Support points of some classes of analytic and univalent functions

Allu, Vasudevarao; Pandey, Abhishek

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

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

$$\left|f^{\prime }\left(z\right){\left(\frac{z}{f\left(z\right)}\right)}^2-1\right|<1\text{for}z\in 𝔻,$$

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

$$\Re \left(1+\frac{z{f}^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)>-\frac{1}{2}\text{for}z\in 𝔻.$$

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

Classification : 30C45, 30C50

[1] Abu-Muhanna, Yusuf; Li, Liulan; Ponnusamy, Saminathan 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] Akesent’ev, Leonid A. Sufficient conditions for univalence of certain integral representations (Russian), Izv. Vyssh. Uchebn. Zaved., Mat., Volume 3(4) (1958), pp. 3-7 | Zbl

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

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

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

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

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

[9] Dunford, Nelson; Schwartz, Jacob T. 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] Firoz, Md Ali; Vasudevarao, Allu; Yanagihara, Hiroshi 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] Grassman, Eckhard; Hengartner, Walter; Schober, Glenn E. Support points of the class of close-to-convex functions, Can. Math. Bull., Volume 19 (1976), pp. 177-179 | DOI | MR | Zbl

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

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

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

[15] Ponnusamy, Saminathan; Sahoo, Swadesh K.; Yanagihara, Hiroshi 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] Thomas, Derek K.; Tuneski, Nikola; Vasudevarao, Allu Univalent functions. A primer, De Gruyter Studies in Mathematics, 69, Walter de Gruyter, 2018 | Zbl

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

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

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