Geometric properties of the Bernatsky integral operator

F. F. Mayer ; M. G. Tastanov ; A. A. Utemisova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 4, pp. 12-19

Résumé

In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection $f(z)\in S^{o}\Leftrightarrow g(z) = zf'(z) \in S^*$ of the classes $S^{o}$ and $S^*$ of convex and star-shaped functions can be considered as mapping using the differential operator $G[f](x) = zf'(z)$ of class $S^{o}$ to class $S^*$, that is, $G: S^{o} \to S^*$ or $G(S^{o}) = S^*$. The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator $G^{-1}[f](x)$, which translates $S^* \to S^{o}$ and thereby “improves” the properties of functions, maps the entire class $S$ of single-leaf functions into itself. At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class $S$ or its subclasses to themselves or to other subclasses. This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition $a < \mathrm{Re}\, zf'(z)/f(z) < b$ ($0 < a < 1 < b$), in the class $K(\gamma)$ of functions, almost convex in order $\gamma$. The results of the article summarize or reinforce previously known effects.

Détail

Keywords: geometric theory of functions of a complex variable, single-leaf functions, Bernatsky integral operator, star-shaped and almost convex functions.
Mots-clés : convex

Bibliographie

[1] F.G. Avkhadiev, L.A. Aksentev, “Osnovnye rezultaty v dostatochnykh usloviyakh odnolistnosti analiticheskikh funktsii”, UMN, 30:4(184) (1975), 3–60 | MR | Zbl

[2] M. Biernacki, “Sur L'Integrale des Fonctions Univalentes”, Bulletin Polish Acad. Sci. Math., Astron. et Phys., 8:1 (1960), 29–34 | MR | Zbl

[3] “Ob odnoi teoreme M. Bernatskogo v teorii odnolistnykh funktsii”, Ukr. matem. zhurn., 17:4 (1965), 63–71 | MR | Zbl

[4] “Integralnye preobrazovaniya v nekotorykh klassakh odnolistnykh funktsii”, Izv. vuzov. Matematika, 1980, no. 12, 45–49

[5] N.N. Pascu, “On a Univalence Criterion II”, Studia Universitatis Babes-Bolyai Mathematica, 6 (1985), 153–154 | MR

[6] D.V. Prokhorov, “Ob oblastyakh znachenii sistem funktsionalov i integrirovanii odnolistnykh funktsii”, Izv. vuzov. Matematika, 1986, no. 10, 33–39 | Zbl

[7] T.P. Sizhuk, “Poryadok zvezdoobraznosti operatora Bernardi v klasse zvezdoobraznykh funktsii”, Vestnik Stavropolskogo gosudarstvennogo universiteta, 2009, no. 4, 76–78

[8] A.V. Kazantsev, “Ob uravnenii Gakhova dlya operatora Bernatskogo”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 2, 2015, 79–92 | MR | Zbl

[9] M.R. Kadieva, F.F. Maier, “Uslovie vypuklosti obobschennogo integrala Bernatskogo dlya odnogo podklassa zvezdoobraznykh funktsii”, Vestnik KazNPU im.Abaya. Seriya: fiziko-matematicheskie nauki, 69:1 (2020), 111–118

[10] M.O. Reade, “The Coefficients of Close-to-Convex Functions”, Duke Math. J., 23:3 (1956), 459–462 | DOI | MR | Zbl

[11] A. Renyi, “Some Remarks on Univalent Functions”, Bulgar. Akad. Nauk., Izv. Mat. Inst., 3 (1959), 111–119 | MR | Zbl

[12] Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966, 628 pp. | MR

[13] E.P. Merkes, D.J. Wright, “On the Univalence of a Certain Integral”, Proc. Amer. Math. Soc., 27:1 (1971), 97–100 | DOI | MR | Zbl