Sufficient conditions for Carathéodory functions and applications to univalent functions
-
Oh Sang Kwon
and
Young Jae Sim
Abstract
In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk 𝔻: = {z ∈ ℂ: |z| < 1}. More precisely, for given β ∈ (–π/2, π/2), γ ∈ [0, cosβ) and δ ∈ (0, π/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e−iβp(z)} > γ or | arg {p(z)–γ} | < δ for all z ∈ 𝔻 by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness.
The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP; Ministry of Science, ICT & Future Planning) (No. NRF-2017R1C1B5076778).
(Communicated by Stanisława Kanas)
References
[1] Attiya, A. A.—Nasr, M. A. M.: On sufficient conditions for Carathéodory functions with applications, J. Inequal. Appl. 2013:191 (2013), 1–10.10.1186/1029-242X-2013-191Search in Google Scholar
[2] Brannan, D. A.—Kirwan, W. E.: On some classes of bounded univalent functions, J. London Math. Soc. 2(1) (1969), 431–443.10.1112/jlms/s2-1.1.431Search in Google Scholar
[3] Darus, M.—Hussain, S.—Raza, M.—Sokół, J.: On a subclass of starlike functions, Result Math. 73(1) (2018), 1–22.10.1007/s00025-018-0771-3Search in Google Scholar
[4] Ebadian, A.—Sokół, J.: On the subordination and superordination of strongly starlike functions, Math. Slovaca 66(4) (2016), 815–822.10.1515/ms-2015-0184Search in Google Scholar
[5] Frasin, B. A.: Sufficient conditions for λ-spirallike and λ-Robertson functions of complex order, J. Math. Art. (2013), Art. ID: 194053.10.1155/2013/194053Search in Google Scholar
[6] Goodman, A. W.: Univalent Functions, Mariner, Tampa, 1983.Search in Google Scholar
[7] Hotta, I.—Nunokawa, M: On strongly starlike and convex functions of order α and typeβ, Mathematica 53(76) (2011), 51–56.Search in Google Scholar
[8] Kim, I. H.—Cho, N. E.: Sufficient conditions for Carathéodory functions, Comput. Math. Appl. 59 (2010), 2067–2073.10.1016/j.camwa.2009.12.012Search in Google Scholar
[9] Kim, Y. C.—Sugawa, T.: The Alexander transform of a spirallike function, J. Math. Anal. Appl. 325(1) (2007), 608–611.10.1016/j.jmaa.2006.01.077Search in Google Scholar
[10] Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions, Proc. Japan. Acad. Ser. A. 69 (1993), 234–237.10.3792/pjaa.69.234Search in Google Scholar
[11] Nunokawa, M.—Kwon, O. S.—Sim, Y. J.—Cho, N. E.: Sufficient conditions for Carathéodory functions, Filomat 32(3) (2018), 1097–1106.10.2298/FIL1803097NSearch in Google Scholar
[12] Nunokawa, M.—Sokół, J.: New conditions for starlikeness and strongly starlikeness of order alpha, Houston J. Math. 43(2) (2017), 333–344.Search in Google Scholar
[13] Nunokawa, M.—SokółL, J.: Some applications of first-order differential subordinations, Math. Slovaca 67(4) (2017), 939–944.10.1515/ms-2017-0022Search in Google Scholar
[14] Miller, S. S.—Mocanu, P. T.: Differential Subordination: Theory and Applications, Marcel Dekker, 2000.10.1201/9781482289817Search in Google Scholar
[15] Miller, S. S.—Mocanu, P. T.: Marx-Strohhächker differential subordination systems, Proc. Amer. Math. Soc. 99 (1987), 527–534.Search in Google Scholar
[16] Montel, P.: Leçons sur les Functions Univalentes on Multivalentes, Gauthier-Villars, Paris, 1933.Search in Google Scholar
[17] Pommerenke, Ch.: Univalent Functions, Vandenhoeck and Ruprecht, 1975.Search in Google Scholar
[18] Robertson, M. S.: A representation of all analytic functions in terms of functions with positive real part, Ann. Math. 38 (1937), 770–783.10.2307/1968833Search in Google Scholar
[19] Sim, Y. J.—Kwon, O. S.—Cho, N. E.—Srivastava, H. M.: Some sets of sufficient conditions for Carathéodory functions, J. Comput. Anal. Appl. 21(7) (2016), 1243–1254.Search in Google Scholar
[20] Špaček, L.: Contribution à la theorie des fonctions univalentes, Časopis Pěst. Mat. 62 (1932), 12–19.10.21136/CPMF.1933.121951Search in Google Scholar
[21] Stankiewicz, J.: Quelques problémes extrémaux dans les classes des fonctions α-angulairement étoilées, Ann. Univ. Mariae Curie-Sklodowska Sect. A. 20 (1966), 59–75.Search in Google Scholar
[22] Tuneski, N.: Some simple sufficient conditions for starlikeness and convexity, Appl. Math. Lett. 22(5) (2009), 693–697.10.1016/j.aml.2008.08.006Search in Google Scholar
[23] Xu, Q. H.—Yang, T.—Srivastava, H. M.: Sufficient conditions for a certain general class of Carathéodory functions, Filomat 3013 (2016), 3615–3625.10.2298/FIL1613615XSearch in Google Scholar
[24] Xu, Q. H.—Lu, S.: The Alexander transformation of a subclass of spirallike functions of type β, J. Inequal. Pure Appl. Math. 10(1) (2009), Art. 17.Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
