Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings

Bulletin of the Malaysian Mathematical Sciences Society, Apr 2016

Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by \({\mathcal {S}}_{{\mathcal {H}}}\). The aim of the paper is to study the properties of a subclass of \({\mathcal {S}}_{{\mathcal {H}}}\), such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch’s constant for co-analytic part.

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Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings

Bull. Malays. Math. Sci. Soc. Coefficient Estimates and Bloch's Constant in Some Classes of Harmonic Mappings S. Kanas 0 1 D. Klimek-Sme¸t 0 1 Mathematics Subject Classification 0 1 B S. Kanas 0 1 0 Department of Mathematics, Maria Curie-Sklodowska University in Lublin , Pl. Marii Curie-Skłodowskiej 5, 20-031 Lublin , Poland 1 Faculty of Mathematics and Natural Sciences, University of Rzeszow , ul. S. Pigonia 1, 35-959 Rzeszów , Poland Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by SH. The aim of the paper is to study the properties of a subclass of SH, such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch's constant for co-analytic part. Univalent harmonic mappings; Convex functions; Bloch's constant; Normal family - 30C75 · 30C45 · 30D45 1 Introduction A complex-valued harmonic function f that is harmonic in a simply connected domain ⊂ C has the canonical representation Communicated by V. Ravichandran. f = h + g, where h and g are analytic in with g(z0) = 0 for some prescribed point z0 ∈ . According to a theorem of Lewy [ 17 ], f is locally univalent, if and only if its Jacobian J f (z) = | fz (z)|2 − | fz¯ (z)|2 = |h (z)|2 − |g (z)|2 does not vanish, and is sense-preserving if the Jacobian is positive. Then h (z) = 0 and the analytic function ω = g / h , called the second complex dilatation of f , has the property |ω| < 1 in . Throughout this paper, we will assume that f is locally univalent and sense-preserving, and we call f a harmonic mapping. Also, we assume = D ⊂ C, and z0 = 0, where D is the open unit disk on the complex plane. The class of all sense-preserving univalent harmonic mappings of D with h(0) = g(0) = h (0) − 1 = 0 is denoted by SH, and 0 (cf. [ 8 ]). Fundamental informations about harmonic imtsaspupbincglasssinf othregp(l0a)ne=ca0nbbyeSfHound in [ 11 ]. Note that each f satisfying (1.1) in D is uniquely determined by coefficients of the following power series expansions then ω(z) = c0 + c1z + c2z2 + · · · , |cn| ≤ 1 − |c0|2, ∞ n=0 ∞ n=1 h(z) = an zn, g(z) = bn zn (z ∈ D), with an ∈ C, n = 0, 1, 2, ..., and bn ∈ C, n = 1, 2, 3, .... When f ∈ SH, then a0 = 0, a1 = 1. α of SH, consisting of all In [ 14 ], the authors studied the properties of a subclass SH univalent anti-analytic perturbations of the identity in the unit disk with |b1| = α, and in [ 15 ], the authors studied the class Sα of all f ∈ SH, such that |b1| = α ∈ (0, 1) and h ∈ CV, where CV denotes the well-known family of normalized, univalent functions which are convex. The classical Schwarz–Pick estimate for an analytic function ω which is bounded by one on the unit disk of the complex plane is the inequality |ω (z)| ≤ 1 − |ω(z)|2 1 − |z|2 (|z| < 1). Ruscheweyh [ 21 ] has obtained the best-possible estimates of higher order derivatives of bounded analytic functions on the disk. Similar estimates were derived by other methods and for different classes of analytic functions in one and several variables by Anderson and Rovnyak [ 1 ] (1 − |z|2)n−1 ω(n)(z) n! ≤ 1 − |ω(z)|2 1 − |z|2 (n = 1, 2, ...). The case z = 0 in (1.4) asserts that if (1.1) (1.2) (1.3) (1.4) (1.5) (1.6) for every n ≥ 1. This result is classical and due to Wiener; see [ 2,16 ]. 2 Bounds of the Fekete–Szegö and Other Functionals Theorem 2.1 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then |bn| ≤ α + (1 − α2)(n − 1) 2 (n = 2, 3, ...). Proof Making use of a relation g = ωh and the power series expansions (1.2) and (1.5), we obtain n−1 p=0 nbn = ( p + 1)a p+1cn− p−1 (n = 2, 3, ...). Since h ∈ CV, |ak | ≤ 1 (k = 1, 2, ...). Applying this for (2.2), we have For the case n = 2, the inequality is sharp, with the equality realized by the function f (z) = 1 −z z + 1 −z z − 11 −+ αα log 11+−αzz . We note that for α close to 1, the above bounds are better than that obtained in [ 15 ]. 1 n−1 p=0 |bn| ≤ n ( p + 1)|cn− p−1|. |bn| ≤ α + n1 np−=20( p + 1)(1 − α2) = α + (1 − α2)(n − 1) 2 . |b2| ≤ α + , |b3| ≤ 1 + α − α2. The fact g = ωh , for the case z = 0, implies that c0 = b1, so that by (1.6), we obtain |cn− p−1| ≤ 1 − |b1|2 = 1 − α2. Therefore, Specially, we get (2.1) (2.2) (2.3) In conclusion, we obtain Corollary 2.2 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then |bn| ≤ min α + (1 − α2)(n − 1), α + (n − α2)(n − 1) 2 n (n = 2,3,...). Theorem 2.3 Let f ∈ Sα, f = h + g¯ with the power series (1.2). Then for μ ∈ R 1 − α2 3 |b3 − μb22| ≤ 3 1 + 4|μ|(1 − α2) + |2 − 3μb1| + α max 31,|1 − μb1| , and Then Proof From the relation (2.2), we have |bn+1 − bn| ≤ 2α + 1 − α2 2n2− 1. 2b2 = c1 + 2a2c0, 3b3 = c2 + 2a2c1 + 3a3c0. (2.4) (2.5) (2.6) 1 2 1 |b3 − μb22| = 3c2 + 3a2c1 + a3c0 − μ 2c1 + a2c0 2 Apply now the estimate that holds for the coefficients of convex functions: |an| ≤ 1 (n = 2,3,...),|a3 − νa22| ≤ max{1/3,|1 − ν|} (ν ∈ R), and the relation (1.6). We obtain then |b3 − μb22| ≤ 3 |c2| + 43 μc12 + |a2||c1| 3 − μc0 + α|a3 − μb1a22| 1 2 1 − α2 3 ≤ 3 1 + 4|μ|(1 − α2) + (1 − α2) 23 − μb1 +α max{1/3,|1 − μb1|}. Next, by (2.2), we have |bn+1 − bn| = 1 n + 1 n+1 p=1 1 n pa pcn+1− p − n 1 ≤ |an+1c0| + |anc0| + n + 1 1 − α2 n ≤ 2α + n + 1 p=1 = 2α + (1 − α2) 2n 2− 1 . p + The proof is now complete; however, the results are not sharp, for example, the function that realizes the accuracy of |b2| in the previous theorem gives |b2−b1| = (1−α2)/2 ≤ 2α + (1 − α2)/2, for any α ∈ (0, 1). The right-hand side is obtained from (2.6) for the case when n = 1. Theorem 2.4 For f ∈ Sα, f = h + g¯ and |z| = r < 1, it holds r 1 − α log 1 + r 1 + r − 1 + α 1 − αr 2r r (1 − α2) 1 + r − |r − α|(1 − αr ) ≤ zg (z) Re 1 + > g (z) ≤ |g(z)| ≤ 1 −r r + 11 +− αα log 11+−αrr , (2.7) zg (z) 2r r (1 − α2) , (2.8) g (z) ≤ 1 − r + |r − α|(1 − αr ) r (α2 − 1) 1 − r |α − r |(1 − αr ) + 1 + r . (2.9) Proof Applying the relation g = ωh , we estimate |g (z)| as follows [ 15 ]: |α − r | α + r (1 − αr )(1 + r )2 ≤ |g (z)| ≤ (1 + αr )(1 − r )2 (|z| = r < 1). Then integrating along a radial line ζ = t eiθ , the right-hand side of (2.7) is obtained immediately [ 15 ]. In order to prove the left-hand side of (2.7), we note first that g is univalent. Let = g({z : |z| = r }) and let ξ1 ∈ be the nearest point to the origin. By a rotation we may assume that ξ1 > 0. Let γ be the line segment 0 ≤ ξ ≤ ξ1 and suppose that z1 = g−1(ξ1) and L = g−1(γ ). With ζ as the variable of integration on L, we have that dξ = g (ζ )dζ > 0 on L. Hence ξ1 = dξ = g (ζ )dζ = |g (ζ )||dζ | ≥ |g (t eiθ )|dt z1 0 z1 0 0 r |α − r | (1 − αr )(1 + r )2 dr = Moreover, ω satisfies [12, p. 320] from which it follows From the relation g = ωh , we obtain Since h is convex, so is univalent, then it holds [13, p. 118] zg (z) g (z) = zωω((zz)) + zhh ((zz)) . 2r 1 + r ≤ zh (z) h (z) We note that |ω(0)| = |c0| = |b1| = α, so that by (2.13) we have |1r −−ααr| ≤ |ω(z)| ≤ 1r ++ααr . Taking into account (2.10), (2.11), and (2.14) and the Schwarz–Pick inequality (1.3), we obtain for |z| = r < 1, zg (z) g (z) zω (z) zh (z) ≤ ω(z) + h (z) r (1 − |ω(z)|2) 2r ≤ |ω(z)|(1 − r 2) + 1 − r r (1 − r 2)(1 − α2) 2r ≤ (1 − r 2)|r − α|(1 − αr ) + 1 − r r (1 − α2) 2r . = |r − α|(1 − αr ) + 1 − r Similarly, we have Moreover and h is convex, therefore By the above, (1.3) and (2.14), we have zg (z) Re 1 + g (z) zω (z) zh (z) = Re ω(z) + Re 1 + h (z) > Re zωω((zz)) + 11 −+ rr . zg (z) Re 1 + g (z) > |α −r(αr|2(1−−1)αr ) + 11 −+ rr , as asserted. 3 Estimates of the Bloch’s Constant A harmonic function f is called the Bloch function if where B f = sup z,w∈D,z=w | f (z) − f (w)| < ∞, (z, w) (z, w) = 21 log z−w ⎞ ⎛ 1 + 1−z¯w ⎝ 1 − 1−z¯w z−w ⎠ = artanh z − w 1 − z¯w denotes the hyperbolic distance in D, and B f is called the Bloch’s constant of f . The harmonic Bloch’s constant was studied by Colonna [ 9 ]. Colonna established that the (2.15) (2.16) (2.17) (3.1) Bloch’s constant B f of a harmonic mapping f = h + g¯ can be expressed in terms of moduli of the derivatives of h and g B f = sup(1 − |z|2) |h (z)| + |g (z)| z∈D = sup(1 − |z|2)|h (z)|(1 + |ω(z)|), z∈D which agrees with the well-known notion of the Bloch’s constant for analytic functions. Moreover, the function f is Bloch if and only if both h and g are, and max(Bh , Bg) ≤ B f ≤ Bh + Bg. Colonna also obtained the best-possible estimate of the Bloch’s constant for the family of harmonic mappings of D into itself. Recently, the Bloch’s constant was studied by many authors, see, for example [ 3,4,19 ]. Very interesting results in this direction were obtained in [ 5–7,18,20,22 ]. Our aim is to determine the bounds for the Bloch’s constant in the classes Sα and Sα. Theorem 3.1 Let f = h + g¯ with h(z) = z/(1 − Bz), −1 < B < 1, and let |B| = A, 0 ≤ A < 1. Then the Bloch’s constant B f is bounded by , where r0 is given by r0 = α(1 + 3 A) − 3 − A + √(1 + α)(1 + A)(9 − 7 A + α(−7 + 9 A)) 4α + 2 A(α − 1) . Proof Applying the distortion theorem 1 |h (z)| ≤ (1 − Ar )2 (|z| = r ), and (3.2), we find Setting (3.2) (3.3) (3.4) B f = sup(1 − |z|2)|h (z)|(1 + |ω(z)|) ≤ (1 + α) sup z∈D 0≤r<1 (1 − Ar )2(1 + αr ) . (1 + r )(1 − r 2) (1 + r )(1 − r 2) q(r ) = (1 − Ar )2(1 + αr ) , we observe that q (r ) = 0, if and only if (1 + r )[(2α + α A − A)r 2 + (3 + A − α − 3α A)r + α − 1 − 2 A] = 0. Remark By the fact that the Bloch’s constant is finite, we already have that the family of harmonic mappings with h(z) ≡ z, and |b1| = α, is a normal family. A function f is normal, if the constant σ f is finite, where , r0 = α − 3 + √ . σ f = sup z∈D (1 − |z|2)| f (z)| 1 + | f (z)| , see [ 10 ]. Indeed, since the quantity | f (z)| in Sα is bounded [ 15 ] therefore, by (3.5) and (3.7), we obtain ⎧ (1 + α) (1+r)(1−r2) σ f ≤ ⎨⎪ 1+αr ⎪ (1 + α) (1+r)(1−r2) ⎩ 1+αr The last equation has solution in the interval (0, 1) at the point r0 given by (3.4), and the function q attains its maximum at r0. Setting B = 0 in the above theorem, we obtain the estimate of B f in the class S¯α, below. Corollary 3.2 For f ∈ S¯α, f = h + g¯, the Bloch’s constant B f is bounded by where r = r0 is given by (3.6), and we see that in both cases σ f is finite. Remark The univalent Bloch functions can be described in terms of geometry of their images; they are precisely those functions whose images do not contain disks of arbitrarily large radius [ 10 ]. Therefore, we suppose that the functions from the class Sα may not be the Bloch functions. Indeed, reasoning similarly as in the Theorem 3.1 we note that in the class Sα we have h(z) = z/(1 − z), then |h (z)| ≤ 1/(1 − r )2. 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S. Kanas, D. Klimek-Smȩt. Coefficient Estimates and Bloch’s Constant in Some Classes of Harmonic Mappings, Bulletin of the Malaysian Mathematical Sciences Society, 2016, 741-750, DOI: 10.1007/s40840-015-0138-9