Abstract: The Geometric Theory of Analytic Functions (GTAF) is the attractive part of complex analysis, which correlates with the rest of the themes in mathematics. Its essential purpose is to formulate numerous classes of geometric analytic functions and to discuss their geometric attributes. In continuation, the association between operator theory and the GTAF area started to take shape and has remained a topic of wide attention today. In the previous century, operator theory was extended to the complex open unit disk and has been applied to propose diverse sorts of generalizations of normalized analytic functions. As a result, the operator theory appears to be a good way to look for things in the GTAF area. Since then, the acquisition of geometric attributes by employing operators has become a significant theme of research studies. The current study centers on and investigates, in the classes of \(\ell\)-uniformly convex and starlike functions of order \(\beta\), the convexity attribute by utilizing a modified Breaz integro-differential operator in the unit disk. Furthermore, in the class of analytic functions, some conditions that make the Breaz operator look like a star are looked into.
For citation: Al-Janaby, H. F. and Ghanim, F. An Analysis of Convexity and Starlikeness Attributes for Breaz Integro-Differential Operator, Vladikavkaz Math. J., 2022, vol. 24, no. 2, pp. 25-34. DOI 10.46698/p4155-0765-8236-d
1. de Branges, L. A Proof of the Bieberbach Conjecture, Acta Mathematica, 1984, vol. 154, no. 1-2, pp. 137-152. DOI: 10.1007/BF02392821.
2. Ghanim, F. and Al-Janaby, H. F. Inclusion and Convolution Features of Univalent Meromorphic Functions Correlating with Mittag-Leffler Function, Filomat, 2020, vol. 34, no. 7, pp. 2141-2150. DOI: 10.2298/FIL2007141G.
3. Ghanim, F., Bendak, S. and Hawarneh, A. A. Certain Implementations in Fractional Calculus Operator Involving Mittag-Leffler Confluent Hypergeometric Functions, Proceedings of the Royal Society A, 2022, vol. 478, no. 2258. DOI: 10.1098/rspa.2021.0839.
4. Oros, G. I. Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function, Symmetry, 2021, vol. 13(2), no. 259, pp. 1-10. DOI: 10.3390/sym13020259.
5. Ghanim, F., Al-Janaby, H. F. and Bazighifan, O. Geometric Properties of the Meromorphic Functions Class Through Special Functions Associated with a Linear Operator, Advances in Continuous and Discrete Models, 2022, vol. 17, pp. 1-15. DOI: 10.1186/s13662-022-03691-y.
6. Lupas, A. A. and Oros, G. I. Fractional Calculus and Confluent Hypergeometric Function Applied in the Study of Subclasses of Analytic Functions, Mathematics, 2022, vol. 10(5), no. 705, pp. 1-9. DOI: 10.3390/math10050705.
7. Atangana, A. and Baleanu, D. New Fractional Derivatives with Nonlocal and non-Singular Kernel: Theory and Application to Heat Transfer Model, Thermal Science, 2016, vol. 20, no. 2, pp. 763-769. DOI: 10.2298/TSCI160111018A.
8. Srivastava, H. M., Fernandez, A. and Baleanuand, D. Some New Fractional-Calculus Connections Between Mittag-Leffler Functions, Mathematics, 2019, vol. 7(6), no. 485, pp. 1-10. DOI: 10.3390/math7060485.
9. Ozarslan, M. A. and Ustaoglu, C. Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators, Mathematics, 2019, vol. 7(5), no. 483, pp. 1-18. DOI: 10.3390/math7050483.
10. Ghanim, F. and Al-Janaby, H. F. An Analytical Study on Mittag-Leffler-Confluent Hypergeometric Functions with Fractional Integral Operator, Mathematical Methods in the Applied Sciences, 2020, vol. 44, no. 5, pp. 3605-3614. DOI: 10.1002/mma.6966.
11. Ghanim, F. and Al-Janaby, H. F. Some Analytical Merits of Kummer-Type Function Associated with Mittag-Leffler Parameters, Arab Journal of Basic and Applied Sciences, 2021, vol. 28, no. 1, pp. 255-263. DOI: 10.1080/25765299.2021.1930637.
