f defined in D admits the canonical decomposition f = h + g where h and g are analytic in D with g() =. We refer to [5 9] for the theory of harmonic mappings. A classical result of Hardy and Littlewood asserts that if p ( ] α ( ) and f is an analytic function in D then ( ( ) ) α M p (r f ) = O as r if and only if where ( ( ) ) α M p (r f) = O as r ( ˆ /p f(re iθ ) dθ) p if p ( ) M p (r f) = max f(z) if p =. z =r doi:.586/aasfm..377 Mathematics Subject Classification: Primary 3C65 3C45; Secondary 3C. Key words: Harmonic mapping harmonic Hardy space coefficient estimate. Corresponding author.

2 7 Shaolin Chen Saminathan Ponnusamy and Xiantao Wang We refer to [ ] for many other related discussions concerning the case of analytic functions. Indeed the above result of Hardy and Littlewood provides a close relationship between the integral means of analytic functions and those of their derivatives [8 3]. In [ Theorem (a)] Girela and Peláez refined the above result for the case α = as follows. Theorem A. If p ( ) and f is an analytic function in D such that ( ( ) ) M p (r f ) = O as r then for all β > () M p (r f) = O ( ( log ) ) β as r. In [ p. 464 Equation (6)] Girela and Peláez asked whether or not β in () can be substituted by /. This problem was affirmatively settled by Girela Pavlovic and Peláez in []. In this paper we first prove that the answer to this problem is affirmative in the setting of harmonic mappings in the unit disk. Then coefficient estimates and a distortion theorem for harmonic mappings in Hardy spaces are also obtained. In order to state our results we need to introduce some notations. For p ( ] the harmonic Hardy space H p h consists of those functions f harmonic in D such that f p < where sup M p (r f) if p ( ) <r< f p = sup f(z) if p =. In addition we let z D f = (f z f z ) and f = ( f z + f z ) /. We now state our first result which generalizes Theorem A. In the case of analytic function f f equals f (z) and therefore the following theorem contains another solution to the open problem of Girela and Peláez [ p. 464 Equation (6)] provided later by Girela Pavlović and Peláez in []. Theorem. If p ( ) and f is a harmonic mapping in D such that ( ( ) ) () M p (r f) = O as r then we have the following: ( ( ) ) / (a) M p (r f) = O log as r ( ( ) ) /p (b) M (r f) = O as r. The following result is a generalization of [ Theorem ] for the harmonic case.

3 Integral means and coefficient estimates on planar harmonic mappings 7 Theorem. If p ( ] and f is a harmonic mapping in D satisfying the condition () then ( ( ) ) /p M p (r f) = O log as r. Moreover the result is sharp and one of the extreme functions is f(z) = /( z) /p where p ( ]. We now state our next two theorems which provide coefficient estimates and a distortion theorem for harmonic Hardy mappings. Theorem 3. Let f be a harmonic mapping in D such that f(z) = a n z n + b n z n n= n= and f H p h for some p [ ]. Then we have the following: (a) a f p ; (b) for p [ ) (c) for p = a n + b n (/p)+ ( + np) n+(/p) (pn) n f p for each n ; a n + b n 4 f for each n. The estimate in this case is sharp and the only extremal functions are f n (z) = α ( ) + βz n f arg βz n where α = β =. Theorem 4. Let f be a harmonic mapping in D with f H p h for some p [ ]. Then for p [ ) and for p = f z (z) + f z (z) (/p)+ ( + p) +(/p) f p( z p ) (3) f z (z) + f z (z) 4 ( z ) f. When p = the estimate (3) is sharp and the only extremal functions are f(z) = α ( ) + φ(z) f arg φ(z) where α = and φ is a conformal automorphism of D. We remark that Theorem 4 is a generalization of [6 Theorems 3 and 4]. The proofs of Theorems 4 are presented in the following section.

