# Research Topics & Guidance

Research Topics and Guidance

Research Topics

• Geometric Properties of Harmonic Mappings Harmonic mappings in the plane are univalent (one-to-one) complex-valued functions of a complex variable that satisfies the Laplace's equation. We construct examples of harmonic mappings by using harmonic shears and also investigate closure properties under Hadamard product (or convolution) of various classes of harmonic univalent function such as the classes of starlike, convex and close-to-convex harmonic functions.
• Radii Problems in Geometric Function Theory: A convex function is necessarily starlike but not conversely. However, the starlike functions maps a smaller disk into convex domains. The largest radius with this property is known as the radius of convexity of starlike functions. For any two geometric properties, onecan talk about radius problems. We have investigated several radii problems associated with starlikeness, convexity and uniform convexity.
• Uniformly Convex Functions and Related Topics: A convex (or starlike) functions need not maps a disk inside the unit disk to a convex (or a starlike) do-main. Such a strong requirement leads to the investigation of uniformly convex (or starlike) functions. Among other things, We have introduced and investigated uniformly spiral functions. Convolutions and Neighborhood problems: An old conjecture of Polya-Schoenberg says that convex, starlike and close-to-convex functions are known to be closed under convolutions with convex functions. This was proved Ruscheweyh and Sheil-Small. We have investigated this convolution problem for several more general classes of functions.
• Starlike/Convex wrt Symmetric and Conjugate Points: Motivated by the investigation of the class of starlike functions with respect to symmetric points by Sakaguchi, we have investigated several related classes of functions. In particular, we have obtained radii results, distortion and growth estimates, Koebe domain, estimates for the coefficients and integral representations for functions starlike and convex with respect to symmetric, conjugate and symmetric conjugate points.
• Differential Subordinations and Superordinations: Concept subordination is the analogue of inequalities in real line to the complex plane. Subordination is defined by containment of regions. A differential subordination is extension of differential inequality while superordination is a dual concept. We have obtained several conditions that ensure analytic functions to have certain geometric properties, like starlikeness, convexity.
• Radii Problems by Fixing Second Coefficient: The second coefficient of univalent functions plays important role in GFT; it leads to growth and distortion estimates, and Koebe domains. We have investigated several radii problems by considering functions where the second coefficient is fixed.
• Subordination Theory for Functions with Preassigned Initial Coefficient: My aim here is to develop the theory of differential subordination in the same line as that of Miller and Mocanu for functions with preassigned initial coefficient. This theory is gives us better results than that of Miller and Mocanu.

