Abstract
In this paper, a class of meromorphic harmonic functions concave in the unit
disc is introduced. Coefficient bounds, distortion inequalities, extreme points, geometric
convolution, integral convolution for the functions belonging to this class are obtained.
Results & Lemmas (7)
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Theorem 2.1.
Theorem 2.1. Let f = h + g be of the form (2). If ∞ ∑ n=1 n2 (|an|+|bn|) ≤1, (4) then, f is harmonic univalent, sense preserving in U 0.
Theorem 2.1. Let f = h + g be of the form (2). If ∞ ∑ n=1 n2 (|an|+|bn|) ≤1, (4) then, f is harmonic univalent, sense preserving in U\{0}.
Theorem 2.2.
Theorem 2.2. Let f = h +g be of the form (2). Then f ∈MHC0 if and only if the inequality (4) holds for the coefficient f = h + g.
Theorem 2.2. Let f = h +g be of the form (2). Then f ∈MHC0 if and only if the inequality (4) holds for the coefficient f = h + g.
Theorem 3.1.
Theorem 3.1. If fk = hk + gk ∈MHC0 and 0 < |z| = r < 1, then |fk(z)| ≤1+r 2 r and |fk(z)| ≥1−r 2 r.
Theorem 3.1. If fk = hk + gk ∈MHC0 and 0 < |z| = r < 1, then |fk(z)| ≤1+r 2 r and |fk(z)| ≥1−r 2 r .
Theorem 3.2.
Theorem 3.2. Let fk = hk + gk where hk and gk are given by (2). Set hk,0 = gk,0 = 1 z hk,n(z) = 1 z + 1 n2 zn, for n = 1,2,3,... and gk,n =…
Theorem 3.2. Let fk = hk + gk where hk and gk are given by (2). Set hk,0 = gk,0 = 1 z hk,n(z) = 1 z + 1 n2 zn, for n = 1,2,3,... and gk,n = 1 z + 1 n2 zn for n = 1,2,3,.... Then, fk ∈MHC0 if and only if fk can be expressed as fk,n = ∞ ∑
Theorem 4.1.
Theorem 4.1. Let fk ∈MHC0 and Fk ∈MHC0. Then the convolution fk ⋆Fk ∈MHC0.
Theorem 4.1. Let fk ∈MHC0 and Fk ∈MHC0. Then the convolution fk ⋆Fk ∈MHC0.
Theorem 4.2.
Theorem 4.2. If fk and Fk of the form (5) and (6) belong to the class MHC0, then the geomet- ric convolution fk ∗Fk also belongs to the…
Theorem 4.2. If fk and Fk of the form (5) and (6) belong to the class MHC0, then the geomet- ric convolution fk ∗Fk also belongs to the class MHC0.
Theorem 4.3.
Theorem 4.3. If fk and Fk of the form (5) and (6) belong to the class MHC0, then the integral convolution fk ⋄Fk also belongs to the class…
Theorem 4.3. If fk and Fk of the form (5) and (6) belong to the class MHC0, then the integral convolution fk ⋄Fk also belongs to the class MHC0.
Function classes studied:
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