Abstract
We show that the convolution of the harmonic function $f=h+\bar{g}$, where $h(z)+{e}^{-2{i}γ}g(z)=z/(1-{e}^{{i}γ}z)$ having analytic dilatation ${e}^{{i}θ} z^n (0\leqθ<2π)$, with the mapping $f_{a,α}=h_{a,α}+\overline{g}_{a,α}$, where $h_{a,α}(z)=(z/(1+a)-{e}^{{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$, $g_{a,α}(z)=(a {e}^{2{i}α}z/(1+a)-{e}^{3{i}α}z^2/2)/(1-{e}^{{i}α}z)^2$ is convex in the direction $-(α+γ)$. We also show that the convolution of $f_{a,α}$ with the right half-plane mapping having dilatation
Results & Lemmas (21)
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Lemma 1.1
Lemma 1.1: [1, Theorem 2, p.491] Let fk ∈S0(Hγk), (k = 1, 2). If f1 ∗f2 is locally univalent and sense-preserving in D, then f1 ∗f2 ∈S0 H…
Lemma 1.1: [1, Theorem 2, p.491] Let fk ∈S0(Hγk), (k = 1, 2). If f1 ∗f2 is locally univalent and sense-preserving in D, then f1 ∗f2 ∈S0 H and is convex in the direction −(γ1 + γ2).
Lemma 1.2
Lemma 1.2: [9, Theorem 7, p.268] Let f1 = h1+¯g1 is a right half-plane mapping given by h1 + g1 = z/(1 −z), and for π/2 ≤α < π, let f2 = h2…
Lemma 1.2: [9, Theorem 7, p.268] Let f1 = h1+¯g1 is a right half-plane mapping given by h1 + g1 = z/(1 −z), and for π/2 ≤α < π, let f2 = h2 + ¯g2 be a strip mapping given by h2(z) + g2(z) = 1 2i sin α log 1+eiαz 1−eiαz . If f1 ∗f2 is locally univalent and sense-preserving, then f1 ∗f2 ∈S0 H and is convex in the direction of the real axis.
Lemma 1.3
Lemma 1.3: [2, Theorem 1.1] Let f ∈S0(Hγ) with dilatation ω(z) = eiθzn (n ∈N, θ ∈R) and f0,0 = h0,0 + ¯g0,0, where h0,0, g0,0 are given by…
Lemma 1.3: [2, Theorem 1.1] Let f ∈S0(Hγ) with dilatation ω(z) = eiθzn (n ∈N, θ ∈R) and f0,0 = h0,0 + ¯g0,0, where h0,0, g0,0 are given by (2). If n = 1, 2, then f0,0 ∗f ∈S0 H and is convex in the direction −γ.
Lemma 1.4
Lemma 1.4: [3, Theorem 2.2] Let fa,0 = ha,0 + ¯ga,0, where ha,α, ga,α are given by (2). If f = h + ¯g is a right half-plane mapping given…
Lemma 1.4: [3, Theorem 2.2] Let fa,0 = ha,0 + ¯ga,0, where ha,α, ga,α are given by (2). If f = h + ¯g is a right half-plane mapping given by h + g = z/(1 −z) with dilatation ω(z) = eiθzn (n ∈N, θ ∈R), then fa,0 ∗f ∈S0 H and is CHD for a ∈[(n −2)/(n + 2), 1). In this paper, we generalize the result in Lemma 1.4 by showing that the con- volution fa,α ∗f is convex in the direction −(γ + α) of the mappings fa,α as given by (2) with f = h + ¯g, satisfying h(z) + e−2iγg(z) = z/(1 −eiγz) with the dilat
Lemma 2.1
Lemma 2.1: Let the function fa,0 = ha,0 + ga,0 be harmonic mapping, where ha,0, ga,0 are given by (2). If ω is the dilatation of slanted…
Lemma 2.1: Let the function fa,0 = ha,0 + ga,0 be harmonic mapping, where ha,0, ga,0 are given by (2). If ω is the dilatation of slanted right half-plane mapping fγ = hγ + gγ ∈S(Hγ), then the dilatation ˜ω of fa,0 ∗fγ is given by ˜ω(z) = 2ω(z)(1 + e−2iγω(z))(a −eiγz) + zω′(z)(a −1)(1 −eiγz) 2(1 −aeiγz)(1 + e−2iγω(z)) + e−2iγzω′(z)(a −1)(1 −eiγz). (4)
Theorem 2.2
Theorem 2.2: Let the function fa,α = ha,α + ga,α be harmonic mapping, where ha,α, ga,α are given by (2). If ω = eiθzn(θ ∈R, n ∈N) is the…
Theorem 2.2 : Let the function fa,α = ha,α + ga,α be harmonic mapping, where ha,α, ga,α are given by (2). If ω = eiθzn(θ ∈R, n ∈N) is the dilatation of slanted right half-plane mapping fγ = hγ + gγ ∈S(Hγ), then the function fa,α ∗fγ ∈S0 H and is convex in the direction −(α + γ) for a ∈[(n −2)/(n + 2), 1).
Lemma 2.3
Lemma 2.3: (Cohn’s rule)[4]. Given a polynomial t(z) = a0 + a1z +... + anzn of degree n, let t∗(z) = znt(1/¯z) = ¯an + ¯an−1¯z +...¯a0¯zn.…
Lemma 2.3: (Cohn’s rule)[4]. Given a polynomial t(z) = a0 + a1z + ... + anzn of degree n, let t∗(z) = znt(1/¯z) = ¯an + ¯an−1¯z + ...¯a0¯zn. Denote by r and s, the number of zeros of t(z) inside and on the unit circle |z| = 1 respectively. If |a0| < |an|, then t1(z) = ¯ant(n) −a0t∗(z) z is of degree n −1 and has r1 = r −1 and s1 = s number of zeros inside and on the unit circle |z| = 1 respectively.
Theorem 2.4
Theorem 2.4: Let the function f = h + g be the harmonic right hal-plane mapping with h(z) + g(z) = z/(1 −z), and the dilatation ω(z) = (a…
Theorem 2.4 : Let the function f = h + g be the harmonic right hal-plane mapping with h(z) + g(z) = z/(1 −z), and the dilatation ω(z) = (a −z2)/(1 − az2), a ∈[0, 1). If the function fa,α = ha,α + ¯ga,α is harmonic right half-plane mapping, where ha,α, ¯ga,α are given by (2), then the function fa,α ∗f ∈S0 H and is convex in the direction of real-axis.
Theorem 2.5
Theorem 2.5: Let the function f = h + g be the harmonic right half-plane mapping with h(z) + g(z) = z/(1 −z), and the dilatation ω(z) = −(a…
Theorem 2.5 : Let the function f = h + g be the harmonic right half-plane mapping with h(z) + g(z) = z/(1 −z), and the dilatation ω(z) = −(a −z)2/(1 − az)2, a ∈[0, 1). If the function fa,α = ha,α + ¯ga,α is harmonic right half-plane
Theorem 2.6
Theorem 2.6: Let the function f = h + g be harmonic mapping given by h(z) + g(z) = 1 2i log 1 + iz 1 −iz , with the dilatation ω(z) = (a…
Theorem 2.6 : Let the function f = h + g be harmonic mapping given by h(z) + g(z) = 1 2i log 1 + iz 1 −iz , with the dilatation ω(z) = (a −z2)/(1 −az2), a ∈(−1, 1). If the function f0,0 = h0,0 + ¯g0,0 is harmonic right-half plane mapping, where h0,0, g0,0 are given by (2), then the function f ∗f0,0 ∈S0 H and is convex in the direction of real axis.
