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Abstract
In this paper, we establish certain new subclasses of meromorphic harmonic
functions using the principles of q-derivative operator. We obtain new criteria of
sense preserving and univalency. We also address other important aspects, such as
distortion limits, preservation of convolution, and convexity limitations. Additionally,
with the help of sufficiency criteria, we estimate sharp bounds of the real parts of the
ratios of meromorphic harmonic functions to their sequences of partial sums.
Results & Lemmas (14)
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Theorem 2.1
Theorem 2.1 If f ∈H is described by the series of the form (1.2) and if ∞ n=1 ρn|an| + σn|bn| ≤L – M, (2.1) then f ∈MS∗ H(q,L,M) with…
Theorem 2.1 If f ∈H is described by the series of the form (1.2) and if ∞ n=1 ρn|an| + σn|bn| ≤L – M, (2.1) then f ∈MS∗ H(q,L,M) with ρn =
Corollary 2.2
Corollary 2.2 Let f ∈H be of the form (1.2). If ∞ n=1 ρn|an| + σn|bn| ≤(1 + q)
Corollary 2.2 Let f ∈H be of the form (1.2). If ∞ n=1 ρn|an| + σn|bn| ≤(1 + q)
Corollary 2.3
Corollary 2.3 Let f ∈H be given in (1.2). If ∞ n=1 n |an| + |bn| ≤1, then f ∈MS∗ H(1,1,–1).
Corollary 2.3 Let f ∈H be given in (1.2). If ∞ n=1 n |an| + |bn| ≤1, then f ∈MS∗ H(1,1,–1).
Theorem 2.4
Theorem 2.4 Let f ∈ϑ0 have expansion (2.6). Then f ∈MS∗ Hϑ (q,L,M) if and only if (2.1) is true.
Theorem 2.4 Let f ∈ϑ0 have expansion (2.6). Then f ∈MS∗ Hϑ (q,L,M) if and only if (2.1) is true.
Corollary 2.6
Corollary 2.6 Let f ∈H be written in the form of Taylor expansion (1.2). If ∞ n=1 [n]q ρn|an| + σn|bn| ≤(L – M), (2.9) then f ∈MSc…
Corollary 2.6 Let f ∈H be written in the form of Taylor expansion (1.2). If ∞ n=1 [n]q ρn|an| + σn|bn| ≤(L – M), (2.9) then f ∈MSc H(q,L,M).
Corollary 2.7
Corollary 2.7 Let f ∈ϑ1 be written in the series form (2.6). Then f ∈MSc Hϑ (q,L,M) if and only if inequality (2.9) is fulfilled.
Corollary 2.7 Let f ∈ϑ1 be written in the series form (2.6). Then f ∈MSc Hϑ (q,L,M) if and only if inequality (2.9) is fulfilled.
Theorem 3.1
Theorem 3.1 Let f have the form (1.2). If f fulfills (2.1), then Re
Theorem 3.1 Let f have the form (1.2). If f fulfills (2.1), then Re
Theorem 3.2
Theorem 3.2 Let f = h+ g, where h and g are given by (1.3). If f fulfills (2.1), then Re
Theorem 3.2 Let f = h+ g, where h and g are given by (1.3). If f fulfills (2.1), then Re
Theorem 3.3
Theorem 3.3 Let f = h+g have the power series form (1.3). If f meets inequality (2.1), then Re
Theorem 3.3 Let f = h+g have the power series form (1.3). If f meets inequality (2.1), then Re
Theorem 3.4
Theorem 3.4 Let f = h + g, where h and g are expressed by (1.3). If f meets (2.1), then Re
Theorem 3.4 Let f = h + g, where h and g are expressed by (1.3). If f meets (2.1), then Re
Theorem 4.1
Theorem 4.1 If f ∈MS∗ Hϑ (q,L,M), then for |z| = r,
Theorem 4.1 If f ∈MS∗ Hϑ (q,L,M), then for |z| = r,
Theorem 4.2
Theorem 4.2 A function f ∈MS∗ Hϑ (q,L,M) if and only if f (z) = ∞ n=1 (Xnhn + Yngn), (4.3)
Theorem 4.2 A function f ∈MS∗ Hϑ (q,L,M) if and only if f (z) = ∞ n=1 (Xnhn + Yngn), (4.3)
Theorem 4.3
Theorem 4.3 Let f1,f2 ∈MS∗ Hϑ (q,L,M). Then f1 ∗f2 ∈MS∗ Hϑ (q,L,M).
Theorem 4.3 Let f1,f2 ∈MS∗ Hϑ (q,L,M). Then f1 ∗f2 ∈MS∗ Hϑ (q,L,M).
Theorem 4.4
Theorem 4.4 The family MS∗ Hϑ (q,L,M) is closed by a convex combination.
Theorem 4.4 The family MS∗ Hϑ (q,L,M) is closed by a convex combination.
Function classes studied:
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