12. Ghanim, F., Al-Janaby, H. F. and Bazighifan, O. Fractional Calculus Connections on Mittag-Leffler Confluent Hypergeometric Functions, Fractal and Fractional, 2021, vol. 5(4), no. 143, pp. 1-10. DOI: 10.3390/fractalfract5040143.
13. Goodman, A. W.Univalent Functions, Florida, Mariner Publishing Company, 1983.
14. Study, E.Vorlesungen uber Ausgewahlte Gegensta der Geometrie, Heft 2, Konforme Abbildung Einfach Zusammenhangender Bereiche, Leipzig, B. G. Teubner, 1913.
15. Alexander, J. W. Functions which Map the Interior of the Unit Circle upon Simple Regions, The Annals of Mathematics, 1915, vol. 17(2), no. 1, pp. 12-22.
DOI: 10.2307/2007212.
16. Nevanlinna, R. Uber die Konforme Abbildund Sterngebieten, Oversikt av Finska-Vetenskaps Societen Forhandlingar, 1921, vol. 63(A), no. 6, pp. 48-403.
17. Robertson, M. S. Certain Classes of Starlike Functions, Michigan Mathematical Journal, 1954, vol. 76, no. 1, pp. 755-758.
18. Shiraishi, H. and Owa, S. Starlikeness and Convexity for Analytic Functions Concerned with Jack's Lemma, International Journal of Open Problems in Computer Science and Mathematics, 2009, vol. 2, no. 1, pp. 37-47. arXiv: 1303.0501.
19. Nunokawa, M., Goyal, S. P. and Kumar, R. Sufficient Conditions for Starlikeness, Journal of Classical Analysis, 2012, vol. 1, pp. 85-90. DOI: 10.7153/jca-01-09.
20. Sokol, J. and Nunokawa, M. On Some Sufficient Conditions for Univalence and Starlikeness, Journal of Inequalities and Applications, 2012, vol. 2012, Article no. 282, pp. 1-9. DOI: 10.1186/1029-242X-2012-282.
21. Nunokawa, M. and Sokol, J. On Some Conditions for Schlichtness of Analytic Functions, Journal of Computational and Applied Mathematics, 2020, vol. 363, pp. 241-248. DOI: 10.1016/j.cam.2019.06.009.
22. Goodman, A. W. On Uniformly Convex Functions, Annales Polonici Mathematici, 1991, vol. 56, no. 1, pp. 87-92. DOI: 10.4064/ap-56-1-87-92.
23. Goodman, A. W. On Uniformly Starlike Functions, Journal of Mathematical Analysis and Applications, 1991, vol. 155, no. 2, pp. 364-370. DOI: 10.1016/0022-247X(91)90006-L.
24. Ronning, F. On Starlike Functions Associated with Parabolic Regions, Annales Universitatis Mariae Curie-Sklodowska. Sectio A. Mathematica, 1991, vol. 45, no. 14, pp. 117-122.
25. Ma, W. C. and Minda, D. Uniformly Convex Functions, Annales Polonici Mathematici, 1992, vol. 57, no. 2, pp. 165-175. URL: http://eudml.org/doc/262507.
26. Ronning, F. Uniformly Convex Functions and a Corresponding Class of Starlike Functions, Proceedings of the American Mathematical Society, 1993, vol. 118, no. 1, pp. 189-196. DOI: 10.2307/2160026.
27. Bharati, R., Parvatham, R. and Swaminathan, A. On Subclasses of Uniformly Convex Functions and Corresponding Class of Starlike Functions, Tamkang Journal of Mathematics, 1997, vol. 28, no. 1, pp. 17-32. DOI: 10.5556/j.tkjm.28.1997.4330.
28. Darus, M. Certain Class of Uniformly Analytic Functions, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 2008, vol. 24, no. 3, pp. 345-353.
29. Breaz, N., Breaz, D. and Darus, M. Convexity Properties for Some General Integral Operators on Uniformly Analytic Functions Classes, Computers & Mathematics with Applications, 2010, vol. 60, no. 12, pp. 3105-3107. DOI: 10.1016/j.camwa.2010.10.012.