Lemma 3.1
Lemma 3.1: [5, Theorem 1, p.304] Suppose f is analytic and non-constant map- ping in D, then Re (1 −z2)f ′(z) > 0, z ∈D if and only if…
Lemma 3.1: [5, Theorem 1, p.304] Suppose f is analytic and non-constant map- ping in D, then Re (1 −z2)f ′(z) > 0, z ∈D if and only if (1) f is univalent in D (2) f is convex in the direction of imaginary axis, and (3) there exists sequences z′ n and z′′ n converging to z = 1 and z = −1, respectively, such that lim n→∞Re(f(z′
Lemma 3.2
Lemma 3.2: [6] A locally univalent harmonic mapping f = h + g on D is uni- valent mapping of D onto a domain convex in the direction of φ…
Lemma 3.2: [6] A locally univalent harmonic mapping f = h + g on D is uni- valent mapping of D onto a domain convex in the direction of φ if and only if h−e2iφg is univalent analytic mapping of D onto a domain convex in the direction of φ. Wang et al. gave a sufficient condition of univalency for the convex combination f3 = tf1 + (1 −t)f2, 0 ≤t ≤1 of two harmonic univalent functions f1 and f2. Indeed, they have proved the following:
Theorem 3.3
Theorem 3.3: [7, Theorem 3, p.455] If the function fi = hi + gi ∈SH satisfies hi(z)+gi(z) = z/(1−z) for i = 1, 2, then the convex…
Theorem 3.3 : [7, Theorem 3, p.455] If the function fi = hi + gi ∈SH satisfies hi(z)+gi(z) = z/(1−z) for i = 1, 2, then the convex combination f3 = tf1+(1−t)f2, 0 ≤t ≤1, is univalent and convex in the direction of real axis. Kumar et al. [8] introduce a locally univalent and sense-preserving harmonic functions fα = hα + gα given by hα(z) + gα(z) = z(1 −αz)/(1 −z2), α ∈[−1, 1], with the dilatation ω = g′ α/h′ α ∈D, and proved the following:
Theorem 3.4
Theorem 3.4: [8, Theorem 2.7] For i = 1, 2, let the function fαi = hαi + gαi be normalized harmonic mapping satisfying hαi(z) + gαi(z) =…
Theorem 3.4 : [8, Theorem 2.7] For i = 1, 2, let the function fαi = hαi + gαi be normalized harmonic mapping satisfying hαi(z) + gαi(z) = z(1 −αiz)/(1 −z2),
Theorem 3.5
Theorem 3.5: [8, Theorem 2.9] For i = 1, 2, let the function fαi = hαi + gαi be normalized harmonic mapping satisfying hαi(z) + gαi(z) =…
Theorem 3.5 : [8, Theorem 2.9] For i = 1, 2, let the function fαi = hαi + gαi be normalized harmonic mapping satisfying hαi(z) + gαi(z) = z(1 −αiz)/(1 −z2), αi ∈[−1, 1]. Let ω1(z) = −z and ω2 be the dilatations respectively of mappings fα1 and fα2, with |ω2(z)| < 1. Let f = tfα1 + (1 −t)fα2, 0 ≤t ≤1, be convex combination of fα1 and fα2. Then, we have (1) If ω2(z) = −z2 and α1 > α2, then f is in SH and is convex in the direction of imaginary axis. (2) If ω2(z) = z2 and |α1| > |α2|, and α1α2 ≥0,
Theorem 3.6
Theorem 3.6: For i = 1, 2, let the function fαi,n = hαi,n + gαi,n be normalized harmonic mapping, satisfying hαi,n(z) + gαi,n(z) = z(1 +…
Theorem 3.6 : For i = 1, 2, let the function fαi,n = hαi,n + gαi,n be normalized harmonic mapping, satisfying hαi,n(z) + gαi,n(z) = z(1 + z2)(1 + z4) . . . (1 + z2n + αiz2n−1)/(1 + z2n+1) ∗log 1/(1 −z), αi ∈[−2( √ 2 −1), 2( √ 2 −1)], n ∈N and |g′ αi,n/h′ αi,n| < 1 in D. Then the convex combination f = tfα1,n + (1 −t)fα2,m, 0 ≤ t ≤1 is in SH and is convex in the direction of of imaginary axis, provided f is locally univalent and sense-preserving.
