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A criterion for univalence extension

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Abstract

This paper presents a study on the univalence extension of a specific class of functions defined by an integral operator. The research provides sufficient conditions for determining this univalence extension, which depend on two arbitrary analytic functions defined in the unit disk, alongside three complex numbers. The main result of the study includes these functions and parameters, and by specializing them, it leads to various well-known univalent conditions. The paper showcases examples to demonstrate how these conditions can be applied.

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References

  1. Ahlfors, L.V.: Sufficient conditions for quasiconformal extension. Ann. Math. Stud. 79, 23–29 (1974)

    MathSciNet  Google Scholar 

  2. Al-shbeil, I., Srivastava, H.M., Arif, M., Haq, M., Khan, N., Bilal, K.: Majorization results based upon the Bernardi integral operator. Symmetry 14, 1404 (2022)

    Article  ADS  Google Scholar 

  3. Becker, J.: Über die Lösungsstruktur einer Differentialgleichung in der konformen Abbildung. J. Reine Angew. Math. 285, 66–74 (1976)

    MathSciNet  Google Scholar 

  4. Becker, J.: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. Reine Angew. Math. 255, 23–43 (1972)

    MathSciNet  Google Scholar 

  5. Çağlar, M., Orhan, H.: Some generalizations on the univalence of an integral operator and quasiconformal extensions. Miskolc Math. 14, 49–62 (2013)

    Article  MathSciNet  Google Scholar 

  6. Deniz, E., Kanas, S., Orhan, H.: Univalence Criteria and Quasiconformal Extension of a General Integral Operator. Ukr. Math. J. 74(1), 27 (2022)

    Article  MathSciNet  Google Scholar 

  7. Deniz, E., Răducanu, D., Orhan, H.: On an improvement of a univalence criteria. Math. Balkanica 24, 33–39 (2010)

    MathSciNet  Google Scholar 

  8. Deniz, E., Orhan, H.: Some notes on extensions of basic univalence criteria. J. Korean Math. Soc. 48(1), 179–189 (2011)

    Article  MathSciNet  Google Scholar 

  9. Deniz, E., Orhan, H.: Löewner chains and univalence criteria related with Ruscheweyh and Salagean derivatives. J. Appl. Anal. Comput. 5(3), 465–478 (2015)

    MathSciNet  Google Scholar 

  10. Duren, P.L.: Univalent functions. Springer Science & Business Media, Berlin (2001)

    Google Scholar 

  11. Eun, C.N., Srivastava, H.M.: Subordinations by \(\eta \)-convex functions for a class of nonlinear integral operators. Bull. Sci. Math. 187, 103304 (2023)

    Article  MathSciNet  Google Scholar 

  12. G.M. Goluzin, Geometric theory of functions of a complex variable, Amer.Math. Soc. Transl. of Math.Monographs 29, Providence, RI, 1969

  13. Kanas, S., Lecko, A.: Univalence criteria connected with arithmetic and geometric means, I. Folia Sci. Univ. Tech. Resov. 20, 49–59 (1996)

    MathSciNet  Google Scholar 

  14. S. Kanas and A. Lecko, Univalence criteria connected with arithmetic and geometric means, II, Proceedings of the Second Int. Workshop of Transform Methods and Special Functions, Varna’96, Bulgar. Acad. Sci. (Sofia)(1996), 201 - 209

  15. Kanas, S., Srivastava, H.M.: Some criteria for univalence related to Ruscheweyh and Salagean derivatives. Complex Var. Elliptic Equ. 38, 263–275 (1997)

    Google Scholar 

  16. Lewandowski, Z.: On a univalence criterion. Bull. Acad. Polon. Sci. Ser. Sci. Math. 29, 123–126 (1981)

    MathSciNet  Google Scholar 

  17. O. Mayer, The Functions Theory of One Variable Complex, Bucureşi, 1981

  18. S.S. Miller P.T. Mocanu, Differential Subordinations,Theory and Applications. Series of Monographs and Textbooks in Pure andApplied Mathematics, vol. 225. Marcel Dekker Inc. NewYork/Basel 2000

  19. Nehari, Z.: The Schwarzian derivate and schlich functions. Bull. Amer. Math. Soc. 55, 545–551 (1949)

    Article  MathSciNet  Google Scholar 

  20. Orhan, H., Răducanu, D., Çağlar, M.: Some sufficient conditions for the univalence of an integral operator. J. Classical Anal. 5, 61–70 (2014)

    Article  MathSciNet  Google Scholar 

  21. Ozaki, S., Nunokawa, M.: The Schwarzian derivative and univalent functions. Proc. Amer. Math. Soc. 33, 392–394 (1972)

