Some Outcomes Involving a Specific Class of Functions over Differential Subordination and Superordination


  • Mustafa I. Hameed Department of Mathematics, College of Education for Pure Sciences, University of Anbar, Ramadi, Iraq.
  • Shaheed Jameel Al-Dulaimi Department of Computer Science, Al-Maarif University College, Ramadi, Iraq.
  • Kayode Oshinubi School of Informatics, Computing and Cyber System, Northern Arizona University, USA.
  • Hussaini Joshua Department of Mathematics, Faculty of Science, University of Kerala, India.
  • Ali F. Jameel Faculty of Education and Arts, Sohar University, Sohar, Sultanate of Oman.
  • Israa A. Ibrahim Department of Science, College of Open Education, Kirkuk Education Directorate, Kirkuk, Iraq.



Differential Operator, Differential Subordination, Generalized Hypergeometric Functions, Meromorphic Functions, Sandwich Theory, Starlike Functions


This work investigates several aspects of differential subordination and superordination, leading to the inclusion of a specific class within the domain of univalent meromorphic functions in a perforated open unit disc and deriving a few sandwich theorems. The purpose of this article is to look into a few of the characteristics of variation subordination for analytic univalent functions over a perforated unit disc. It additionally aims to shed insight into geometric characteristics like coefficient inequality, Hadamard product characteristics, and the Komatu integral operator. A few interesting findings have been discovered for variations in subordination as well as superordination in analytic univalent functions. The outcomes about variations in subordination, including linear algebra operators, were presented employing convolutions involving two linear operators. Everyone evaluates and investigates subordinations as well as superordinations about convolutions using includes from the Komatu integral operator. The convolution operator as a tool was used for obtaining multiple findings over differential subordination within the perforated unit disk employing a generalized hypergeometric function. Appropriate classes of acceptable functions are examined, and the two-dimensional real estate of the differential subordinations is explained by utilizing the linear operator, a technique that Srivastava introduced as well as examined. This leads to the establishment of several sandwich-type theorems for a class of univalent analytical functions. The current work examines several subclasses of star-like functions that are defined by subordination. Additionally, our team provides some relevant links between the results reported here and those acquired previously.


Rogosinski W. On Subordination Functions. Math Proc Camb Philos Soc. 1939; 35(1): 1 – 26.

Littlewood JE. Lectures on the Theory of Functions. UK: Oxford University Press; 1944. 244 p.

Srivastava HM, Owa S. Some Applications of the Generalized Hypergeometric Function Involving Certain Subclasses of Analytic Functions. Publ Math Debr. 1987; 34(3-4): 299 – 306.

Miller SS, Mocanu PT. Second-order Differential Inequalities in the Complex Plane. J Math Anal Appl. 1978; 65(2): 298–305.

Sambo B, Lasode AO. Differential Subordination and Superordination for a Family of Analytic Functions Defined by a New Multiplier Differential Operator. Ann Math Comput Sci. 2023; 17: 40-49.‏

Hameed MI, Shihab BN, Jassim KA. An Application of Subclasses of Goodman-Salagean-Type Harmonic Univalent Functions Involving Hypergeometric Function. 1st International Conference on Advanced Research in Pure and Applied Science (ICARPAS 2021): Third Annual Conference of Al-Muthanna University/College of Science 24–25 March 2021, Al-Samawah, Iraq. AIP Conf. Proc. 2022; 2398(1): 060012. .

Bulboacă T. Classes of First Order Differential Superordinations. Demonstr Math. 2002; 35(2): 287-292. .

Miller SS, Mocanu PT. Subordinates of Differential Superordinations. Complex Var. 2003; 48(10): 815-826. .

Cotîrlă L-I, Cătaş A. Differential Sandwich Theorem for Certain Class of Analytic Functions Associated with an Integral Operator. Stud Univ Babes-Bolyai Math. 2020; 65(4): 487-494. .

Avdiji S, Tuneski N. Sufficient Conditions for Starlikeness Using Subordination Method. Adv Math Sci J. 2020; 9(12): 10707-10716.‏ .

Lupaş AA. Applications of a Multiplier Transformation and Ruscheweyh Derivative for Obtaining New Strong Differential Subordinations. Symmetry. 2021; 13(8): 1312. .

Ali M, Saeed M. On A Differential Subordination and Superordination of New Class of Meromorphic Functions. Matematiche. 2014; 69(1): 259-274. .

Hameed MI, Ali MH, Shihab BN. A Certain Subclass of Meromorphically Multivalent Q-Starlike Functions Involving Higher-Order Q-Derivatives.‏ Iraqi J Sci. 2022; 63(1): 251-258.‏ .

Shehab NH, Juma ARS. Third Order Differential Subordination for Analytic Functions Involving Convolution Operator. Baghdad Sci J. 2022; 19(3): 0581-0581.‏ .

Alb Lupaş A, Oros GI. On Special Differential Subordinations Using Fractional Integral of Sălăgean and Ruscheweyh Operators. Symmetry. 2021; 13(9): 1553.‏ .

Sokol J. Convolution and Subordination in the Convex Hull of Convex Mappings. Appl Math Lett. 2006; 19(4): 303-306. .

Patel J, Cho NE, Srivastava HM. Certain Subclasses of Multivalent Functions Associated with a Family of Linear Operators. Math Comput Model. 2006; 43(3-4): 320-338. .

Frasin B. A New Differential Operator of Analytic Functions Involving Binomial Series. Bol Soc Paran Mat. 2020; 38(5): 205-213.‏ .

Royster WC. On the Univalence of a Certain Integral. Mich Math J. 1965; 12(4): 385–387. .

Rashid AM, Juma ARS. Some Subclasses of Univalent and Bi-Univalent Functions Related to K-Fibonacci Numbers and Modified Sigmoid Function. Baghdad Sci J. 2023; 20(3): 0843-0843.‏ .





How to Cite

Some Outcomes Involving a Specific Class of Functions over Differential Subordination and Superordination. Baghdad Sci.J [Internet]. [cited 2024 Jun. 14];21(12). Available from: