Support points of some classes of analytic and univalent functions

Vasudevarao Allu; Abhishek Pandey

doi:10.5802/crmath.181

Complex analysis and geometry

Support points of some classes of analytic and univalent functions

Vasudevarao Allu ¹ ; Abhishek Pandey ¹

¹ School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha (State), India

Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 465-473.

Abstract

Let $𝒜$ denote the class of analytic functions in the unit disk $𝔻 : = {z \in ℂ : | z | < 1}$ satisfying $f (0) = 0$ and $f^{'} (0) = 1$ . Let $𝒰$ be the class of functions $f \in 𝒜$ satisfying

|f^{'} (z) {(\frac{z}{f (z)})}^{2} - 1| < 1 for z \in 𝔻,

and $𝒢$ denote the class of functions $f \in 𝒜$ satisfying

ℜ (1 + \frac{z f^{''} (z)}{f^{'} (z)}) > - \frac{1}{2} for z \in 𝔻 .

In the present paper, we characterize the set of support points of the classes $𝒰$ and $𝒢$ .

Received: 2020-11-02
Accepted: 2021-01-19
Published online: 2021-06-17

Zbl

DOI: 10.5802/crmath.181

Classification: 30C45, 30C50

Author's affiliations:

Vasudevarao Allu ¹; Abhishek Pandey ¹

¹ School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha (State), India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Vasudevarao Allu and Abhishek Pandey},
     title = {Support points of some classes of analytic and univalent functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {465--473},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {4},
     year = {2021},
     doi = {10.5802/crmath.181},
     zbl = {07362166},
     language = {en},
}

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Vasudevarao Allu; Abhishek Pandey. Support points of some classes of analytic and univalent functions. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 465-473. doi : 10.5802/crmath.181. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.181/

References
Cited by

[1] Yusuf Abu-Muhanna; Liulan Li; Saminathan Ponnusamy Extremal problems on the class of convex functions of order $- 1 / 2$ , Arch. Math., Volume 103 (2014) no. 6, pp. 461-471 | MR | Zbl

[2] Leonid A. Akesent’ev Sufficient conditions for univalence of certain integral representations (Russian), Izv. Vyssh. Uchebn. Zaved., Mat., Volume 3(4) (1958), pp. 3-7 | Zbl

[3] James W. Alexander Functions which maps the interior of the unit circle upon simple regions, Ann. Math., Volume 17 (1915), pp. 12-22 | DOI | MR | Zbl

[4] S. Vaidhyanathan Bharanedhar; Saminathan Ponnusamy Uniform close-to-convexity radius of sections of functions in the close-to-convex family, J. Ramanujan Math. Soc., Volume 29 (2014) no. 3, pp. 243-251 | MR | Zbl

[5] Louis Brickman; Thomas H. MacGregor; Donald R. Wilken Convex hull of some classical family of univalent functions, Trans. Am. Math. Soc., Volume 156 (1971), pp. 91-107 | DOI | Zbl

[6] Daouh H. Bshouty; Abdallah Lyzzaik close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory, Volume 5 (2011) no. 3, pp. 767-774 | DOI | MR | Zbl

[7] Paul C. Cochrane; Thomas H. MacGregor Fretchet differentiable functionals and support points for families of analytic functions, Trans. Am. Math. Soc., Volume 236 (1978), pp. 75-92 | Zbl

[8] Hua Deng; Saminathan Ponnusamy; Jinjing Qiao Extreme points and support points of families of harmonic Bloch mappings, Potential Anal. (2020) | DOI | Zbl

[9] Nelson Dunford; Jacob T. Schwartz Linear operators. Part I: General theory (With the assistence of William G. Bade and Robert G. Bartle), Pure and Applied Mathematics, 7, Interscience Publishers, 1958 | Zbl

[10] Md Ali Firoz; Allu Vasudevarao; Hiroshi Yanagihara On a class of univalent functions defined by a differential inequality, J. Ramanujan Math. Soc., Volume 35 (2020) no. 3, pp. 217-226 | MR | Zbl

[11] Eckhard Grassman; Walter Hengartner; Glenn E. Schober Support points of the class of close-to-convex functions, Can. Math. Bull., Volume 19 (1976), pp. 177-179 | DOI | MR | Zbl

[12] David J. Hallenbeck; Thomas H. MacGregor Linear problem and convexity techniques in geometric function theory, Monographs and Studies in Mathematics, 22, Pitman Advanced Publishing Program, 1984 | MR | Zbl

[13] David J. Hallenbeck; Shelton Perera; Donald R. Wilken Subordination, Extreme points and support points, Complex Variables, Theory Appl., Volume 11 (1989) no. 2, pp. 111-124 | DOI | MR | Zbl

[14] Shigeo Ozaki; Mamoru Nunokawa The Schwarzian derivative and univalent functions, Proc. Am. Math. Soc., Volume 33 (1972), pp. 392-394 | DOI | MR | Zbl

[15] Saminathan Ponnusamy; Swadesh K. Sahoo; Hiroshi Yanagihara Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Anal., Theory Methods Appl., Volume 95 (2014), pp. 219-228 | DOI | MR | Zbl

[16] Derek K. Thomas; Nikola Tuneski; Allu Vasudevarao Univalent functions. A primer, De Gruyter Studies in Mathematics, 69, Walter de Gruyter, 2018 | Zbl

[17] Toshio Umezama Analytic functions convex in one direction, J. Math. Soc. Japan, Volume 4 (1952), pp. 194-202 | MR | Zbl

[18] Allu Vasudevarao; Abhishek Pandey The Zalcman conjecture for certain analytic and univalent functions, J. Math. Anal. Appl., Volume 492 (2020) no. 2, 124466 | MR | Zbl

[19] Donald R. Wilken; R. J. M. Hornblower On the support points of close-to-convex functions, Houston J. Math., Volume 10 (1984) no. 4, pp. 593-599 | MR | Zbl

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