Abstract
In this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.
(Communicated by Stanisława Kanas)
References
[1] Booth, A.: Researches on the geometrical properties of elliptic integrals, Philos. Trans. Roy. Soc. London 142 (1852), 311–416; 144 (1854), 53–69.10.1098/rstl.1852.0019Search in Google Scholar
[2] Booth, A.: A Treatise on Some New Geometrical Methods, Longmans Green, London, 1877.Search in Google Scholar
[3] Fenchel, W.: Bemerkungen über die in Einheitskreis meromorphen schlichten Funktionen, Preuss. Akad. Wiss. Phys. - Math. Kl. 22/23 (1931), 431–436.Search in Google Scholar
[4] Goluzin, G. M. On typically real functions, Mat. Sbornik N.S. 27(69) (1950), 201–218 (in Russian).Search in Google Scholar
[5] Goodman, W.: Univalent Functions, Vol. I – II, Mariner Publishing Company, Inc., Tampa, Florida, 1983.Search in Google Scholar
[6] Janowski, W.: Extremal problem for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970), 159–177.10.4064/ap-23-2-159-177Search in Google Scholar
[7] Jenkins, J. A.: On a conjecture of Goodman concerning meromorphic univalent functions, Michigan Math. J. 9 (1962), 25–27.10.1307/mmj/1028998616Search in Google Scholar
[8] Kanas, S.—Tatarczak, A.: Generalized Meixner-Pollaczek polynomials, Adv. Diff. Equ. 2013 (2013), 131.10.1186/1687-1847-2013-131Search in Google Scholar
[9] Kanas, S.—Tatarczak, A.: Generalized typically real functions, Filomat 30 (2016), 1697–1710.10.2298/FIL1607697KSearch in Google Scholar
[10] Kanas, S.—Tatarczak, A.: Constrained coefficients problem for generalized typically real functions, Compl. Vari. Ellip. Equ. 61 (2016), 1052–1063.10.1080/17476933.2015.1135551Search in Google Scholar
[11] Koczan, L.—Szapiel, W.: Sur certaines classes de fonctions holomorphes definies parune integrale de Stieltjes, Ann. Univ. Mariae Curie-Skłodowska Sect. A 28 (1974), 39–51.Search in Google Scholar
[12] Koczan, L.—Zaprawa, P.: Domains of univalence for typically-real odd functions, Complex Var. Theory Appl. 48 (2003), 1–17.10.1080/0278107021000037039Search in Google Scholar
[13] Loria, G.: Spezielle algebraische und transzendente Ebene Kurven, Theorie und Geschichte, Vol. I, transl. F. Schütte, Teubner, Leipzig, 1910.Search in Google Scholar
[14] Naraniecka, I.—Szynal, J.—Tatarczak, A.: The generalized Koebe function, Trudy Petrozawodskogo Universiteta, Matematika 17 (2010), 62–66.Search in Google Scholar
[15] Naraniecka, I.—Szynal, J.—Tatarczak, A.: An extension of typically-real functions and associated orthogonal polynomials, Ann. UMCS, Mathematica 65 (2011), 99–112.10.2478/v10062-011-0017-2Search in Google Scholar
[16] Pommerenke, C.: Linear-invariant Familien analytischer Funktionen, Mat. Ann. 155 (1964), 108–154.10.1007/BF01344077Search in Google Scholar
[17] Pommerenke, C.: Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975.Search in Google Scholar
[18] Richardson, S.: Some Hele-Shaw flows with time-dependent free boundaries, J. Fluid Mech. 102 (1981), 263–278.10.1017/S0022112081002632Search in Google Scholar
[19] Robertson, M. S.: On the coefficients of typically real functions, Bull. Amer. Math. Soc. 41 (1935), 565–572.10.1090/S0002-9904-1935-06147-6Search in Google Scholar
[20] Rogosinski, W.: Über positive harmonische Entwicklungen und typische-reelle Potenzreihen, Math. Z. 35 (1932), 93–121.10.1007/BF01186552Search in Google Scholar
[21] Struik, D.: Lectures on Classical Differential Geometry, 2nd ed., Dover, New York, 1961.Search in Google Scholar
[22] Thomas, D. K.—Tuneski, N.—Vasudevarao, A.: Univalent functions. de Gruyter St. Math. 69, 2018.10.1515/9783110560961Search in Google Scholar
[23] Zwikker, C.: The Advanced Geometry of Plane Curves and their Applications, Dover, New York, 1963.Search in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
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- Factorization of polynomials over valued fields based on graded polynomials
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- On reverse Hölder and Minkowski inequalities
- Coefficient inequalities related with typically real functions
- Existence of wandering and periodic domain in given angular region
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- Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
- Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
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- 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
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- The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
- Multi-opponent James functions
- An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
- A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
- On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
- Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
Articles in the same Issue
- Regular papers
- Some relative normality properties in locales
- Upper bounds of some special zeros for the Rankin-Selberg L-function
- Factorization of polynomials over valued fields based on graded polynomials
- Varieties of ∗-regular rings
- On reverse Hölder and Minkowski inequalities
- Coefficient inequalities related with typically real functions
- Existence of wandering and periodic domain in given angular region
- The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
- Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
- Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
- Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
- 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
- More on closed non-vanishing ideals in CB(X)
- The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
- Multi-opponent James functions
- An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
- A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
- On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
- Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations