Some new applications of Integral and Differential operator for new subclasses of analytic functions

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  • Naeem Ahmad Qassim University


Analytic functions, q-Calculus, Integral operators, Convolution, Starlike and Convex functions, Differential subordination..


In this paper, we consider convolution operatorz D_x h(z)=h(z)*z/(1-xz)(1-z) ,    and define a new differential operator D_(λ,x)^m  on the analytic functions in the complex plane, for all x,|x|≤1.   For x=1   the operator becomes the Al Oboudi differential operator and for real number x=q    and λ=1,  0 < q < 1, we obtain the Salagean q-differential operator. Furthermore, we consider this operator D_(λ,x)^m   and then define two new integral operators. We discuss some interesting applications of these operators by introducing several new subclasses of analytic functions. Also geometric properties are investigated for integral operators on new subclasses of analytic functions and some subordination results are discussed for differential operatorD_(λ,x)^m .  .


Agrawal, S. Sahoo, S.K. Geometric properties of basic hypergeometric functions. J. Di¤. Equ. Appl. 2014, 20, 15021522.

Agrawal, S. Sahoo, S.K. A generalization of starlike functions of order alpha. Hokkaido Math. J. 2017, 46, 1527.

Al-Oboudi, F.M. On univalent functions de ned by a generalized Salagean operator. Ind. J. Math. Math. Sci. 2004, 27, 14291436.

Alb Lupas, A. Oros, G.I. Di¤erential subordination and superordination results using fractional integral of conuent Hypergeometric function. Symmetry 2021, 13, 327.

Alb Lupas, A. Oros, G.I. On special di¤erential subordinations using fractional integral of Salagean and Ruscheweyh operators. Symmetry 2021, 13, 1553.

Alexander, J.W. Functions which map the interior of the unit circle upon simple regions. Ann. Math. 1915, 17, 1222.

Breaz, D. Breaz, N. Two integral operators. Studia Universitatis Babes Bolyai,Mathematica, Cluj-Napoca, 3(3002):13-21, 2002.

Breaz, D. Owa, S. Breaz, N. A new integral univalent operator. Acta Univ. Apul. 2008, 16, 1116.

Catas, A. Sendrutiu, R. Iambor, L.F. Certain subclass of harmonic multivalent functions de ned by derivative operator. J. Comput. Anal. Appl. 2021, 29, 775785.

Ghazy, A. Alamoush, Darus, M. On Certain subclasses of analytic multivalent functions using generalized Salagean operator, International Journal of Di¤erential Equations Volume 2015, Article ID 910124, 7 pages

Govindaraj, M. Sivasubramanian, S. On a class of analytic functions related to conic domains involving q-calculus, Analysis Mathematica. 43, 2017, 475487. 21

Hussain, S. Khan, S. Zaighum, M.A. Darus. M. Applications of a q-Salagean type operator on multivalent functions, Journal of Inequalities and Applications 2018, 2018:301

Hussain, S. Khan, S. Zaighum, M.A. Darus. M. Certain subclass of analytic functions related with conic domains and associated with Salagean q-di¤erential operator, AIMS Mathematics, 2017, 2(4), 622-634 DOI:10.3934/Math.2017.4.622.

Hussain, S. Khan, S. Zaighum, M.A. Darus. M. A subclass of un ormaly convex func tions and a corresponding subclass of starlike function with xed coe¢ cient associated with q-analogue of Ruscheweyh operator. Math. Slovaca 69, 2019, No. 4, 825832, DOI:


Ibrahim, R.W. Darus, M. New symmetric di¤erential and integral operators de ned in the Complex domain. Symmetry 2019, 11, 906.

Ibrahim, R.W. Elobaid, R.M. Obaiys, S.J. Geometric inequalities via a symmetric di¤erential operator de ned by quantum calculus in the open unit disk. J. Funct. Spaces 2020, V(2020), 6932739.

Ismail, M.E.H., Merkes, E. Styer, D. A generalization of starlike functions. Complex Var.1990, 14, 7784.

Jackson F.H. On q-functions and a certain di¤erence operator, Transactions of the Royal Society of Edinburgh. 46,1908, 253281.

Jackson, F.H. On q-de nite integrals. Pure and Applied Mathematics Quarterly. 41, 1910, 193-203.

Kanas, S. Raducanu, D. Some class of analytic functions related to conic domains, Math. Slovaca, 64(5), 2014, 11831196.

Khan, B. Srivastava, H.M. Tahir, M. Darus, M. Ahmad, Q.Z. Khan, N. Applications of a certain q-integral operator to the subclasses of analytic and bi-univalent functions. AIMS Math. 2021, 6, 10241039.

Libera, R.J. Some classes of regular univalent functions. Proc. Am. Math. Soc. 1965, 16, 755-758.

Livingston, A. On the radius of univalence of certain analytic functions. Proc. Am. Math. Soc. 1966, 17, 352-357.

Miller, S.S. Mocanu, P.T. Di¤erential subordinations: Theory and Applications. In Series on Monographs and Textbooks in Pure and Applied Mathematics; No. 225; Marcel Dekker: New York, NY, USA; Basel, Switzerland, 2000.

Oros, G.I. New di¤erential subordinations obtained by using a di¤erential-integral

Ruscheweyh-Libera operator. Miskolc Math. Notes 2020, 21, 303317.

Oros, G.I. Study on new integral operators de ned using conuent hypergeometric function. Adv. Di¤er. Equ. 2021, V(2021), 342.

Park, J.H. Srivastava, H.M. Cho, N.E. Univalence and convexity conditions for certain integral operators associated with the Lommel function of the rst kind. AIMS Math. 2021, 6, 1138011402.

Páll-Szabó, A.O. Wanas, A.K. Coe¢ cient estimates for some new classes of bi-Bazilevic functions of Ma-Minda type involving the Salagean integro-di¤erential operator. Quaest. Math. 2021, 44, 495502.

Raghavendar, K. Swaminathan, A. Close-to-convexity of basic hypergeometric functions using their Taylor coe¢ cients. J. Math. Appl. 2012, 35, 111125.

Ravichandran, V. Darus, M. Khan, H.M. Subramanian, K.G. Di¤erential subordination associated with linear operators de ned for multivalent functions. Acta Math. Vietnam. 2005, 30, 113121.

Ravichandran, V. Certain applications of rst order di¤erential subordination. Far East J. Math. Sci. 2004, 12, 4151.

Rønning, F. A Szegö quadrature formula arising from q-starlike functions: Continued Frac tions and Orthogonal Functions, Theory and Applications. Marcel Dekker Inc., New York 1994, pp. 345352. 22

Sahoo, S.K. Sharma, N.L. On a generalization of close-to-convex functions. Ann. Polon. Math. 113, 2015, 93108.

Salagean, G.S. Subclasses of univalent functions. In Lecture Notes in Math; Springer: Berlin, Germany, 1983, Volume 1013, 362372.

Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-Calculus and their applications. Iran. J. Sci. Technol. Trans. A Sci. 2020, 44, 327344.

Srivastava, H.M. Bansal, D. Close-to-convexity of a certain family of q-MittagLe­ er functions. J. Nonlinear Var. Anal. 2017, 1, 6169.

Srivastava, H.M. Khan, N. Darus, M. Khan, S. Ahmad, Q.Z. Hussain, S. Fekete-Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points. Mathematics, 8, (2020) 8, 842.

Srivastava, H.M. Khan, N. Khan, S. Ahmad, Q.A. Khan, B. A class of k-symmetric har monic functions involving a certain q-derivative operator, Mathematics




How to Cite

Ahmad, N. (2023). Some new applications of Integral and Differential operator for new subclasses of analytic functions. Results in Nonlinear Analysis, 6(4), 97–115. Retrieved from