The fractional integrodifferential operator and its univalence and boundedness features according to Pre-Schwarzian derivative structur
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Regular function, Locally univalent, Pre-Schwarzian derivative, Fractional calculus. 2010 Mathematics Subject Classifications: 30C45, 30C50, 30C10Abstract
Complex-valued regular functions that are normalized in the open unit disk are vastly studied. The current study introduces a new fractional integrodifferential (non-linear) operator. Based on the pre-Schwarzian derivative, certain appropriate stipulations on the parameters included in this constructed operator to be univalent and bounded are investigated and determined.
References
J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1), (1915), 12–22.
H. F. Al-Janaby and F. Ghanim, Geometric study on modified Koebe functions imposed by confluent hypergeometric functions, Italian Journal of Pure and Applied Mathematics, 48, (2022), 580–594.
S. M. Amsheri and V. Zharkova, Subclasses of valent starlike functions defined by using certain fractional derivative operator, Int. J. Math. Sci. Edu., 4 (1), (2011), 17–32.
J. Becker, Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte functionen, J. Reine Angew. Math., 255, (1972), 23–43.
J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354, (1984), 74–94.
J.H. Choi, Y.C. Kim, S. Ponnusamy, T. Sugawa, Norm estimates for the Alexander transforms of convex functions of order alpha, J. Math. Anal. Appl., 303 (2005) 661–668.
P. L. Duren, Univalent functions, New York: Springer, 1983.
H. A. Farzana, B. A. Stephen and M. P. Jeyaraman, Third-order differential subordination of analytic function defined by functional derivative operator, Ann. Alexandru Ioan Cuza Univ. Math., 2014, (2014), 1–16.
F. Ghanim and H. F. Al-Janaby, An analytical study on Mittag-Leffler-confluent hypergeometric functions with fractional integral operator, Math Meth Appl Sci., (2020), 1–10.
F. Ghanim and H. F. Al-Janaby, Inclusion and Convolution Features of Univalent Meromorphic Functions Correlating with Mittag-Leffler Function, Filomat, 34 (7), (2020), 2141–2150.
F. Ghanim and H. F. Al-Janaby, Some Analytical Merits of Kummer-Type Function Associated with Mittag-Leffler Parameters, Arab Journal of Basic and Applied Sciences, 28 (1), (2021), 255–263.
F. Ghanim and H. F. Al-Janaby, O. Bazighifan, Some new extensions on fractional differential and integral properties for Mittag-Leffler confluent hypergeometric function, Fractal Fract., 5 (143), 2021, 1–12.
F. Ghanim, S. Bendak and A. A. Hawarneh, Certain implementations in fractional calculus operator involving MittagLeffler confluent hypergeometric functions, Proceedings of the Royal Society A, 478 (2258), 2022, https://doi.org/10.1098/rspa.2021.0839.
A. W. Goodman, Univalent functions, Florida: Mariner Publishing Company, 1983.
H. Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen, Monatsh. Math., 73, (1969), 36–45.
Y.C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math., 32, (2002), 179–200.
Y.C. Kim, S. Ponnusamy, T. Sugawa, Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives, J. Math. Anal. Appl., 299 (2004) 433–447.
S. Owa, On the distortion theorems I, Kyungpook Math. J., 18 (1), (1978), 53–59.
Z. Orouji and R. Aghalary, The norm Estimates of pre-Schwarzian derivatives of Spirallike Functions and Uniformly Convex α-spiralike Functions, Sahand commun. math. anal. (SCMA), 12 (1), (2018), 89–96.
H. Rahmatan, S. Najafzadeh and A. Ebadian, The norm of pre-Schwarzian derivatives of certain analytic functions with bounded positive real part, Stud. Univ. Babeş-Bolyai Math., 61 (2), (2016), 155–162.
H. M. Srivastava, Y. Ling and G.Bao, Some distortion inequalities associated with the fractional derivatives of analytic and univalent function, J. Ineq. Pure and Appl. Math., 2 (2), (2001), 1–6.
H.M. Srivastava, S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, John Wiley and Sons, New York, Chichester, Brisban, and Toronto, 1989.
L. Stanciu and D. Breaz, Some properties of a general integral operator, Bull. Iran. Math. Soc., 40, 6 (2014), 1433–1439.
S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J., 28, (1999), 217–230.