Research Guidance

1. Janani B B, Ph.D., NITT, 2021--
2. Priya G Krishnan, Ph.D., NITT, 2020--09-2024, Subordination and radius problems for analytic and meromorphic functions
3. S. Madhumitha, PhD, NIT-T, 01-07-2019 ---31-01-2024, Subordination and radius problems for analytic functions
4. Shalu Yadav, PhD, NIT-T, 01-07-2019 ---28-06-2024, Radii Constants, Differential Subordination and Convolution for Certain Ma-Minda Classes
5. Kanaga, PhD, NIT-T, 01/2019 - 29-12-2022, Radii constants, differential subordination and averaging operator on certain Ma-Minda classes
6. Somya, PhD, NIT-T, 01-07-2019 ---2022, Convolution, Subordination and Radius Problems for Analytic Functions
7. Asha Sebastian, PhD, NIT-T, 07/2018 -2022, Starlikeness of certain analytic functions
8. Vibha, PhD, University of Delhi, 9/2017-2021, Starlikeness and Convexity of Certain Univalent and Entire Functions
9. Prachi Gupta, PhD, University of Delhi, 2016-2021, Radius Constants and Differential Subordination for Certain Subclasses of Analytic Functions
10. Shweta Gandhi, PhD, University of Delhi, 2015- 04/2018, Subordination for Starlike Functions
11. Subzar Ahmad Beig, PhD, Universisty of Delhi, 10/2014- 04/2018, Convolution and Linear Combinations of Harmonic Mappings
12. Kanika Khatter, Ph.D.(co-supervisor)Delhi Techonological University, 2014-2018, Coefficient Estimates and Subordination for Univalent Functions
13. Nisha Bohra, PhD, University of Delhi, 2015-11/12/2017 , Geometric properties of univalent, bi-univalent and some specical functions
14. Shelly Verma, PhD, University of Delhi, 9/6/2014-11/7/2017, Coefficient and radius estimates of normalized analytic functions
15. Sushil Kumar, PhD, University of Delhi, 2013-2016, Coefficient estimates and subordination for univalent functions
16. Kanika Sharma, PhD, University of Delhi, 2013-2016, Differential subordination criteria for starlike functions
17. Rajni Kapoor, PhD, University of Delhi, 2009-2014, Subordination and radius problems for some classes of univalent functions
18. Sumit Nagpal, PhD, University of Delhi, 2011 - 2013 , Close-to-convex planar harmonic univalent mappings
19. Naveen Kumar Jain, PhD, University of Delhi, 2009 - 2013 , Radius constants for geometric properties of univalent functions
20. Shamani a/p Supramaniam, PhD (Field-supervisor), Universiti Sains Malaysia, 2009-2014, Differential subordination and coefficient problems for certain analytic functions
21. Mahnaz Moradi Nargesi, PhD, (Field-supervisor), Universiti Sains Malaysia, 2013, Inclusion properties of linear operators and analytic functions
22. Chandra Shekar, PhD, (Field-supervisor), Universiti Sains Malaysia, 2012, Subordination and convolution of analytic, meromorphic harmonic functions
23. Abeer O. Badghaish, PhD, (Field-supervisor), Universiti Sains Malaysia, 2011, Subordination and convolution of multivalent functions and starlikeness of integral transforms
24. N. Seenivasagan, PhD, (Co-supervisor), Universiti Sains Malaysia, 2007, Differential subordination and superordination for analytic and meromorphic functions defined by linear operators
M Phil
25. Mansi Sharma, M.Phil (2016-17), University of Delhi,Nonvanishing divided difference of order N
26. Neeru Bala, M.Phil (2015-16), University of Delhi, Univalence of integral operators
27. Abdul Wakil Baidar (2014-16), (International Student from Afghanistan), University of Delhi, First order differential subordinations for starlike functions
28. Prachi Gupta, MPhil (2014-15), University of Delhi, Functions with positive real part
29. Soma Das, MPhil (2013-14), University of Delhi, Univalent Functions with Negative Coefficients
30. Sumit Nagpal, MPhil (2009-10), University of Delhi, First and Second Order Differential Subordinations and Radius Problems for Caratheodary Functions
31. Chandrashekhar, MPhil (2008-09), University of Delhi, Sufficient Conditions for Univalency, Starlikeness and Convexity of Analytic Functions
32. Bikram Singh, MPhil, University of Delhi, (2007-2008), Radii of Univalence, Convexity, and Starlikeness of Analytic Functions
33. Mukesh Aggarwal , MPhil (2007-08), University of Delhi, Properties of Uniformly Starlike and Uniformly Convex Functions
34. Shamani Supramanian, MSc (by Research), (field-supervisor), Universiti Sains Malaysia, 2009, Convolution and coefficient problems for multivalent functions defined by subordination (emphBest MSc Thesis 2009, awarded by the Malaysian Mathematical Sciences Society, December 8, 2010)
35. Maisarah Hj Mohamd, MSc (by Research), (field-supervisor), Universiti Sains Malaysia, 2009, Convolution, coefficient and radius problems of certain univalent functions
36. Chandra Shekar, MSc(Mixed Mode), Universiti Sains Malaysia, 2007, Applications of first order differential subordination to univalent function theory