A note on some elementary properties and applications of certain operators to certain functions analytic in the unit disk

Hüseyin Irmak


This scientific note relates to introducing certain elementary operators
defined in the unit disk in the complex plane, then determining various
applications (specified by those operators) to certain analytic functions, and
also revealing a number of possible implications of them.


fractional calculus; unit open disk; complex plane; analytic functions; normalized functions; Taylor-Mclaurin series expansion; the Tremblay operator; fractional derivative operator; differential inequalities in the complex plane

Mathematics Subject Classification (2010)

30C45; 30C50; 26A33; 26D05; 26D10; 26D15; 33E12; 33D15; 30C80; 26E35


Altıntas, Osman, and Hüseyin Irmak, and Hari Mohan Srivastava. "Fractional calculus and certain starlike functions with negative coefficients." Comput. Math. Appl. 30, no. 2, (1995): 9-15.

Chen, Ming-Po, and Hüseyin Irmak, and Hari Mohan Srivastava. "Some families of multivalently analytic functions with negative coefficients." J. Math. Anal. Appl. 214, no. 2 (1997): 674-690.

Chen, Ming-Po, and Hüseyin Irmak, and Hari Mohan Srivastava. "A certain subclass of analytic functions involving operators of fractional calculus." Comput. Math. Appl. 35, no. 2 (1998): 83-91.

Duren, Peter L. Univalent functions. Vol. 259 of Grundlehren der Mathematischen Wissenschaften. New York: Springer-Verlag, 1983.

Esa, Zainab et al. "A novel subclass of analytic functions specified by a family of fractional derivatives in the complex domain." Filomat 31, no. 9, (2017): 2837-2849.

Goodman, Adolph W. Univalent functions. Vols. I and II. Tampa, FL: Mariner Publishing Co., Inc., 1983.

Kilbas, Anatoly A., and Hari Mohan Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations. Vol. 204 of North-Holland Mathematics Studies. Amsterdam: Elsevier Science B.V., 2006.

Ibrahim, Rabha W., and Jay M. Jahangiri. "Boundary fractional differential equation in a complex domain." Bound. Value Probl. 2014, (2014): art. no. 66.

Irmak, Hüseyin. "Certain complex equations and some of their implications in relation with normalized analytic functions." Filomat 30, no. 12 (2016): 3371-3376.

Irmak, Hüseyin. "Some novel applications in relation with certain equations and inequalities in the complex plane." Math. Commun. 23, no. 1, (2018): 9-14.

Irmak, Hüseyin. "Certain basic information related to the Tremblay operator and some applications in connection therewith." Gen. Math. 27, no. 2 (2019): 13-21.

Irmak, Hüseyin. "Geometric properties of some applications of the Tremblay operator." (Submitted)

Irmak, Hüseyin. "Notes on certain elementary operator created by fractional calculus and some of their implications." (Submitted)

Irmak, Hüseyin, and Praveen Agarwal. "Comprehensive inequalities and equations specified by the Mittag-Leffler functions and fractional calculus in the complex plane." In: Advances in mathematical inequalities and applications, 295–310. Singapore: Birkhäuser, Springer, 2018.

Irmak, Hüseyin, and Basem Aref Frasin. "An application of fractional calculus and its implications relating to certain analytic functions and complex equations." J. Fract. Calc. Appl. 6, no. 2 (2015): 94-100.

Irmak, Hüseyin, and Olga Engel. "Some results concerning the Tremblay operator and some of its applications to certain analytic functions." Acta Univ. Sapientiae Math. 11, no. 2 (2019): 296-305.

Jack, I.S. "Functions starlike and convex of order alpha." J. London Math. Soc. 3, (1971): 469-474.

Miller, Sanford S., and Petru T. Mocanu. "Second-order differential inequalities in the complex plane." J. Math. Anal. Appl. 65, no. 2 (1978): 289-305.

Nunokawa, Mamoru. "On properties of non-Carathéodory functions." Proc. Japan Acad. Ser. A Math. Sci. 68, no. 6 (1992): 152-153

Owa, Shigeyoshi "On the distortion theorems. I." Kyungpook Math. J. 18, no. 1 (1978): 53-59.

San, Müfit, and Hüseyin Irmak. "A note on some relations between certain inequalities and normalized analytic functions." Acta Univ. Sapientiae Math. 10, no. 2 (2018): 368-374.

Srivastava, Hari Mohan, and Shigeyoshi Owa (eds.). Univalent functions, fractional calculus, and their applications. Chichester: Ellis Horwood Ltd.; New York: Halsted Press, 1989.

Full Text: PDF

Download statistics: 320

Licencja Creative Commons
This article by Hüseyin Irmak is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.

e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej