Some operators of fractional calculus and their applications regarding various complex functions analytic in certain domains

Hüseyin Irmak

Abstract


In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complex-exponential forms, and also in the light of certain special information in \cite{8}, \cite{18}, \cite{19} and \cite{37}, an extensive result
together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out.


Keywords


Complex plane, Analytic functions, complex exponentials, complex series expansions, Fractional calculus, operators of fractional order,

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References


Abdulnaby, Z.E., Ibrahim, R.W., Kilicman, A., On boundedness and compactness of a generalized Srivastava-Owa fractional derivative operator, Journal of King Saud University-Science, 30(2018), 153-157.

Abro, K.A., Solangi, M.A., Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo-Fabrizoi fractional derivatives, Punjab Univ. J. Math. (Lahore), 49(2017), 113-125.

Al-Ameedee, S.A., Atshan, W.G., Al-Maamori, F.A., Yal c n, S., On third-order sandwich results of multivalent analytic functions involving Catas Operator, An. Univ. Oradea Fasc. Mat, 28(2021), 111-129.

Burkill, J.C., Fractional orders of integrability, J. Lond. Math. Soc., 11(1936), 220-226.

Chen, M.P., Irmak, H., Srivastava, H.M., A certain subclass of analytic functions involving operators of fractional calculus, Comput. Math. Appl., 35(1998), 83-91.

Davis, H.T., The application of fractional operators to functional equations, Amer. J. Math., 49(1927), 123-142.

Debnath, L., A brief historical introduction to fractional calculus, Internat. J. Math. Ed. Sci. Tech., 35(2004), 487-501.

Esa, Z., Srivastava, H.M., Kilicman, A., Ibrahim, R.W., A novel subclass of analytic functions specified by a family of fractional derivatives in the complex domain, Filomat, 31(2017), 2837-2849.

Frasin, B.A., Murugusundaramoorthy, G., Fractional calculus to certain family of analytic functions defined by convolution, Indian J. Math., 51(2009), 537-548.

Goodman, A.W., Univalent Functions, Vols. I and II, Mariner Publishing Co., Inc., Tampa, FL, 1983.

Grozdev, S., On the appearance of the fractional calculus, J. Theoret. Appl. Mech., 27(1997), 11-20.

Guney, H. O., Acu, M., Breaz, D., Owa, S., Applications of fractional derivatives for Alexander integral operator, Afr. Mat., 32(2021), 673-683.

Hardy, G.H., Littlewood, J.E., Some properties of fractional integrals. II, Math. Z., 34(1932), 403-439.

Hussain, M., Application of the Srivastava-Owa fractional calculus operator to Janowski spiral-like functions of complex order, Punjab Univ. J. Math. (Lahore), 50(2018), 33-43.

Ibrahim, R.W., Jahangiri, J.M., Boundary fractional differential equation in a complex domain, Bound. Value Probl., 66(2014), 1-11.

Irmak, H., Certain complex equations and some of their implications in relation with normalized analytic functions, Filomat, 30(2016), 3371-3376.

Irmak, H., Certain basic information related to the Tremblay operator and some applications in connection therewith, Gen. Math., 27(2020), 13-21.

Irmak, H., Characterizations of some fractional-order operators in complex domains and their extensive implications to certain analytic functions, An. Univ. Craiova Ser. Mat. Inform., 48(2021), 349-357.

Irmak, H., An extensive note on various fractional-order type operators and some of their effects to certain holomorphic functions, Ann. Univ. Paedagog. Crac. Stud. Math.,

(2022), 7-15.

Irmak, H., Engel, O., Some results concerning the Tremblay operator and some of its applications to certain analytic functions, Acta Univ. Sapientiae Math., 11(2019,) 296-

Irmak, H., Y ld z, T. H., Comprehensive notes on various effects of some operators of fractional-order derivatives to certain functions in the complex domains and some of related implications, Punjab Univ. J. Math. (Lahore), 54(2022), 285-296.

Isife, K.I., Existence of solution for some two-point boundary value fractional differential equations, Turkish J. Math., 42(2018), 2953-2964.

Jack, I.S., Functions starlike and convex of order , J. Lond. Math. Soc., 2(1971), 469-474.

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier, Amsterdam, London and New York, 2006.

Kumar, D., Ayant, F.Y., Fractional calculus pertaining to multivariable I-function defined by Prathima, J. Appl. Math. Stat. Inform., 15(2019), 61-73.

Liu, Y., Yin, B., Li, H., Zhang, Z., The unified theory of shifted convolution quadrature for fractional calculus, J. Sci. Comput., 89(2021), Paper No. 18.

Macphail, M.S., Some iterated integrals in the fractional calculus, Bull. Amer. Math. Soc., 44(1938), 707-715.

Nehari, Z., Conformal Mapping, McGraw-Hill, New York, 1952.

Nguyen, V.T., Fractional calculus in probability, Probab. Math. Statist., 3(1984), 173-189.

Nishimoto, K., Fractional calculus of products of elementary functions, I. J. College Engrg. Nihon Univ. Ser. B, 28(1987), 21-31.

Owa, S., Saigo, M., Megumi, K., Kiryakova, V., Inequalities for Saigo's fractional calculus operator, J. Approx. Theory Appl., 3(2007), 53-62.

Pishkoo, A., Darus, M., Fractional differintegral transformations of univalent Meijer's G-functions, J. Inequal. Appl., 36(2012), 10 pp.

Raza, N., Unsteady rotational flow of a second grade fluid with non-integer Caputo time fractional derivative, Punjab Univ. J. Math. (Lahore), 49(2017), 15-25.

Ross, B., Origins of fractional calculus and some applications, Internat. J. Math. Statist. Sci., 1 (1992), 21-34.

Srivastava, H.M., Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60(2020), 73-116.

Srivastava, H.M., Owa, S., Univalent Functions, Fractional Calculus and Their Applications, Halsted Press, John Wiley and Sons, New york, Chieschester, Brisbane, Toronto,

Taberski, R., Contributions to fractional calculus and exponential approximation, Funct. Approx. Comment. Math., 15(1986), 81-106.

Tremblay, R., Une Contribution e la Theorie de la Derivee Fractionnaire, Ph.D. Thesis, Universite Laval, Quebec, Canada, 1974.

Vinagre, J., Blas, M., Calderon, G., Antonio, J., Suarez, M., Jose, I., Monje, M., Concepcion, A., Control theory and fractional calculus, Rev. R. Acad. Cienc. Exactas Fis. Nat., 99(2005), 241-258.

Watanabe, S., A fractional calculus on Wiener space. Stochastic processes, Springer, New York, 1993, 341-348.




DOI: http://dx.doi.org/10.24193/subbmath.2024.3.01

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