Lemma 3.7
Lemma 3.7: For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be the normalized harmonic mapping, such that hαi,n(z) + gαi,n(z)…
Lemma 3.7: For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be the normalized harmonic mapping, such that hαi,n(z) + gαi,n(z) = z(1 + z2)(1 + z4) . . . (1+ z2n + αiz2n−1)/(1+ z2n+1) ∗log 1/(1 −z), αi ∈[−2( √ 2−1), 2( √ 2 −1)] and ωi = g′ αi,n/h′ αi,n, with |ωi(z)| < 1 in D. Then for n ≥m, the dilatation ˜ω of the convex combination f = tfα1,n + (1 −t)fα2,m, 0 ≤t ≤1 is given by
Theorem 3.8
Theorem 3.8: For i = 1, 2 and n ∈N, let the function fi,n = hi,n + gi,n be the normalized harmonic mapping such that hi,n(z) + gi,n(z) =…
Theorem 3.8 : For i = 1, 2 and n ∈N, let the function fi,n = hi,n + gi,n be the normalized harmonic mapping such that hi,n(z) + gi,n(z) = z(1 + z2)(1 + z4) . . . (1+z2n +αz2n−1)/(1+z2n+1) ∗log 1/(1 −z), α ∈[−2( √ 2−1), 2( √ 2−1)] and having dilatation ωi = g′ i,n/h′ i,n. If |ωi(z)| < 1 in D, then the convex combination f = tf1,n + (1 −t)f2,n, 0 ≤t ≤1 belongs to SH and is convex in the direction of imaginary axis.
Theorem 3.9
Theorem 3.9: For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be normalized harmonic mapping satisfying hαi,n(z) + gαi,n(z) =…
Theorem 3.9 : For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be normalized harmonic mapping satisfying hαi,n(z) + gαi,n(z) = z(1 + z2)(1 + z4) . . . (1+z2n +αiz2n−1)/(1+z2n+1) ∗log 1/(1 −z), αi ∈[−2( √ 2−1), 2( √ 2−1)]. If ω1(z) = −z2n−1 and ω2(z) = z2n−1 are dilatations respectively of the functions fα1,n and fα2,n, then the convex combination f = tfα1,n + (1 −t)fα2,n, 0 ≤t ≤1 belongs to SH and is convex in the direction of imaginary axis provided α1 ≥α2.
Theorem 3.10
Theorem 3.10: For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be the normalized harmonic mapping satisfying hα,n(z) + gα,n(z)…
Theorem 3.10 : For i = 1, 2 and n ∈N, let the function fαi,n = hαi,n + gαi,n be the normalized harmonic mapping satisfying hα,n(z) + gα,n(z) = z(1 + z2)(1 + z4) . . . (1 + z2n + αz2n−1)/(1 + z2n+1) ∗log 1/(1 −z), αi ∈[−2( √ 2 −1), 2( √ 2 −1)]. If ω1 = −z2n−1 and ω2(z) (|ω2| < 1) are dilatations respectively of fα1,n and fα2,n, then for the convex combination f = tfα1,n + (1 −t)fα2,n, 0 ≤t ≤1, we have (1) If ω2(z) = −z2n and α1 > α2, then the function f belongs to SH and is convex in the direc
Theorem 3.11
Theorem 3.11: For i = 1, 2 and n ∈N − 1, let the function fαi,n = hαi,n + gαi,n, be the normalized harmonic mapping satisfying hα,n(z) +…
Theorem 3.11 : For i = 1, 2 and n ∈N −{1} , let the function fαi,n = hαi,n + gαi,n, be the normalized harmonic mapping satisfying hα,n(z) + gα,n(z) = z(1 + z2)(1+z4) . . . (1+z2n+αz2n−1)/(1+z2n+1) ∗log 1/(1 −z), αi ∈[−2( √ 2−1), 2( √ 2− 1)]. If ω1(z) = −z2n−2 and ω2(z) = z2n−1 are dilatations respectively of functions fα1,n and fα2,n, then the convex combination f = tfα1,n + (1 −t)fα2,n, 0 ≤t ≤1 belongs to SH and is convex in the direction of imaginary axis, provided α1 ≤α2.