4 7 Shaolin Chen Saminathan Ponnusamy and Xiantao Wang. Proofs It seems that the standard technologies from the theory of analytic functions are not useful to prove Theorem and therefore we use some ideas from Green s theorem in their proofs. Green s theorem (cf. [3 6]) states that if g C (D) i.e. twice continuously differentiable on D then (4) g(re iθ ) dθ = g() + ˆ D r g(z) log r z da(z) for r ( ) where da(z) denotes the normalized area measure in D. Moreover the proof of Theorem (b) relies on the following lemma. Lemma B. [8 Lemma 3 p. 84] If a ( ) and ρ = ( + r)/ then Proof of Theorem. (a) Let ρe it r a dt = O ( () a) as r. A(r f) = Then Hölder s inequality yields f(re iθ ) p f(re iθ ) dθ. (5) A(r f) Mp (r f) Mp p (r f). By the identity (4) the inequality (5) the subharmonicity of f and the fact M p (t f) being an increasing function of t we have Mp p (r f) = f() p + ˆ ( f(z) p ) log r D r z da(z) ˆ [ = f() p + p (p/ ) f(z) p 4 f z (z)f(z) + f(z)f z (z) D r ] + f(z) p f(z) log r z da(z) ˆ f() p + p(p ) f(z) p f(z) log r D r z da(z) = f() p + p(p ) f() p + p(p ) which in particular implies that M p (r f) f() + p(p ) f() + p(p ) ˆ r ˆ r ˆ r ˆ r f(te iθ ) p f(te iθ ) t log r dt dθ t Mp (t f)mp p (t f)t log r t dt M p (t f)t log r t dt M p (t f)(r t) dt ˆ = f() + p(p )r Mp (rt f)( t) dt f() + C log

5 and hence or equivalently Integral means and coefficient estimates on planar harmonic mappings 73 M p (r f) = O ( ( log M p (r f) = O ( ( log ) ) as r ) ) / as r where C is a positive constant. The proof of Part (a) in Theorem is complete. (b) Let z D ρ be arbitrary with ρ ( ). Then by the Poisson integral formula and Hölder s inequality there is a positive constant C such that f(z) ρ z z ρe iθ f(ρeiθ ) dθ ( ˆ dθ CM p (τ f) z ρe iθ p p ) p p. ρ + z z ρe iθ f(ρeiθ ) dθ Now we set ρ = ( + z )/ and apply Lemma B. Obviously there exists a positive constant C such that ( ) + z [ ] (6) f(z) CM p f ( z ) p p p p = CM p( + z f) ( z ). /p By computations we have (7) f(ρe iθ ) f() + df(te iθ ) dt dt f() + By the well-known Minkowski inequality and (7) we have ( M p (ρ f) = { ˆ f() + C [ f() + ) /p f(ρe iθ ) p dθ p } /p f(te iθ ) dt] dθ ( ˆ /p f(te iθ ) dθ) p dt = f() + C where C is a positive constant. In particular M p (ρ f) f() + C M p (t f) dt. f(te iθ ) dt. By (6) (with p in place of p) and letting ρ = ( + r)/ we get [ ˆ ] C (+r)/ (8) M (r f) f() + C () /(p) M p (t f) dt where C and C are positive constants. M p (t f) dt

6 74 Shaolin Chen Saminathan Ponnusamy and Xiantao Wang (9) On the other hand M p (t f) = = ( ) /(p) f(te iθ ) p dθ ( ˆ f(te iθ ) p f(te iθ ) p dθ ) /(p) M / (t f) Mp / (t f). For α [ ] and z D let F α (z) = f z (z) + e iα f z (z). Then for each z D r F α (z) r z ˆ z re iθ F α(re iθ r z ) dθ z re iθ f(reiθ ) dθ which gives f(z) max α [] F α(z) A procedure similar to the proof of (6) shows that () f(z) CM p( + z f) ( z ) /p for some constant C > and (9) implies () M p (s f) CM p( +s f) where s [ +r ( s) p r z z re iθ f(reiθ ) dθ. C ( s) + p ]. Therefore by (8) () and () we get ( ( ˆ +r M (r f) O () p and the proof of Theorem (a) follows. ) ) ( ( ( s) p ) ) /p ds = O We now recall the following well-known result which we need to prove Lemma (and hence Theorem ). then Lemma C. [7 Lemma.9] Let a b [ ) and p [ ). Then we have a p + b p (a + b) p p (a p + b p ). Lemma. Let f be harmonic in D and p ( ]. If ˆ () f p p C(p) () p M p p (r f) dr < ( f() p + ˆ ) () p Mp p (r f) dr where C(p) is a positive constant which depends only on p.

