On certain subclasses of analytic and harmonic mappings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915932579823616 |
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| author | Biswas, Raju |
| author_facet | Biswas, Raju |
| contents | Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(α, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-α)h'(z)+αzh''(z)-1+α\right|\leq M+\left|(1-α)g'(z)+αzg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $α\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(α, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain subclasses of analytic and harmonic mappings Biswas, Raju Complex Variables 30C45, 30C50, 30C80, 31A05 Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(α, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-α)h'(z)+αzh''(z)-1+α\right|\leq M+\left|(1-α)g'(z)+αzg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $α\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(α, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$. |
| title | On certain subclasses of analytic and harmonic mappings |
| topic | Complex Variables 30C45, 30C50, 30C80, 31A05 |
| url | https://arxiv.org/abs/2505.19160 |