30. Sokol, J. and Trojnar-Spelina, L. On a Sufficient Condition for Strongly Starlikeness, Journal of Inequalities and Applications, 2013, vol. 2013, Article no. 283, pp. 1-11.
DOI: 10.1186/1029-242X-2013-383.
31. Magesh, N. Certain Subclasses of Uniformly Convex Functions of Order \(\alpha\) and Type \(\beta\) with Varying Arguments, Journal of the Egyptian Mathematical Society, 2013, vol. 21, no. 3, pp. 184-189. DOI: 10.1016/j.joems.2013.02.005.
32. Al-Janaby, H. F., Ghanim, F. and Darus, M. Some Geometric Properties of Integral Operators Proposed by Hurwitz-Lerch Zeta Function, IOP Conf. Series: Journal of Physics: Conference Series, 2019, vol. 1212, pp. 1-6. DOI: 10.1088/1742-6596/1212/1/012010.
33. Libera, R. J. Some Classes of Regular Univalent Functions, Proceedings of the American Mathematical Society, 1965, vol. 16, pp. 755-758. DOI: 10.1090/S0002-9939-1965-0178131-2.
34. Bernardi, S. D. Convex and Starlike Univalent Functions, Transactions of the American Mathematical Society, 1969, vol. 135, pp. 429-446. DOI: 10.1090/S0002-9947-1969-0232920-2.
35. Miller S. S., Mocanu, P. T. and Reade, M. O. Starlike Integral Operators, Pacific Journal of Mathematics, 1978, vol. 79, pp. 157-168. DOI:10.2140/PJM.1978.79.157.
36. Ruscheweyh, S. New Criteria for Univalent Functions, Proceedings of the American Mathematical Society, 1975, vol. 49, pp. 109-115. DOI: 10.1090/S0002-9939-1975-0367176-1.
37. Salagean, G. S. Subclasses of Univalent Functions, Lecture Notes in Mathematics,
1983, vol. 1013, pp. 362-372. DOI: 10.1007/BFb0066543.
38. Carlson, B. C. and Shaffer, D. B. Starlike and Prestarlike Hypergeometric Functions, SIAM Journal on Mathematical Analysis, 1984, vol. 15, pp. 737-745. DOI: 10.1137/0515057.
39. Srivastava, H. M. and Attiya, A. A. An Integral Operator Associated with the Hurwitz-Lerch Zeta Function and Differential Subordination, Integral Transforms and Special Functions, 2007, vol. 18, no. 3, pp. 207-216. DOI: 10.1080/10652460701208577.
40. Ghanim, F., Al-Shaqsi, K., Darus, M. and Al-Janaby, H. F. Subordination Properties of Meromorphic Kummer Function Correlated with Hurwitz-Lerch Zeta-Function, Mathematics, 2021, vol. 9, no. 192, pp. 1-10. DOI: 10.3390/math9020192.
41. Pascu, N. N. and Pescar, V. On the Integral Operators of Kim-Merkes and Pfaltzgraff, Mathematica, Universitatis Babes-Bolyai Cluj-Napoca, 1990, vol. 32(55), no. 2, pp. 185-192.
42. Breaz, D. and Breaz, N. Two Integral Operators, Studia Universitatis Babes-Bolyai, Mathematica, Clunj-Napoca, 2002, vol. 47, no. 3, pp. 13-19.
43. Breaz, D., Owa, S. and Breaz, N. A New Integral Univalent Operator, Acta Universitatis Apulensis, 2008, vol. 16, pp. 11-16.
44. Frasin, B. A. Univalence Criteria for General Integral Operator, Math. Commun., 2011, vol. 16, no. 1, pp. 115-124.
45. Deniz, E. Univalence Criteria for a General Integral Operator, Filomat, 2014, vol. 28, no. 1, pp. 11-19. DOI: 10.2298/FIL1401011D.
46. Barbatu, C. and Breaz, D. Univalence Criteria for a General Integral Operator, General Math., 2019, vol. 27, no. 2, pp. 43-57. DOI: 10.2478/gm-2019-0014.
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