7 Integral means and coefficient estimates on planar harmonic mappings 75 Proof. If p [ ] then the proof of the inequality () follows from Lemma C [ Theorem B] and the fact that f has the canonical decomposition f = h + g where h and g are analytic in D. Therefore it suffices to prove the theorem for the case p ( ). Let r < r <. Then we see that ˆ r f(r e iθ ) f(r e iθ ) = d dt f(teiθ ) dt ˆ r f(te iθ ) dt and therefore which gives (3) r f(r e iθ ) f(r e iθ ) p p/ (r r ) p max f(te iθ ) p r t r Mp p (r f) Mp p (r f) f(r e iθ ) f(r e iθ ) p dθ p/ (r r ) p max f(te iθ ) p. r t r For α [ ] and z D let K α (z) = f z (z) + e iα f z (z). By using Hardy Littlewood maximal theorem to K α there is a positive constant C such that max r t r K α (te iθ ) p C The arbitrariness of α in [ ] implies that max f(te iθ ) p p/ C r t r K α (r e iθ ) p dθ p/ C r f(r e iθ ) p dθ. f(r e iθ ) p dθ. Without loss of generality we assume f() = so that M p ( f) =. n {...} let r n = n. Applying (3) we deduce that n+ [ Mp p (r n+ f) = M p p (r k f) Mp p (r k f) ] n+ C (r k r k ) p Mp p (r k f) C k= n ˆ rk+ k(p ) Mp p (r f) dr C k= r k k= ˆ rn+ For any () p M p p (r f) dr where the positive constant C is not necessarily the same at each occurrence in the above three inequalities. Then the inequality () follows if we let n. Proof of Theorem. Without loss of generality we assume that f() =. For r ( ] let f r (z) = f(rz) z D. By using () and Lemma there is a positive constant C such that ˆ Mp p (r f) C ( t) p Mp p (rt f) dt (ˆ r ( t) p ˆ ) C (t) dt + ( t) p p r (t) dt p (ˆ r ˆ C t dt + ) ( ( ( t) p ) ) dt = O log. () p The proof of this theorem is complete. r

8 76 Shaolin Chen Saminathan Ponnusamy and Xiantao Wang We can now prove our final two results concerning the coefficient estimates and a distortion theorem for harmonic Hardy mappings. Let s recall the following result which is referred to as Jensen s inequality (cf. [9]). Lemma D. Let (Ω A µ) be a measure space such that µ(ω) =. If g is a realvalued function that is µ-integrable and if ϕ is a convex function on the real line then (ˆ ) ˆ ϕ g dµ ϕ g dµ. Ω Ω Proof of Theorem 3. For s < by the Poisson integral formula we have f(z) = s z z se iθ f(seiθ ) dθ for z D s. Using Jensen s inequality (see Lemma D) we have f(z) p This in particular gives f(z) < s z z se iθ f(seiθ ) p dθ s s z M p p (s f). /p ( z ) /p f p for z D. For ζ D and a fixed r ( ) let F (ζ) = f(rζ)/r = H(ζ) + G(ζ) so that H(ζ) = a r + A n ζ n and (ζ) = B n ζ n n= where A n = a n r n and B n = b n r n. It is not difficult to see that ( ˆ /p a = f() dθ) p f p and /p F (ζ) < r() f /p p = M(r) for ζ D. In particular the proof of Theorem 3(a) follows. For the proof of the remaining two cases we let n= T (ζ) = F (ζ) M(r) = H (ζ) + G (ζ) where H(ζ) = H (ζ)m(r) and G(ζ) = G (ζ)m(r). Then for any ζ D T (ζ) <. As in [] (see also []) it follows easily that (4) A n + B n 4M(r) for n =.... For the sake of completeness we include the details from []. For θ [ ) let and observe that v θ (ζ) = Im (e iθ T (ζ)) v θ (ζ) = Im (e iθ H (ζ) + e iθ G (ζ)) = Im (e iθ H (ζ) e iθ G (ζ)).