    Article  MathSciNet  Google Scholar 

  22. Ovesea, H.: A generalization of Ruscheweyh’s univalence criterion. J. Math. Anal. Appl. 258, 102–109 (2001)

    Article  MathSciNet  Google Scholar 

  23. Ovesea, H.: A generalization of the univalence criteria of Becker, of Nehari and Lewandoski (I). Demo Math 28(4), 953–960 (1995). https://doi.org/10.1515/dema-1995-0421

    Article  MathSciNet  Google Scholar 

  24. Ovesea, H.: An extension of the univalence critreia of Becker, of Nehari and of Lewandowski. Stud. Cerc. Math. 49, 217–223 (1997)

    Google Scholar 

  25. Ovesea, H., Pascu, N.N., Pascu, R.N.: A generalization of the univalence criteria of Nehari, of Ahlfors and Becker and of Lewandowski. General Math Sibiu 1, 1–7 (1993)

    Google Scholar 

  26. Ovesea-Tudor, H., Owa, S.: An extension of the univalence criteria of Nehari and Ozaki and Nunokawa. Hokkaido Math. J. 34, 533–539 (2005)

    Article  MathSciNet  Google Scholar 

  27. Pascu, N.N.: On a univalence criterion, II, Itinerant seminar on functional equations approximation and convexity. Cluj-Napoca 6, 153–154 (1985)

    Google Scholar 

  28. N. N. Pascu, An improvement of Becker’s univalence criterion, in Proc. Commem. Sess. Simion Stoilow, Brasov/Rom., 1988, vol. 1987, pp. 43–48

  29. Pescar, V.: A new generalization of Ahlfor’s and Becker’s criterion of univalence. Bull. Malays. Math. Soc. 19, 53–54 (1996)

    MathSciNet  Google Scholar 

  30. Pescar, V.: On the univalence of some integral operators. General Math. 14(2), 83–90 (2006)

    MathSciNet  Google Scholar 

  31. Pescar, V.: On the univalence of an integral operator. Appl. Math. Lett. 23(5), 615–9 (2010)

    Article  MathSciNet  Google Scholar 

  32. ch. Pommerenke,Univalent Functions, Vandenhoeck Ruprecht in Göttingen, 1975

  33. Răducanu, D., Orhan, H., Deniz, E.: On some sufficient conditions for univalence. An. Şt. Univ. Ovidius Constanta Ser. Mat. 18(2), 217–222 (2010)

    MathSciNet  Google Scholar 

  34. Răducanu, D., Tudor, H., Shigeyoshi, O.: An extension of a basic univalence criterion. Tamkang J. Math. 44(4), 417–430 (2013)

    Article  MathSciNet  Google Scholar 

  35. Singh, V., Chichra, P.N.: An extension of Becker’s criterion for univalence. J. Indian Math. Soc. 41, 353–361 (1977)

    MathSciNet  Google Scholar 

  36. H. M. Srivastava, K. Alshammari, M. Darus, A new \(q\)-fractional integral operator and its applications to the coefficient problem involving the second hankel determinant for \(q\)-starlike and \(q\)-convex functions, J. Nonlinear Var. Anal., 7 (6) (2023)

  37. Srivastava, H.M., Alavi, R., Shams, S., Aghalary, R., Joshi, S.B.: New criteria for starlikness and convexity of a certain family of integral operators. Mathematics 11(18), 3919 (2023)

    Article  Google Scholar 

  38. Srivastava, H.M., Bansal, M.K., Harjule, P.: A Class of Fractional Integral Operators Involving a Certain General Multiindex Mittag-Leffler Function. Ukrainian Math. J. 75, 1255–1271 (2024)

    Article  MathSciNet  Google Scholar 

  39. H. M. Srivastava, A. K. Nain, R. K. Vats, P. Das, A theoretical study of the fractional-order \(p\)-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability, Rev. Real Acad. Cienc. ExactasFís. Natur. Ser. A Mat. (RACSAM), 117 (4) (2023), 160

  40. Tudor, H.: An extension of Ozaki and Nunokawa’s univalence criteria. J. Inequal. Pure Appl. Math. 9, 1–4 (2008)

    Google Scholar 

  41. H. Tudor, A connection between basic univalence criteria, Abstr. Appl. Anal. 2013 Jan 1 (Vol. 2013). Hindawi

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Acknowledgements

The authors thank the editor and the anonymous referees for constructive and pertinent suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, Tehran, Iran

    Vali Soltani Masih, Hormoz Rahmatan & Seyedeh Kolsum Taheri

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  1. Vali Soltani Masih
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  2. Hormoz Rahmatan
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  3. Seyedeh Kolsum Taheri
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All the authors contributed equally to each part of this work.

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Masih, V.S., Rahmatan, H. & Taheri, S.K. A criterion for univalence extension. Afr. Mat. 35, 31 (2024). https://doi.org/10.1007/s13370-024-01172-x

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