Abstract
We study a family of harmonic univalent functions in the
open unit disc defined by using post quantum calculus operators. We first
obtained a coefficient characterization of these functions. Using this, co-
efficients estimates, distortion and covering theorems were also obtained.
The extreme points of the family and a radius result were also obtained.
The results obtained include several known results as special cases.
1
Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 3
Theorem 3 Let the function f = h + g be such that the functions h and g are given by (7). Also, let the (p, q)-coefficient inequality ∞ X k=2…
Theorem 3 Let the function f = h + g be such that the functions h and g are given by (7). Also, let the (p, q)-coefficient inequality ∞ X k=2 λk[k]m p,q −αuk[k]n p,q 1 −α |ak| + ∞ X k=1 µk[k]m
Theorem 3
Theorem 3 also yields the following corollary.
Theorem 3 also yields the following corollary.
Corollary 4
Corollary 4 For the function fm = h + gm given by (14), we have |ak| ≤ 1 −α λk[k]m p,q −αuk[k]np,q, k ≥2 and |bk| ≤ 1 −α µk[k]m p,q…
Corollary 4 For the function fm = h + gm given by (14), we have |ak| ≤ 1 −α λk[k]m p,q −αuk[k]np,q , k ≥2 and |bk| ≤ 1 −α µk[k]m p,q −(−1)n+j−(m+i)αvk[k]np,q , k ≥1. The result is sharp for each k.
Theorem 5
Theorem 5 Let the function fm = h + gm be given by (14). Then the func- tion fm ∈clco T SH(m, n, Φi, Ψj, p, q, α) if and only if fm(z) = ∞…
Theorem 5 Let the function fm = h + gm be given by (14). Then the func- tion fm ∈clco T SH(m, n, Φi, Ψj, p, q, α) if and only if fm(z) = ∞ P k=1 (xkhk(z) + ykgmk(z)), where h1(z) = z, hk(z) = z − 1 −α λk[k]m p,q −αuk[k]np,q zk, k ≥2, gmk(z) = z + (−1)m+i−1
Theorem 6
Theorem 6 Let the function fm ∈T SH(m, n, Φi, Ψj, p, q, α), γk = λk[k]m p,q − αuk[k]n p,q, k ≥2 and φk = µk[k]m p,q −(−1)n+j−(m+i)αvk[k]n…
Theorem 6 Let the function fm ∈T SH(m, n, Φi, Ψj, p, q, α), γk = λk[k]m p,q − αuk[k]n p,q, k ≥2 and φk = µk[k]m p,q −(−1)n+j−(m+i)αvk[k]n p,q, k ≥1. If {γk} and {φk} are non-decreasing sequences, then we have |fm(z)| ≤(1 + |b1|)|z| + 1 −α β 1 −µ1 −(−1)n+j−(m+i)αv1 β |b1| |z|2
Corollary 7
Corollary 7 Under the hypothesis of Theorem 6, we have
Corollary 7 Under the hypothesis of Theorem 6, we have
Theorem 8
Theorem 8 If the function fm ∈T SH(m, n, Φi, Ψj, p, q, α), then the function fm is convex in the disc |z| ≤min k
Theorem 8 If the function fm ∈T SH(m, n, Φi, Ψj, p, q, α), then the function fm is convex in the disc |z| ≤min k
Related Papers