9 Integral means and coefficient estimates on planar harmonic mappings 77 Because v θ (ζ) < it follows that (5) e iθ H (ζ) e iθ G (ζ) K(ζ) = λ + log ( + ζξ ζ where ξ = e iim(λ) λ = e iθ H () e iθ G () and denotes the subordination [9 p. 7]. The superordinate function K(ζ) maps D onto a convex domain with K() = λ and K () = ( + ξ) and therefore by a theorem of Rogosinski [8 Theorem.3] it follows that A n e iθ B n M(r) + ξ 4M(r) for n =.... By the arbitrariness of θ in [ ) we have (4) whence we deduce that [ ] a n + b n +(/p) f p inf <r< r n () /p = +(/p) f p ) [ [ ] = +(/p) f p ( + pn) n+(/p) max r n () /p (pn) n <r< Thus the proof of Theorem 3(b) follows. Finally by a simple calculation it can be easily seen that /p ( + pn) n+(/p) lim = p (pn) n and thus the proof of (c) follows from (b) as a limiting case. p we conclude that ]. Indeed by letting (6) a n + b n 4 f for n =... for f Hh. The estimates of (6) are sharp. By subordination the equality sign occurs in (6) if and only if f n (z) = α f Im (log + ) βzn α = β = βz n and the images of D under f n are confined to a diametral segment of the disk D f = {z : z < f }. The proof of this theorem is complete. Proof of Theorem 4. For a fixed z D let F (w) = f(φ(w)) where φ(w) = z+w +zw is a conformal automorphism of D. By calculations we have where ζ = φ(w). This gives By Theorem 3 we have F w (w) = f ζ (φ(w))φ (w) and F w (w) = f ζ (φ(w))φ (w) F w (w) + F w (w) = ( f ζ (φ(w)) + f ζ (φ(w)) ) φ (w). F w () + F w () = ( f ζ (z) + f ζ (z) )( z ) (/p)+ ( + p) +(/p) f p. p Then f ζ (z) + f ζ (z) (/p)+ ( + p) +(/p) f p. p( z )

10 78 Shaolin Chen Saminathan Ponnusamy and Xiantao Wang If we allow p then the last inequality implies that 4 (7) f ζ (z) + f ζ (z) ( z ) f The estimates of (7) are sharp. The only extremal functions are f(z) = α f ( ) + φ(z) arg φ(z) where α = and φ is a conformal automorphism of D. Acknowledgements. The authors would like to thank the referee for his/her careful reading of this paper and many useful suggestions. The research was partly supported by NSF of China (No. 763) and Hunan Provincial Innovation Foundation for Postgraduate (No. 5-43). The work was carried out while the first author was visiting IIT Madras under RTFDCS Fellowship. This author thanks Centre for Cooperation in Science & Technology among Developing Societies (CCSTDS) for its support and cooperation. The research was also partly supported by the Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province. References [] Chen Sh. S. Ponnusamy and X. Wang: Bloch constant and Landau s theorems for planar p-harmonic mappings. - J. Math. Anal. Appl [] Chen Sh. S. Ponnusamy and X. Wang: Coefficient estimates and Landau Bloch s theorem for planar harmonic mappings. - Bull. Malays. Math. Sci. Soc. () [3] Chen Sh. S. Ponnusamy and X. Wang: On planar harmonic Lipschitz and planar harmonic Hardy classes. - Ann. Acad. Sci. Fenn. Math [4] Clunie J. G. and T. H. MacGregor: Radial growth of the derivative of univalent functions. - Comment. Math. Helv [5] Clunie J. G. and T. Sheil-Small: Harmonic univalent functions. - Ann. Acad. Sci. Fenn. Ser. A I Math [6] Colonna F.: The Bloch constant of bounded harmonic mappings. - Indiana Univ. Math. J [7] Duren P. L.: Univalent functions. - Grundlehren Math. Wiss. 59 New York Berlin Heidelberg Tokyo Spring-Verlag 983. [8] Duren P.: Theory of H p spaces. - Dover Mineola N.Y. nd edition. [9] Duren P.: Harmonic mappings in the plane. - Cambridge Univ. Press 4. [] Girela D. M. Pavlović and J. A. Peláez: Spaces of analytic functions of Hardy Bloch type. - J. Anal. Math [] Girela D. and J. A. Peláez: Integral means of analytic functions. - Ann. Acad. Sci. Fenn. Math [] Hardy G. H. and J. E. Littlewood: Some properties of conjugate functions. - J. Reine Angew. Math [3] Hardy G. H. and J. E. Littlewood: Some properties of fractional integrals II. - Math. Z [4] Holland F. and J. B. Twomey: On Hardy classes and area function. - J. London Math. Soc

11 Integral means and coefficient estimates on planar harmonic mappings 79 [5] Noonan J. W. and D. K. Thomas: The integral means of regular functions. - J. London Math. Soc [6] Pavlović M.: Green s formula and the Hardy Stein indentities. - Filomat [7] Ponnusamy S.: Foundations of functional analysis. - Alpha Science International Publishers UK ; Narosa Publishing House India. [8] Rogosinski W.: On the coefficients of subordinate functions. - Proc. London Math. Soc. 48: [9] Rudin W.: Real and complex analysis. - McGraw-Hill 987. Received 5 November