Schwarzian Norm Estimates for Some Classes of Analytic Functions
Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathca...
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| Autor*in: |
Ali, Md Firoz [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Mediterranean journal of mathematics - Springer International Publishing, 2004, 20(2023), 6 vom: 30. Aug. |
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| Übergeordnetes Werk: |
volume:20 ; year:2023 ; number:6 ; day:30 ; month:08 |
| Links: |
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| DOI / URN: |
10.1007/s00009-023-02496-x |
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OLC2145300104 |
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| 520 | |a Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. | ||
| 650 | 4 | |a Univalent functions | |
| 650 | 4 | |a starlike functions | |
| 650 | 4 | |a convex function in some direction | |
| 650 | 4 | |a Schwarzian norm | |
| 650 | 4 | |a two-point distortion | |
| 700 | 1 | |a Pal, Sanjit |4 aut | |
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10.1007/s00009-023-02496-x doi (DE-627)OLC2145300104 (DE-He213)s00009-023-02496-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ali, Md Firoz verfasserin aut Schwarzian Norm Estimates for Some Classes of Analytic Functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. Univalent functions starlike functions convex function in some direction Schwarzian norm two-point distortion Pal, Sanjit aut Enthalten in Mediterranean journal of mathematics Springer International Publishing, 2004 20(2023), 6 vom: 30. Aug. (DE-627)389869848 (DE-600)2149653-5 (DE-576)12119308X 1660-5446 nnns volume:20 year:2023 number:6 day:30 month:08 https://doi.org/10.1007/s00009-023-02496-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2088 AR 20 2023 6 30 08 |
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10.1007/s00009-023-02496-x doi (DE-627)OLC2145300104 (DE-He213)s00009-023-02496-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ali, Md Firoz verfasserin aut Schwarzian Norm Estimates for Some Classes of Analytic Functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. Univalent functions starlike functions convex function in some direction Schwarzian norm two-point distortion Pal, Sanjit aut Enthalten in Mediterranean journal of mathematics Springer International Publishing, 2004 20(2023), 6 vom: 30. Aug. (DE-627)389869848 (DE-600)2149653-5 (DE-576)12119308X 1660-5446 nnns volume:20 year:2023 number:6 day:30 month:08 https://doi.org/10.1007/s00009-023-02496-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2088 AR 20 2023 6 30 08 |
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10.1007/s00009-023-02496-x doi (DE-627)OLC2145300104 (DE-He213)s00009-023-02496-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ali, Md Firoz verfasserin aut Schwarzian Norm Estimates for Some Classes of Analytic Functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. Univalent functions starlike functions convex function in some direction Schwarzian norm two-point distortion Pal, Sanjit aut Enthalten in Mediterranean journal of mathematics Springer International Publishing, 2004 20(2023), 6 vom: 30. Aug. (DE-627)389869848 (DE-600)2149653-5 (DE-576)12119308X 1660-5446 nnns volume:20 year:2023 number:6 day:30 month:08 https://doi.org/10.1007/s00009-023-02496-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2088 AR 20 2023 6 30 08 |
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10.1007/s00009-023-02496-x doi (DE-627)OLC2145300104 (DE-He213)s00009-023-02496-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ali, Md Firoz verfasserin aut Schwarzian Norm Estimates for Some Classes of Analytic Functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. Univalent functions starlike functions convex function in some direction Schwarzian norm two-point distortion Pal, Sanjit aut Enthalten in Mediterranean journal of mathematics Springer International Publishing, 2004 20(2023), 6 vom: 30. Aug. (DE-627)389869848 (DE-600)2149653-5 (DE-576)12119308X 1660-5446 nnns volume:20 year:2023 number:6 day:30 month:08 https://doi.org/10.1007/s00009-023-02496-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2088 AR 20 2023 6 30 08 |
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10.1007/s00009-023-02496-x doi (DE-627)OLC2145300104 (DE-He213)s00009-023-02496-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ali, Md Firoz verfasserin aut Schwarzian Norm Estimates for Some Classes of Analytic Functions 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. Univalent functions starlike functions convex function in some direction Schwarzian norm two-point distortion Pal, Sanjit aut Enthalten in Mediterranean journal of mathematics Springer International Publishing, 2004 20(2023), 6 vom: 30. Aug. (DE-627)389869848 (DE-600)2149653-5 (DE-576)12119308X 1660-5446 nnns volume:20 year:2023 number:6 day:30 month:08 https://doi.org/10.1007/s00009-023-02496-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2088 AR 20 2023 6 30 08 |
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Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract Let $${\mathcal {A}}$$ denote the class of analytic functions f in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$$ normalized by $$f(0)=0$$, $$f'(0)=1$$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $${\mathcal {G}}(\beta )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]<1+\beta /2\}$$, where $$\beta >0$$ and $${\mathcal {F}}(\alpha )=\{f\in {\mathcal {A}}:\mathrm{Re\,}[1+zf''(z)/f'(z)]>\alpha \}$$, where $$-1/2\le \alpha \le 0$$. We also establish two-point distortion theorem for functions in the classes $${\mathcal {G}}(\beta )$$ and $${\mathcal {F}}(\alpha )$$. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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Schwarzian Norm Estimates for Some Classes of Analytic Functions |
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