Research

Differential Subordinations: The differential subordination is the complex analogue of the differential inequality in the real line and differential superordination is its dual concept. We have developed the theory of differential subordination for functions with preassigned initial coefficient by making appropriate modifications and improvements to the existing Miller and Mocanu’s subordination theory. This new theory has several interesting applications in univalent function theory. 

Geometric Properties of functions defined by subordination: In 1992, Ma and Minda unified various subclasses of starlike functions in terms of subordination. Using this notion, we have introduced and defined various classes of starlike function which are associated with exponential function, right half of the shifted lemniscate of Bernoulli and cardioid. Various inclusion relations, coefficient bounds and radius problems are derived for these newly defined classes of univalent functions. 

Harmonic Univalent Mappings: A planar harmonic univalent mapping is a complex-valued function that does not take the same value twice and whose real and imaginary parts have continuous second partial derivatives satisfying the Laplace equation. The study of planar harmonic univalent mappings initiated by Clunie and Sheil-Small in 1984, is a fairly active area of research. We investigate the properties of various subclasses of harmonic univalent functions defined by natural geometric conditions such as the classes of starlike, convex and close-to-convex harmonic functions.  Apart from that, we have recently introduced a new product for harmonic mappings which is expected to play a more important role than the existing concept of harmonic convolution.  

• Gandhi, Shweta; Gupta, Prachi; Nagpal, Sumit; Ravichandran, V. Starlike functions associated with an epicycloid. Hacet. J. Math. Stat. 51 (2022), no. 6, 1637--1660. MR4510346 

• Raj, Ankur; Nagpal, Sumit. Radius of convexity for analytic part of sense-preserving harmonic mappings. Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2665--2679. MR4489583 

• Kumar, Virendra; Nagpal, Sumit; Cho, Nak Eun. Coefficient functionals for non-Bazilevič functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 1, Paper No. 44, 14 pp. MR4344855 

• Ravichandran, V.; Nagpal, Sumit. A survey on univalent functions with fixed second coefficient. Math. Newsl. 33 (2022), no. 1, 17--28. MR4518875 

• Gupta, Prachi; Nagpal, Sumit; Ravichandran, Vaithiyanathan. Inclusion relations and radius problems for a subclass of starlike functions. J. Korean Math. Soc. 58 (2021), no. 5, 1147--1180. MR4303808 

• Gupta, Prachi; Nagpal, Sumit; Ravichandran, V. Marx-Strohhäcker theorem for multivalent functions. Afr. Mat. 32 (2021), no. 7-8, 1421--1434. MR4327479 

• Raj, Ankur; Nagpal, Sumit; Ravichandran, V. On the product of planar harmonic mappings. Comput. Methods Funct. Theory 21 (2021), no. 3, 427--452. MR4299907 

• Naz, Adiba; Nagpal, Sumit; Ravichandran, V. Exponential starlikeness and convexity of confluent hypergeometric, Lommel, and Struve functions. Mediterr. J. Math. 17 (2020), no. 6, Paper No. 204, 22 pp. MR4172977 

• Naz, Adiba; Nagpal, Sumit; Ravichandran, V. Star-likeness associated with the exponential function. Turkish J. Math. 43 (2019), no. 3, 1353--1371. MR3962536 

• Ahuja, Om P.; Nagpal, Sumit; Ravichandran, V. A technique of constructing planar harmonic mappings and their properties. Kodai Math. J. 40 (2017), no. 2, 278--288. MR3680562 

• Kumar, Sushil; Nagpal, Sumit; Ravichandran, V. Coefficient inequalities for Janowski starlikeness. Proc. Jangjeon Math. Soc. 19 (2016), no. 1, 83--100. MR3469654 

• Nagpal, Sumit; Ravichandran, V. Convolution properties of the harmonic Koebe function and its connection with 2-starlike mappings. Complex Var. Elliptic Equ. 60 (2015), no. 2, 191--210. MR3298078 

• Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38 (2015), no. 1, 365--386. MR3394060 

• Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. Radii of starlikeness and convexity for analytic functions with fixed second coefficient satisfying certain coefficient inequalities. Kyungpook Math. J. 55 (2015), no. 2, 395--410. MR3367954

• Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. Second-order differential superordination for analytic functions with fixed initial coefficient. Southeast Asian Bull. Math. 39 (2015), no. 6, 851--864. MR3444396 

• Nagpal, Sumit; Ravichandran, V. A subclass of close-to-convex harmonic mappings. Complex Var. Elliptic Equ. 59 (2014), no. 2, 204--216. MR3170754 

• Nagpal, Sumit; Ravichandran, V. Univalence and convexity in one direction of the convolution of harmonic mappings. Complex Var. Elliptic Equ. 59 (2014), no. 9, 1328--1341. MR3210304

• Mendiratta, Rajni; Nagpal, Sumit; Ravichandran, V. A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli. Internat. J. Math. 25 (2014), no. 9, 1450090, 17 pp. MR3266533

• Nagpal, Sumit; Ravichandran, V. Construction of subclasses of univalent harmonic mappings. J. Korean Math. Soc. 51 (2014), no. 3, 567--592. MR3206405 

• Ahuja, Om P.; Nagpal, Sumit; Ravichandran, V. Radius constants for functions with the prescribed coefficient bounds. Abstr. Appl. Anal. 2014, Art. ID 454152, 12 pp. MR3261223 

• Nagpal, Sumit; Ravichandran, V. A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions. Stud. Univ. Babeş-Bolyai Math. 59 (2014), no. 1, 41--55. MR3197914 

• Nagpal, Sumit; Ravichandran, V. Starlikeness, convexity and close-to-convexity of harmonic mappings. Current topics in pure and computational complex analysis, 201--214, Trends Math., Birkhauser/Springer, New Delhi, 2014. MR3329718 

• Nagpal, Sumit; Ravichandran, V. Fully starlike and fully convex harmonic mappings of order $\alpha$. Ann. Polon. Math. 108 (2013), no. 1, 85--107. MR3021270 

• Nagpal, Sumit; Ravichandran, V. Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions. Ann. Polon. Math. 105 (2012), no. 3, 225--238. MR2950658 

• Ali, Rosihan M.; Nagpal, Sumit; Ravichandran, V. Second-order differential subordination for analytic functions with fixed initial coefficient. Bull. Malays. Math. Sci. Soc. (2) 34 (2011), no. 3, 611--629. MR2823592 

PH.D. SUPERVISION (COMPLETED)

Ms. Prachi Gupta, Department of Mathematics, University of Delhi (July 2022) with the title of thesis “Radius Constants and Differential Subordination for Certain Subclasses of Analytic Functions”

PH.D. SUPERVISION (THESIS SUBMITTED)

Mr. Ankur Raj, Department of Mathematics, University of Delhi with the title of thesis “Radius Problems and Construction Techniques for Univalent Harmonic Mappings”

PH.D. SUPERVISION (IN PROGRESS)

• Ms. Gurpreet Kaur, Mata Sundri College, University of Delhi

• Ms. Adiba Naz, Department of Mathematics, University of Delhi

M.PHIL. SUPERVISION (COMPLETED)

Ms. Adiba Naz, Department of Mathematics, University of Delhi (August 2018) with the title of dissertation “Certain Special First and Second order Differential Subordinations.”

• Principal investigator of a one-year Innovation Project “Private Coaching verses Classroom Teaching in Schools/ Universities” awarded by University of Delhi. The grant for this project was Rs. 3.5 Lakhs.

• Principal Investigator of three-year Star Innovation Project “Taking the work of Ramanujan to next level: An Innovation Project in Cryptography” by the University of Delhi. The grant for the same was more than 10 Lakhs.

Collaborated with 7 national and international researchers for research work:

• Prof Dato' Indera Dr. Rosihan M. Ali, Universiti Sains Malaysia 

• Prof. Om P. Ahuja, Department of Mathematical Sciences, Kent State University, Burton, Ohio, USA 

• Nak Eun Cho, Department of Applied Mathematics, Pukyoung National University, Busan, South Korea

• Dr. Rajni Mendiratta, Associate Professor, Department of Mathematics, Keshav Mahavidyalaya, University of Delhi

• Dr. Sushil Kumar, Head, Department of Applied Science,  Bharati Vidyapeeth’s College of Engineering, Delhi

• Dr. Virendra Kumar, Assistant Professor, Department of Mathematics, Ramanujan College, University of Delhi

• Dr. Shweta Gandhi, Assistant Professor, Department of Mathematics, Miranda House, University of Delhi